Content-Type: multipart/mixed; boundary="-------------0303281937898" This is a multi-part message in MIME format. ---------------0303281937898 Content-Type: text/plain; name="03-142.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-142.keywords" Dirichlet forms, Markovian semigroups, Z_2 graded algebras, quasi-free states. ---------------0303281937898 Content-Type: application/postscript; name="CAR#mp-arc#.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="CAR#mp-arc#.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: CAR(mp-arc).dvi %%CreationDate: Fri Mar 28 16:10:23 2003 %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 MSBM10 CMR12 CMR8 CMR10 CMBX10 CMR7 CMBX12 CMSY10 %%+ CMMI12 EUFM10 CMMI8 CMSY8 CMEX10 CMTI12 MSAM10 CMR6 CMBXTI10 CMSY6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 CAR(mp-arc) %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.03.28:1607 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e67a35269b877f161 ead3b9314f0ebb23d1a803608fb8bed47522b2a7814eba409f33973ca88ebb86 148d3d0d91d8056e573e23f17b04c8dfeb3d040cab25f1da3caad77b904b64f1 98dc18b1abbbeb700e3bcad39189c15c31bdc0009f94094d223f303408ab2800 533c624b77a8b576fcb13a492a5aac1035d7766bf5e2ba738c1de82bcb395cb4 344d6f452463521d7a6d8d979625e6900ac2b993de3d6a15cf4473176be41862 e61b3f9e109d500405e6f01936bd4cd77ca687b490df2a2e501e141fbecc7082 86f85fa428a8121af362ef3bee57af 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d8c1e5c02115408a 277a0db93878d14fcdfef5998e8b3b5c55f7bce51275cff17d1c2c5e44dedd6b c925936d7ad89ab523d5ce20b1497ce800185ef81b62987f48869ca4e6a51ec7 40a1f0c03468ed6933a676503c495a4d8e789a51964d1793f72453b112952ab6 02e6550e64e00bc7c14e2b22b3057e016c3eb494838f927d10ce9cef075b52ff 01581d98998a33a422fcbfeb02a79a584fb452017cc13c790b7e9593a9ba98ec cf3c0258dc0ae2968e07e069d15c46a9f64a41c7cfbe131e8d7d020e3db6a8c1 db32a075f4b431b7d81838a30021522408a3cb035c53f35b420283e87f0bb611 d85bdccd7782f3f6ee3b87e6582babf6de007a76e7aa9b8244d78ae8b6332ee1 dfb5dcfec098943819302e9490f5cbaf0832a273419a15dfb45ef5fa1eabd10b 55844f54506a302e47caa8de7f7d9dd278b030a14042b316e05fbd7414d08c6d aefbcd4fc6fc1d5117b6f41d2a3351e15d202d26ea18567be720c3245ce4142a f1ee020b9b3d0ed631f09e9c53843c0f42b0a8339de73ca33b52c80f4a8063b1 033c895f6bcaa075ee366ec1a1ab3d00431ceac7edd2f209341de2000d1b0be9 f3c8eed13395db1bcd0270918fdefef6a63bd464fe3738a4f701de4cf44a08ae b491af90f3c75e6374da9314afc7f35673bd7b8d2d9ec4f2fc6485653e85a558 a6c52cac5af2bc444d3eabf7e87f61d5c4516071955a8ca161196a4be12393d4 92b0dccf61b3bb03abecf595ecd831276992fe837a9947bc1d93dce02846d582 9e8e450be90d265c7b3f678e049a3574470d1974f20e15f17334a09a97dcf339 acd36d0ac62b7043f86826b6f0ca37cba13cda88bf66bd8cb7c0afa0b88a8496 0238275ccca62b2e74c601b6950b173f0c82bf5b2cf4cedd437d341ff872f18c 8f9dcb1410232ad6262480444d05ecccfa34a7a2d99bcc040d25e359484905f2 d65183daa9d7aeba84a97655ba8c228822fa4cbe1c592addc84ff60bea8b5bf6 d1fc45a023046fceac0dd2c6b94c819699765c26a64febcdb78da1d980d97a48 c5e1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 44f6ae92facec57658999c8bf01f60611560043962c0695e1bb87d090e0fdaab 659d6557208a660be9aeef8ae55f71b8bc76394d792967e41bd8023b6e7f8c3d 9987d5e54c53bf2bbb32855b946473b44833bd78873df8e4c7e668ffff25aabb 8da4148f744b9d0524d6cb8281247b3052cc5d66f3f1bcf8a56a3fb446ea587b d8be30257652cee035628aeddfd8283b5ad21b8b1480ad24d898970ac8a1e33e 543ce4e3c48f9ee621c5ce3e8ae930f890011b630866c988606068d9e92396eb f51bb3db204505fc3d09d4701ac703a14d92bbfe27bd7411b363a88e33a8e9a8 025ac31f21e73dc52b333b918d0b1270c54873a490222d041a66e0192612be6d 2c88958a8f70c10390146ee5091df020ab97675e8e6acf98b28b539cd0a20d24 a42f535ae44b8f5e9290b38fc6b02e1bbbdf1aca96558c2dd5afbec53f03ab51 fbce958ed4a65b17fbc03f9defd4aea4a15229f3c320876235f2d8b69bda5f48 03a2798790d35b394c63295997dd487bc603cdd10d18f3e3fcc7f17d2a6afc77 f1d7f6cf69d2a86b104b9d3b260e4a78716927a485ad8a650d610e68378f4514 3610b087981f0a0655090163dd2dd4d0030faf6a953896a9cb2cd033ec98811f 435755bc9fbc0d438f230046183582ba27c4978944f6c9b68033288372e22490 2d09886f519de56ec6afb6cec695920c555ec4300cdc69afb573c02db905eb04 cb49197ef0bcaf2c4a5c50d42e067b8a9cc41553a9abc6d08d8db2e547a08545 37c00ec442560c59571e716b5683edda2f4b000e98551720d7583fb75d481a03 cb125074974b8c907dd83847701dd344043f6fda98500514331763447c61d1d3 e6713fb078628b7d4d0c7c3f93ad48487ed7660760d18ef53bfa18887df6af14 1db008934185634e57e1d013a4259a697e948ddb5c550b3318a22a3fde9d45a1 0e293d5149cefe06255d318fec99dad2aeacf2e30731855424374f9669be9e31 6d75d26174045f05f5813ef8831d82ad507b9c3a9cfc8d7e94ed0a8a2267a7da d965d04dfeedbd149e7326c32ada4ff1121b3a9c084c702836ecd64c4ca1d086 3da878d6ef25ffa9cf71e59cb66ec669bc9c981348d48a8c84a0ddb5e0172b28 41aafcd57aa1619a9979546c9a8913f842449974491445c24b4c7bf87f4c2bc0 84b9f0b5bf172cacfdc07516e6804cc87d83249277cd54c64973d0a7ab44554a 12100e29c526bc81f87ee65f163956b2eb0e8bacc6f94d3511b9d7a285986156 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2175320dbf90b56fdf64bbe1a7d5e722cd48ba89962ac2614efc3e8d09491e45 fee17bef2c7958181bfc14573a7d7a788a0de7fc23080bde46f42b9edf626a41 3589eac7fa79f4ce667e20419a869d78692bfdcffd4a94 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 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f907c93368484081dce0daebfff7d6e70874f4b2ba348f9c0b4ff7b9c31fbd80 f5b634648b5f837b6952f599ad9dce8d81603266e509f34884c454edb342aee1 e05a04ee9aadf6d7f2df8616f81ddec34daec925855feda73f230d5d2eff3a94 87033afa73230c0bd1a04f04d70696265c9e7c741e65ac6af42ae68e5e33fd74 a4b14be99ca0cf7586e96744c0ad0c498dfdf4210ae2fc3fb1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 26 /braceleftbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 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0caec1cf9681d4512cf828bd73199cc69c60798bfa27ce0b11b317e6f7e556ca 23143fbe198336126fdcfb3f16cf067a78e6947202881b57c5cc600460698bb9 0a1a8a1fc380b757f14b5aa2380fc79f7a240dd5eaa8b67547c9100fcd8bff2c 46f07811a4a1f082a146d2993c63a9e6582e7a2d53119dbc24f3da3ebac20f92 a1e9a39e8aa76cd2d041cd0e4e5d7ef8d2ff64365e900c3cffd9c5a7625b4aab ccfc4a646201d17cf226a3d1fcbb04040a970f66cb4fc40b9a146640f1f2db2c 71de306c1dd87cd39e0869348618d06ce685c00623cd4cdcfd44283f90d3e790 113cd7f6e323b6ed50173647e67d45fe1296a24274ba2156a2d67dd744042b25 e9f608a3b386bc046f7b75517bc345a6b1f08ca370ca7daf23870f6f1424fbf3 7a2311f36950ed6a1e2143e0fc4e621cbdff82b4a734344d5cff7a12614bdc50 22dcc2a1c53555c0dafd892f014a8caa94051898ebe139074b9940a0d059442c 1d73c29030443707ba4bed7237a46510cf05 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 72 /H put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03acaa5e71066673665ebc99c1671730a922347e0df8497c3ca74d55d3 32625a2d47d41d7ab8ee776eb4fab79116f2e8018fb519002418aa46a669068f 27c27cae5f5aa453d8de98319e56ce64386181c9c8069aa47ff27668714ebb39 b4c3b424b32b4a704f3efd9a7812a1472bf6d7844fb99821211bad1be564f7e4 cb8df6f8bc30406de33d229796cdef744f8eeda755f47e55b10b11fad13586c0 ce9c8daee0e895ee322a0c6970f9f6a4e737e00f3cc403cb0e5022d51695acec 6341712f39ee25174d3d9b4174faec9c6fd3c1a1c827823e7977015a340289fb 5319e41405430cde003f786a133e2bb84e95660c4ef7ab2bb2f7cab852670090 d1ee899abd24e6fa74bd79de569918378066501431d7823bef29a96bf8e7559b 83f88d6caa105b0c2df2e0937d24b032884674af0c4a498c5126c473a3eb5165 f51707f802554ad9e99a1ba206f13c337732eb6057732c448aa56ef330eb8208 334f0c15c41272e0240a9eab5f6738357bedde47cc5362dd391300586799cbdb 5b44f29054ef05163e13c38ab53b95b55bf90393d38bbc0b1bf3b14250cc2d32 44b97a366a04848b3056fa2e47bbe721365dc2f7337d413d7c63b50cf2467541 7c4105f597f7751b885165b23d0e7aa9c35280f3efca68e23fb2397e10e6ba81 f1fc1e8e5caa4f17fbc67a0f28062b28bec983b648a8e4aba420723c520fd571 e493a70e8f525f1c4bb1cd239b3852cbb023295e9465486d85810bf34c1f938d 90c951fc30136187711e51b500ac59736fd27753c1ba9f90893c9aba09da3ac8 d3381ed819b3c8f01bf675a8e15188b30a0c9617609a9cf33b4f8743edb292c2 d1bdbfb6b53babfbde58f0295b5264929e8b5a49befad429518b955b71a7d41f 969d819ec33cfd5a20fb5fa0049aa1b319077c685df14ef826721bd9f6a53ced 9ea6e385ec2b61116cae3cb6d5c0d85e5c8ea7317ccc7646229780463c7a9e61 c1a4c356efd43532840e6b6bd2e51beedd13f31a0c4aa2daa907a6cbe41f4b82 664a9fb5be2be7e9d66b36efef92f499a8f3993abdc7451ee2aaa9151c2bc2a6 5bdc01529edd7f7b20107930c201d0b65c85d35a53872ecc18a87c600624be52 56d0359264ba52a1edfbd77156b7df3fe6a3545e31a38a3dc60822d36beecba0 5ea270ff61114c5ffc63ef0593d98aa196af1c4a9e3b4d7bcc91c39a8a64d66c e6d7cbee124b5504022e31ea4982b6aad713e2981c2d20e137ecf85169e00707 23499a68301896f382f66b48a94c016d160e3a47cc5bfaa53e8ef903b253c4ea 947bb3be541b15e36690bbbd82e9d8db0a3ca0940e8f146e3344218182bd231a de11a28c698df0e77bc7819018c60b368ae9817d97f79cfaaf67934496e98136 6ab18c9236a1536752275cc4518791b06a9567fe99ce41dfbb0208a85b1c72b3 ebf0f4b3bc34c293293865def1eac363cfa90dec6ba888e07485f49dafe5428e 48da2a517be02720cc80585fa4bee83f2352b99398583e071fd752c24eb0f22c 475fecbf3c05174787f0bb4af0c81ee3f043c68791602b44cf25061e3b905377 0d596cdf101be050afcf6141ec7add64cc8d76032b5d44f9504166a018f02fd4 ebf77e01532ffc23321ab4c957fec8180a8501c580c0b0a7b47ac8d720d12cba cbc70787ed67c270a6ae856aa88c4626db7b54df16b6c307302560a3873d0ba7 7a6a43bac753ea4d0dac9f07df127006cff3d2e0f42acb1cf497ad5bdc3fed18 1ce16c45f968127688e30e8c89ce9c20716a309a8bc3cf04c411432419885393 04661030f0e519b5e58fadafb0853a76884af18af10199a2fba979df0de2f29a 74cb0991052a0be10726d22c944cf98fc4fa8214838ccf0fa10ae3c3e07b4e05 e6a9ad87f340c5f1528e19ef11147e2f78204b28db456168bd2cecef78d60bd6 dbc7ca5959114a0d8584ee 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c458ba9cb7568b924f70eaf52ec0167e9a9af3b6a4e5d3fd4d 62b615142bc52f2bee5557f7483d9377ec4d123091a300b3a6ae990d5ba252ab 95e88d1ac5df908a57a9896b043e7a4cf1efce44489d118414cccec128cebd1c c8c0f3e48c52461da294731af5936aee45de7f41d43e2762f740a2479405c947 e94092c2b06bc2a6483af211e9afe88a7c8995973b084f63152c2a446b291ee5 a63630e14f08283a6a6f922785e61b9cae0ce7aed3a6f99c1fa52bbc3e370bdb 0ca8bdff4bc0b21fd84de65439f4908c35dffe10d6356ed0851a66ee3d33066c c7912ef1b9f929375ec464e5c98bbfa5d5ab59d4f9d191336426a4fbccd95ef3 bf73184faef62d83670a072ad2d80d40f452cdae68c3f649ce52c9d71ce34365 e658685bd459cd7e1d24fde49e45db47d70b3afd7d67925dcb41ff49f2c46ad2 7af0421249bf3142ddccb4b7fa81fc5cc51e540fb4922cc06be13731474c0a1f 2277f51a0c4b4cf330ef97343829d409f6fc994ae5a7667ec13ba110c9ecf68d 303492d8c4557e18a8ff42773ad7ddfe9cefb66d2e0d052ae7e17ced55033659 975ed42dfd1a98e5bfa1ac1d1e90237e3a16bacba571cd69e6a83ccb61171b9f 4e47613a264cb8d7ca3c8087303677ab8a99bd0c357d1d35afacd6de4fdf043e d50b70a0937fb00586f8c6759183651359c4d58d61bc367ad8a6516d682fea39 16543cfa52f3c973bb8ad7488f71f4b5a9de0557f7013723d23d6d17ba4b59ff 536456b4468b2fee2faf016a98d119fe5a956a7b305c6f27b1f67b91bc10d1c6 42f90000653b722b294bf89e35fe6fb13b6b65d33636213f55ac244ce1894a72 a7fd546c68793b292f914631b84f3547a445edf11fc76d3fb3cac8ce266d40eb bcd73952194ba1b88f1031f4a5c3cca1bc84556fb63f4dec3e68b5df6a634d9c 00fabddb46ca95e3b540a46fb8d5925d139c484f0e06e802ed954ebde20663b3 65a1ada2365d2e9fd52f4874c80c4cc8205b5cb16f683214a4a4028968227e39 bccdca40e26c7980f1bece1a0c6ecf6880420270c918a52a4e3bd433b1a07ada 0a2efc8bd401ea8ae25a3bb44f3df89577aeedb61e7dbce9ac6688be223dee78 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 21 /lambda put dup 22 /mu put dup 28 /tau put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 72 /H put dup 73 /I put dup 74 /J put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 104 /h put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueXX 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 13a84eb181d2946b5d0075f71db8a991dc1a809cd3cfec573514392567a82e75 946ccf6e14f4a968313b3b6d727c83018ed58b882b96db208ebd27f7d30ffbc8 ab11e10a0f81425a62bdf7eed9f618c746103dc2d1c3e1f37d27083f4068a30e 099c92685e320266ff8047e81140658a63a6b52e4cd63e95d1b83fde2655d8b5 7dc0c494c0f8ff83dcb17169fa300050816a1c407324d081e187a52da4a31e0e bff67357bee1d19362b342d1a18d2df456cf4b91ffe04ff375304de3d365e20f f9450f9ae5532da629071700a78544b7a2cf132c861408540020e469db016153 fc7e43064654eb4f3b5b1faaa17868b8db070b1d6a7756c07206f2e13561752f 16c41ed094c3c3a069d97383d20b0d72860c5865113f3bd74502ef099853f3d4 2e56a4d94da73652e4efe6e274cb145e291ed8fea94fb5cfa865d4a14d6aeeca 493a53d68aa4e2db4d2a1e9a67ef676178517e7e964186efc28b9d755a5d11c2 d067b6b82145540773d2f18f6ff501e1be27bb1372dc1edf6cb01f026de4db37 1addcdee54b0f4ab80aa3034dba3bab7de9f35af7c353ab5d26233f2e630f622 53b3a05ff10267f1372da857df92cb77f8c3a9b0b74718ab7d2590b04c45444f 22cfde2c4cfe6f5bbfa932c05b1994d2eeb99cd2199068fa557a9a38ab3c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 120 /x put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 55 /mapsto put dup 56 /universal put dup 59 /emptyset put dup 63 /perpendicular put dup 65 /A put dup 69 /E put dup 70 /F put dup 72 /H put dup 77 /M put dup 80 /P put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4300265fd2347a3a851a9cb00aa30bbb57598cca6485e51c30bb78bf8307 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d70d79908a99279c0e0399d0fa0a80f66cc75693e3f9df6a297cf1a54018b805 939e44aaa16323dec4180ec7b1a7fc35afef328ca51c8a974670332647ec13e0 696481619cdc225b34a353cf29f5140d3ede0857273e844a9b31228fb25378be b076a0a51adb8f50d9e578bfb9651e8a65d9fa8fb311b5ea37f144762a74cafd fcf5604c88df111b0a1932395a0941d36df5304bd6cefda2b44f96e022987fe9 9790aad6dc4708610e664cf322af8615650baf671e9253402e418631af91 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 64 /at put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784da87b5d0b13f35180e 58efc5cf12de78792a88665f1d5e60d4569e5c614792303616a6d0bc41d20457 507b75a7973223073a715fbc7fe4b888df49c5b1841fb540bdb8dd0128f1626a 801f8d3871c2b4d5e5d8bccca9cb92d282289128d69c0f75f990aece52ea6ab5 cfd2d5dd01ce4ff38d21301e6014664c0c937b57a7af2be7e8aae980929cd77e 66694fbd58c24e5531988dfde0d8ed096c25719fddaa11dd9354bb2e379cc716 b80194087a8d4822d289871d6510625aaf06726d6833cbff90afccf2af6d3def 95fcfdb1aa0e890467164da5107dbf2243e48acc60c1b304f73ab4a3209fb636 060f9c812fe5b50da3682923ee15290c5aa0a9895bb2e7836549c403770d825a 4f4092ca19ca195e16fbbaab55c6868efc2e341b1f90ba3da1379c374194ad97 570658336de5a24600f4dafed3d142e7c94d9e959dcfc983a65a6217132bd1dd e2873796c347663955f778f8f887a6d6426a88f82a01129a12f9f0973bce51ac 1ff4aebce6f82f86218b5dd3f4a69957fd9cbfe75420e3e9e435f8c88fb1eec0 29c19d1fdee97387ad365f6b1d2f271ccc612444eb45b3a687626fc3b0507649 bdde0c6fc6aa18807d3d792b3d20bd263e4608dd80082dfdc97a83713ec740b3 b98281ee6e2ebf13735f48c2d41a70b84db608c3a0fbb6687419e864012f5742 73b91877dfdc859488f6f80bb8116e89d022825d409c5e6670870c38fad9c3e0 884c8ee63f1abfa5874b88524b2f09c3375a0dbb7f0296839b31443c40eef3d7 c0417fe2a23156aae7c2350b99c930e346c40da420c0737dd71aa62eb30614da fc309f7cc5668984ad2bf770e5bdc8439cfe623e905a116b0eb1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 35 /numbersign put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e50bd6bf02070d34a2 265e57d706e67f56b5c8b53d91beeeabefea967e90d35429a32daede5e3313ab e1b108d02a0d3a0b1d8f35a59ad8cc8be7ede1546a7bda96d7ff80474307192b 224c6b619ab1a582fa020c5fb3539cb26abbedd47034003925acfb4ea775dbf1 81d0c92974b16ec4f86afc6c515e0b2f4ec6d71ebb6aa09eac01fbe9654b6cfb da6aac14cc19179844a9927b105673a954cf2664da855a3ad7e3cf21f0746426 95063939eb59fac4df347b094e987e618c4b6191cb31b44497359ac625e9b770 3e570d45416de00c7a81a4bbcec71f7ac41277cdc86d74b7f273380009094429 05b1217abbc1289e4c3342cfc02fc9fb08bace29b7598e906dcc37ea1d9058f8 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46432269ed13042e9a5c2a1651327c46664c2bf1091a249a20da8c4226bc4990 0ca2b6abb4b0b31be3426b02806ecd5ceb22fd8c50799c7ca5bf60a01db54b23 dbc5671b14d33719577398a8d3a162d22cce72d35611fcf5590ac205bd832218 510082ff57d46374e5e48e81b2d4bc49a856f8fce032331b2c032bca78a6ee74 3d010b708b4c99e37b927029e05d9027c3089f39aad5935eae546d4062f1bef4 a37df142036edd600a6d3409fe1b8be58bebe1e495026e3871 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 28 /oslash put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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ad37f85f1effbe3b8d283bfc03c437f628ed607bc71058708202e740b7e1caca 467cc8c687b87aea61467896fcb2c4356ff8b5ed66f92f5627258290ea1ee0af 0df542cbfc617fbdb391c6195f797b76ba9b4bfd13e25d8a6b3d822c34978386 ee67bf24ac90b42369a8c86ee88138cb36282e96e9120ba56586e04a86d2a9f7 128d0fb4c983b5f50d51f8ab3b7733fe5b8edfc92c9eb3e1a1d0ee364fbf7437 cfc532a7c9be45dff393fd9053 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 68 /D put dup 77 /M put dup 78 /N put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (CAR(mp-arc).dvi) @start /Fa 206[785 49[{}1 1328.35 /CMBXTI10 rf /Fb 207[243 48[{}1 664.176 /CMSY6 rf /Fc 206[523 49[{}1 885.568 /CMBXTI10 rf /Fd 203[406 406 406 406 49[{}4 664.176 /CMR6 rf /Fe 252[1033 3[{}1 1328.35 /MSAM10 rf /Ff 135[789 1079 1[830 581 589 610 1[830 747 830 1245 415 789 1[415 830 747 457 682 830 664 1[726 12[1039 1[1115 1[1021 3[898 2[557 2[939 981 1145 2[1128 7[747 747 747 747 747 747 747 747 747 747 1[415 4[581 581 27[830 12[{}43 1328.35 /CMBX12 rf /Fg 133[531 631 598 863 598 697 432 531 548 598 664 664 731 1063 332 598 399 399 664 598 399 598 664 598 598 664 3[399 1[399 797 5[930 731 948 996 881 996 966 1165 815 999 682 501 966 1[848 881 981 930 1[966 6[399 664 4[664 664 664 664 664 1[399 465 399 1[664 531 531 27[731 12[{}61 1328.35 /CMTI12 rf /Fh 144[886 886 11[738 7[738 1697 1919 5[627 1255 3[2007 1476 6[812 812 7[1181 1[1181 1[1181 29[996 8[793 793 14[609 609{}20 1328.35 /CMEX10 rf /Fi 183[801 21[627 941 255 26[732 14[732 2[470 2[732{}8 885.568 /CMSY8 rf /Fj 165[1275 90[{}1 1912.83 /MSBM10 rf /Fk 133[438 5[340 435 1[420 474 455 571 832 281 490 381 320 2[452 436 484 408 1[500 22[514 408 773 10[470 1[261 25[585 4[418 5[566 549 7[485 528 12[{}28 885.568 /CMMI8 rf /Fl 151[692 104[{}1 1328.35 /EUFM10 rf /Fm 133[604 1[739 3[470 613 1[577 653 1[776 1138 387 676 538 444 749 623 641 603 676 560 553 683 11[883 762 800 5[1258 885 1[719 574 1078 1026 2[1080 931 988 975 2[1012 650 1012 361 361 24[810 5[739 1[738 568 1[783 759 2[606 642 2[576 675 734 827 11[{}50 1328.35 /CMMI12 rf /Fn 143[1107 4[664 369 517 517 664 664 9[886 886 10[924 2[1595 4[1122 1[955 701 3[1061 1[1033 3[664 2[738 0 4[886 1328 15[1328 6[1033 4[1033 1033 2[1033 2[664 3[1033 6[664 1033 369 1033{}32 1328.35 /CMSY10 rf /Fo 165[886 7[959 3[959 10[959 67[{}4 1328.35 /MSBM10 rf /Fp 136[1554 1136 1196 837 849 879 1136 1196 1076 1196 1793 598 1136 1[598 1196 1076 658 982 1196 956 1196 1046 12[1497 1[1605 1[1470 1[1683 2041 3[801 1683 3[1649 1554 1[1625 6[598 5[1076 1076 1076 1076 2[598 717 45[{}39 1912.83 /CMBX12 rf /Fq 134[584 3[615 430 437 434 1[615 553 615 922 307 584 1[307 615 2[492 1[492 615 553 27[753 4[861 5[307 11[307 369 45[{}22 1106.96 /CMR10 rf /Fr 204[441 441 441 49[{}3 774.872 /CMR7 rf /Fs 165[808 90[{}1 1212.12 /MSBM10 rf /Ft 139[542 550 574 14[620 774 678 31[1054 65[{}7 1212.12 /CMBX10 rf /Fu 134[640 640 875 640 673 471 478 475 640 673 606 673 1010 337 640 1[337 673 606 370 539 673 539 673 606 3[337 1[337 1[909 1[1246 1[909 1[673 892 1[825 1[909 1111 1[943 1[438 4[926 875 1[909 7[606 1[606 2[606 1[606 606 606 1[337 404 337 44[{}48 1212.12 /CMR10 rf /Fv 162[261 1[261 29[732 8[470 470 470 470 470 4[732 1[366 366 4[784 33[784 1[{}13 885.568 /CMR8 rf /Fw 128[650 650 3[578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 3[361 1[361 795 975 975 1336 975 975 939 723 957 1012 885 1012 975 1192 813 1011 668 469 975 1021 849 885 993 939 921 975 3[1012 1[361 361 650 650 650 650 650 650 650 650 650 650 1[361 434 361 1012 1[506 506 6[361 4[650 13[1084 723 723 759 939 8[1084 813{}84 1328.35 /CMR12 rf /Fx 205[780 50[{}1 1594.02 /CMR12 rf /Fy 165[1531 90[{}1 2295.84 /MSBM10 rf /Fz 134[1113 2[1113 1173 813 825 813 1[1173 1053 1173 1773 573 1113 1[573 1173 1053 633 933 1173 933 1173 1053 18[1592 1952 8[1622 22[693 45[{}25 2295.84 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 7858 12675 a Fz(Diric)-60 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Fw(-real,)395 b(real-v)-72 b(alued,)396 b(densely)385 b(de\014ned)f(quadratic)i(form)g(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))385 b(is)h(called)g Fg(Markovian)2717 64472 y Fw(with)434 b(resp)36 b(ect)433 b(to)h Fm(\030)12289 64671 y Fv(0)13184 64472 y Fn(2)368 b(P)543 b Fw(if)13851 67461 y Fm(\021)417 b Fn(2)368 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))19112 66912 y Fk(J)20198 67461 y Fw(implies)434 b Fm(\021)25255 67660 y Fk(I)26156 67461 y Fn(2)369 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))433 b(and)g Fn(E)119 b Fw([)p Fm(\021)35144 67660 y Fk(I)35676 67461 y Fw(])369 b Fn(\024)h(E)119 b Fw([)p Fm(\021)48 b Fw(])p Fm(:)8460 b Fw(\(2.1\))2717 70449 y(A)434 b(closed)g(Mark)-36 b(o)g(vian)434 b(form)g(is)g(called)h(a)e Fg(Dirichlet)464 b(form)p Fw(.)4668 72110 y(Next,)634 b(w)-36 b(e)594 b(collect)h(main)e(results)h(of)g([Cip1)q(].)1058 b(Let)593 b(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))594 b(b)36 b(e)593 b(a)h Fm(J)123 b Fw(-real,)634 b(real)594 b(v)-72 b(alued,)26801 78590 y(4)p eop %%Page: 5 5 5 4 bop 2717 7306 a Fw(densely)434 b(de\014ned)e(closed)i(form.)579 b(Assume)433 b(that)g(the)g(follo)-36 b(wing)436 b(prop)36 b(erties)433 b(hold:)12082 10295 y(\()p Fm(a)p Fw(\))1106 b Fm(\030)15451 10494 y Fv(0)16346 10295 y Fn(2)368 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))p Fm(;)27953 b Fw(\(2.2\))12146 12287 y(\()p Fm(b)p Fw(\))1172 b Fn(E)119 b Fw(\()p Fm(\030)64 b(;)221 b(\030)64 b Fw(\))370 b Fn(\025)f Fw(0)443 b(for)g Fm(\030)434 b Fn(2)368 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))p Fm(;)12143 14280 y Fw(\()p Fm(c)p Fw(\))1168 b Fm(\030)433 b Fn(2)369 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))20087 13731 y Fk(J)21394 14280 y Fw(implies)655 b Fm(\030)26598 14479 y Fi(\006)27754 14280 y Fn(2)369 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))876 b(and)442 b Fn(E)119 b Fw(\()p Fm(\030)37265 14479 y Fv(+)38052 14280 y Fm(;)221 b(\030)39202 14479 y Fi(\000)39990 14280 y Fw(\))369 b Fn(\024)g Fw(0)p Fm(:)2717 17269 y Fw(Then)442 b Fn(E)560 b Fw(is)442 b(a)h(Diric)-36 b(hlet)442 b(form)g(if)h(and)e (only)h(if)h Fn(E)119 b Fw(\()p Fm(\030)64 b(;)221 b(\030)29153 17468 y Fv(0)29679 17269 y Fw(\))383 b Fn(\025)g Fw(0)443 b(for)f(all)h Fm(\030)k Fn(2)383 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))300 b Fn(\\)g(P)109 b Fw(.)604 b(The)441 b(ab)36 b(o)-36 b(v)g(e)2717 18929 y(result)433 b(follo)-36 b(ws)436 b(from)e(Prop)36 b(osition)434 b(4.5)g(\(b\))f(and)g(Prop)36 b(osition)434 b(4.10)h(\(ii\))f(of)g([Cip1].)4668 20589 y(The)439 b(follo)-36 b(wing)440 b(is)f(Theorem)f(4.11)i(of)f([Cip1)q (])f(:)588 b(Let)438 b Fn(f)p Fm(T)32433 20788 y Fk(t)32829 20589 y Fn(g)33493 20788 y Fk(t)p Fi(\025)p Fv(0)35528 20589 y Fw(b)36 b(e)439 b(a)f Fm(J)123 b Fw(-real,)440 b(strongly)f(con)-36 b(tin-)2717 22250 y(uous,)479 b(symmetric)469 b(semigroup)h(on)f Fn(H)483 b Fw(and)469 b(let)h(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))469 b(b)36 b(e)469 b(the)g(asso)36 b(ciated)471 b(densely)f(de\014ned)2717 23910 y Fm(J)123 b Fw(-real,)434 b(real)g(v)-72 b(alued)434 b(quadratic)g(form.)578 b(Then)434 b(the)f(follo)-36 b(wings)435 b(are)f(equiv)-72 b(alen)-36 b(t.)17566 26899 y(\()p Fm(a)p Fw(\))1106 b Fn(f)p Fm(T)21793 27098 y Fk(t)22189 26899 y Fn(g)22853 27098 y Fk(t)p Fi(\025)p Fv(0)25327 26899 y Fw(is)434 b(sub-Mark)-36 b(o)g(vian)p Fm(:)13441 b Fw(\(2.3\))17631 28892 y(\()p Fm(b)p Fw(\))1171 b(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))876 b(is)434 b(a)g(Diric)-36 b(hlet)434 b(form)g(.)2717 31880 y(W)-108 b(e)434 b(refer)f(the)h(reader)f(to)g([Cip1)q(])h(for)g (the)f(details.)4668 33541 y(Let)553 b Fm(\015)866 b Fw(:)795 b Fn(M)573 b(!)g(M)554 b Fw(b)36 b(e)553 b(a)h Fo(Z)20013 33740 y Fv(2)20538 33541 y Fw(-grading)g(of)g Fn(M)p Fw(,)584 b(so)553 b(that)g Fm(\015)626 b Fw(is)553 b(a)h Fn(\003)p Fw(-automorphism)f(of)h Fn(M)2717 35201 y Fw(whic)-36 b(h)451 b(is)g(in)-36 b(v)g(olutiv)g(e:)615 b Fm(\015)15031 34719 y Fv(2)15955 35201 y Fw(=)399 b(id)p Fm(:)451 b Fw(Th)-36 b(us)451 b Fn(M)g Fw(is)g(a)h(direct)e(sum)h(of)h Fn(M)36615 35400 y Fk(e)37506 35201 y Fw(:=)398 b Fn(f)p Fm(A)h Fn(2)g(M)g Fw(:)g Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))398 b(=)h Fm(A)p Fn(g)2717 36862 y Fw(and)442 b Fn(M)6850 37061 y Fk(o)7745 36862 y Fw(:=)384 b Fn(f)p Fm(A)g Fn(2)g(M)g Fw(:)h Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))383 b(=)h Fn(\000)p Fm(A)p Fn(g)p Fw(.)605 b(The)443 b(elemen)-36 b(ts)442 b(of)h Fn(M)34766 37061 y Fk(e)35701 36862 y Fw(are)g(called)g Fg(even)e Fw(and)h(those)g(of)2717 38522 y Fn(M)4312 38721 y Fk(o)5211 38522 y Fg(o)-66 b(dd)p Fw(.)564 b(The)388 b(iden)-36 b(tit)g(y)388 b(elemen)-36 b(t)389 b Ff(1)369 b Fn(2)f(M)389 b Fw(is)g(ev)-36 b(en.)563 b(A)389 b Fg(sup)-66 b(er)g(derivation)387 b Fm(\016)438 b Fw(on)389 b(a)g Fo(Z)44336 38721 y Fv(2)44861 38522 y Fw(-graded)f(v)-36 b(on)2717 40183 y(Neumann)433 b(algebra)h(\()p Fn(M)p Fm(;)221 b(\015)72 b Fw(\))434 b(is)g(a)g(linear)g(map)f (satisfying)i(that)f(for)g(all)g Fm(A;)221 b(B)437 b Fn(2)369 b(M)p Fw(,)18928 43171 y(\()p Fm(a)p Fw(\))221 b Fm(\016)50 b Fw(\()p Fm(AB)67 b Fw(\))368 b(=)h Fm(\016)50 b Fw(\()p Fm(A)p Fw(\))p Fm(B)362 b Fw(+)294 b Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))p Fm(\016)50 b Fw(\()p Fm(B)67 b Fw(\))p Fm(;)11544 b Fw(\(2.4\))18928 45164 y(\()p Fm(b)p Fw(\))221 b Fm(\016)50 b Fw(\()p Fm(\015)72 b Fw(\()p Fm(A)p Fw(\)\))368 b(=)g Fn(\000)p Fm(\015)72 b Fw(\()p Fm(\016)50 b Fw(\()p Fm(A)p Fw(\)\))p Fm(:)2717 48153 y Fw(A)434 b(sup)36 b(erderiv)-72 b(ation)433 b Fm(\016)483 b Fw(is)434 b(called)g Fg(inner)e Fw(if)j(there)e(is)h Fm(x)369 b Fn(2)f(M)434 b Fw(suc)-36 b(h)433 b(that)18381 51142 y Fm(\016)50 b Fw(\()p Fm(A)p Fw(\))369 b(=)f Fm(xA)296 b Fn(\000)f Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))p Fm(x;)443 b Fn(8)221 b Fm(A)369 b Fn(2)g(M)p Fm(:)12990 b Fw(\(2.5\))2717 54130 y(The)480 b(elemen)-36 b(t)479 b Fm(x)h Fw(whic)-36 b(h)480 b(de\014nes)f(an)g(inner)g(sup)36 b(erderiv)-72 b(ation)480 b(as)g(in)g(\(2.5\))g(is)g(o)36 b(dd.)717 b(See)479 b(Lemma)2717 55791 y(1.1)435 b(of)f([DL2].)4668 57451 y(In)g(this)f(pap)36 b(er)433 b(w)-36 b(e)434 b(assume)f(that)g Fm(\030)22274 57650 y Fv(0)23234 57451 y Fw(is)h(in)-36 b(v)-72 b(arian)-36 b(t)433 b(under)g Fm(\015)505 b Fw(in)433 b(the)g(sense)h(that)17666 60440 y Fn(h)p Fm(\030)18751 60639 y Fv(0)19276 60440 y Fm(;)221 b(A\030)21401 60639 y Fv(0)21928 60440 y Fn(i)369 b Fw(=)f Fn(h)p Fm(\030)25279 60639 y Fv(0)25805 60440 y Fm(;)221 b(\015)72 b Fw(\()p Fm(A)p Fw(\))p Fm(\030)29689 60639 y Fv(0)30215 60440 y Fn(i)1300 b Fm(A)369 b Fn(2)g(M)p Fm(:)12275 b Fw(\(2.6\))2717 63429 y(Then)433 b(there)g(exists)i(a)e(unitary)h(op)36 b(erator)434 b Fm(U)24823 63628 y Fk(\015)25849 63429 y Fw(on)f Fn(H)447 b Fw(suc)-36 b(h)432 b(that)18981 66418 y Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))1117 b(=)g Fm(U)25844 66617 y Fk(\015)26437 66418 y Fm(AU)28434 65869 y Fi(\000)p Fv(1)28295 66746 y Fk(\015)29692 66418 y Fm(;)443 b(A)369 b Fn(2)g(M)p Fm(;)13811 b Fw(\(2.7\))19206 68410 y Fm(U)20089 68609 y Fk(\015)20681 68410 y Fn(P)1216 b(\032)1107 b(P)109 b Fm(;)19145 70403 y(U)20028 70602 y Fk(\015)20620 70403 y Fm(\030)21188 70602 y Fv(0)22832 70403 y Fw(=)1117 b Fm(\030)25529 70602 y Fv(0)26055 70403 y Fm(;)20166 72395 y(U)21188 71847 y Fv(2)21049 72724 y Fk(\015)22832 72395 y Fw(=)g Ff(1)q Fm(:)26801 78590 y Fw(5)p eop %%Page: 6 6 6 5 bop 2717 7306 a Fw(In)366 b(fact)i Fm(U)7657 7505 y Fk(\015)8249 7306 y Fm(A\030)9792 7505 y Fv(0)10687 7306 y Fw(=)h Fm(\015)72 b Fw(\()p Fm(A)p Fw(\))p Fm(\030)15370 7505 y Fv(0)15895 7306 y Fw(,)380 b(where)367 b Fm(A)i Fn(2)f(M)p Fw(.)557 b(See)366 b(the)g(pro)36 b(of)367 b(of)g(Corollary)i(2.5.32)f(of)f([BR)q(].)556 b(Here)2717 8966 y(w)-36 b(e)418 b(ha)-36 b(v)g(e)418 b(used)g(the)f(prop)36 b(ert)-36 b(y)417 b Fm(\015)18820 8484 y Fv(2)19715 8966 y Fw(=)g(id)h(for)h(the)e(last)h(relation.)574 b(In)418 b(this)f(case,)422 b(it)c(can)g(b)36 b(e)418 b(c)-36 b(hec)g(k)g(ed)2717 10627 y(that)433 b Fm(U)6418 10826 y Fk(\015)7444 10627 y Fw(comm)-36 b(utes)434 b(with)f(\001)h(and)f Fm(J)123 b Fw(,)433 b(whic)-36 b(h)434 b(implies)g(for)g(an)-36 b(y)434 b Fm(t)368 b Fn(2)h Fo(R)p Fm(;)886 b(A)369 b Fn(2)g(M)p Fm(;)20343 13423 y(\015)72 b Fw(\()p Fm(\033)22335 13622 y Fk(t)22730 13423 y Fw(\()p Fm(A)p Fw(\)\))1106 b(=)h Fm(\033)29187 13622 y Fk(t)29582 13423 y Fw(\()p Fm(\015)72 b Fw(\()p Fm(A)p Fw(\)\))p Fm(;)15173 b Fw(\(2.8\))20864 15415 y Fm(\015)72 b Fw(\()p Fm(j)j Fw(\()p Fm(A)p Fw(\)\))1106 b(=)h Fm(j)75 b Fw(\()p Fm(\015)d Fw(\()p Fm(A)p Fw(\)\))p Fm(:)4668 18211 y Fw(In)446 b(order)f(to)i(express)f(Diric)-36 b(hlet)446 b(forms,)k(let)c(us)f(in)-36 b(tro)36 b(duce)445 b(the)h(notion)g(of)g(admissible)h(func-)2717 19872 y(tions)434 b(in)f([P)-36 b(ar].)2717 22474 y Ff(De\014nition)500 b(2.1)651 b Fg(A)-33 b(n)483 b(analytic)h(function)f Fm(f)547 b Fw(:)406 b Fm(D)442 b Fn(!)406 b Fo(C)485 b Fg(on)g(a)f(domain)h Fm(D)520 b Fg(c)-66 b(ontaining)482 b(the)i(strip)2717 24134 y Fm(I)104 b(m)443 b(z)428 b Fn(2)369 b Fw([)p Fn(\000)p Fw(1)p Fm(=)p Fw(4)p Fm(;)221 b Fw(1)p Fm(=)p Fw(4])468 b Fg(is)d(said)g(to)g(b)-66 b(e)464 b(admissible)g(if)g(the)h(fol)66 b(lowing)464 b(pr)-66 b(op)g(erties)464 b(hold:)3592 26737 y(\(a\))651 b Fm(f)142 b Fw(\()p Fm(t)p Fw(\))368 b Fn(\025)h Fw(0)1329 b Fg(for)f Fn(8)p Fm(t)369 b Fn(2)f Fo(R)p Fm(;)3658 29449 y Fg(\(b\))651 b Fm(f)142 b Fw(\()p Fm(t)294 b Fw(+)h Fm(i=)p Fw(4\))h(+)e Fm(f)142 b Fw(\()p Fm(t)295 b Fn(\000)g Fm(i=)p Fw(4\))370 b Fn(\025)f Fw(0)1328 b Fg(for)g Fn(8)p Fm(t)369 b Fn(2)g Fo(R)p Fm(;)3658 32161 y Fg(\(c\))651 b(ther)-66 b(e)464 b(exist)g Fm(M)508 b(>)368 b Fw(0)466 b Fg(and)f Fm(p)368 b(>)h Fw(1)465 b Fg(such)h(that)f(the)f(b)-66 b(ound)21472 34957 y Fn(j)p Fm(f)142 b Fw(\()p Fm(t)294 b Fw(+)h Fm(is)p Fw(\))p Fn(j)368 b(\024)i Fm(M)139 b Fw(\(1)295 b(+)g Fn(j)p Fm(t)p Fn(j)p Fw(\))34773 34409 y Fi(\000)p Fk(p)5969 37753 y Fg(holds)466 b(uniformly)e(in)g Fm(s)368 b Fn(2)h Fw([)p Fn(\000)p Fw(1)p Fm(=)p Fw(4)p Fm(;)221 b Fw(1)p Fm(=)p Fw(4])p Fm(:)2717 40356 y Fw(W)-108 b(e)655 b(remark)g(that)f (there)g(exists)h(an)g(admissible)g(function)f(satisfying)i (De\014nition)f(2.1.)1242 b(See)2717 42016 y(Lemma)434 b(3.1)g(of)g([P)-36 b(ar)q(].)4668 43676 y(W)-108 b(e)706 b(are)h(ready)g(to)f(giv)-36 b(e)707 b(a)g(construction)f(of)h(Diric) -36 b(hlet)706 b(forms)h(on)f(the)g(standard)g(form)2717 45337 y(\()p Fn(M)p Fm(;)221 b Fn(H)13 b Fm(;)221 b Fn(P)109 b Fm(;)221 b(J)123 b Fw(\))609 b(asso)36 b(ciated)609 b(with)f(pair)g(\()p Fn(M)p Fm(;)221 b(\030)26292 45536 y Fv(0)26818 45337 y Fw(\))p Fm(;)608 b Fw(where)g Fn(M)g Fw(is)g(a)g Fo(Z)38054 45536 y Fv(2)38579 45337 y Fw(-graded)f(v)-36 b(on)608 b(Neumann)2717 46997 y(algebra)385 b(b)-36 b(y)385 b Fm(\015)72 b(:)384 b Fw(Denote)h(b)-36 b(y)385 b Fn(M)18236 47196 y Fk(an)19746 46997 y Fw(the)f(dense)g(subset)g(of)h Fn(M)g Fw(consisting)g(of)g Fm(\033)40922 47196 y Fk(t)41317 46997 y Fw(-analytic)g(elemen)-36 b(ts)2717 48658 y(on)464 b(a)h(domain)f(con)-36 b(taining)464 b(the)g(strip)f Fm(I)22594 48864 y Fv(1)p Fk(=)p Fv(2)24482 48658 y Fw(:=)421 b Fn(f)p Fm(z)702 b Fw(:)643 b Fn(j)p Fw(Im)p Fm(z)59 b Fn(j)421 b(\024)34212 48135 y Fv(1)p 34212 48352 471 54 v 34212 49116 a(2)34816 48658 y Fn(g)p Fw([BR].)671 b(By)464 b(Prop)36 b(osition)465 b(2.5.21)2717 50318 y(of)442 b([BR],)h(an)-36 b(y)442 b(elemen)-36 b(t)441 b Fm(A)382 b Fn(2)g(M)19108 50517 y Fk(an)20676 50318 y Fw(is)442 b(strongly)g(analytic.)603 b(F)-108 b(or)441 b(a)h(giv)-36 b(en)442 b(admissible)g(function)f Fm(f)2717 51979 y Fw(and)433 b(o)36 b(dd)433 b(elemen)-36 b(t)434 b Fm(x)369 b Fn(2)f(M)16574 52178 y Fk(o)17380 51979 y Fn(\\)295 b(M)20156 52178 y Fk(an)21282 51979 y Fw(,)434 b(de\014ne)e(a)i(sesquilinear)h(form)f Fn(E)487 b Fw(:)370 b Fn(H)308 b(\002)295 b(H)381 b(\000)-221 b(!)370 b Fo(C)434 b Fw(b)-36 b(y)6499 54775 y Fn(E)119 b Fw(\()p Fm(\021)48 b(;)221 b(\030)64 b Fw(\))38627 b(\(2.9\))6499 57358 y(=)7880 55550 y Fh(Z)9430 55883 y(D)10242 56282 y(\000)10850 57358 y Fm(\033)11589 57564 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)13978 57358 y Fw(\()p Fm(x)p Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)19210 57564 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)21598 57358 y Fw(\()p Fm(x)22843 56809 y Fi(\003)23369 57358 y Fw(\)\))p Fm(U)25264 57557 y Fk(\015)25856 56282 y Fh(\001)26465 57358 y Fm(\021)48 b(;)27737 56282 y Fh(\000)28346 57358 y Fm(\033)29085 57564 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)31474 57358 y Fw(\()p Fm(x)p Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)36706 57564 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)39094 57358 y Fw(\()p Fm(x)40339 56809 y Fi(\003)40865 57358 y Fw(\)\))p Fm(U)42760 57557 y Fk(\015)43352 56282 y Fh(\001)43961 57358 y Fm(\030)44593 55883 y Fh(E)45405 57358 y Fm(f)142 b Fw(\()p Fm(t)p Fw(\))p Fm(dt)7800 60752 y Fw(+)9033 58944 y Fh(Z)10583 59278 y(D)11395 59676 y(\000)12003 60752 y Fm(\033)12742 60959 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)15131 60752 y Fw(\()p Fm(x)16376 60204 y Fi(\003)16902 60752 y Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)20889 60959 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)23277 60752 y Fw(\()p 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Fg(such)2717 33319 y(that)465 b(the)f(b)-66 b(ound)21713 36308 y Fw(sup)19992 37452 y Fk(s)p Fi(2)p Fv([)p Fi(\000)p Fv(1)p Fk(=)p Fv(4)p Fk(;)p Fv(1)p Fk(=)p Fv(4])25393 36308 y Fn(k)p Fm(\033)26796 36507 y Fk(t)p Fv(+)p Fk(is)28679 36308 y Fw(\()p Fm(x)p Fw(\))p Fn(k)368 b(\024)i Fm(M)14089 b Fw(\(2.12\))2717 40168 y Fg(holds)466 b(uniformly)e(in)g Fm(t)369 b Fn(2)f Fo(R)p Fg(.)598 b(Then)464 b(the)h(fol)66 b(lowing)464 b(pr)-66 b(op)g(erties)464 b(hold:)2717 41828 y(\(a\))i Fm(H)103 b(\030)6658 42027 y Fv(0)7552 41828 y Fw(=)369 b(0)p Fg(,)2717 43489 y(\(b\))466 b Fn(E)119 b Fw([)p Fm(J)k(\030)64 b Fw(])369 b(=)g Fn(E)119 b Fw([)p Fm(\030)64 b Fw(])p Fm(;)444 b Fn(8)p Fm(\030)433 b Fn(2)368 b(H)13 b Fg(,)1793 b(\(J-r)-66 b(e)g(al)464 b(\))2717 45149 y(\(c\))i Fn(E)119 b Fw(\()p Fm(\030)6737 45348 y Fv(+)7524 45149 y Fm(;)221 b(\030)8674 45348 y Fi(\000)9461 45149 y Fw(\))369 b Fn(\024)g Fw(0)1152 b Fg(for)686 b Fn(8)p Fm(\030)433 b Fn(2)369 b(H)19966 44667 y Fk(J)20618 45149 y Fm(:)2717 46810 y Fg(F)-100 b(urthermor)-66 b(e)465 b Fw(\()p Fn(E)119 b Fm(;)221 b Fn(H)13 b Fw(\))464 b Fg(is)h(a)g(Dirichlet)f(form.)4668 49577 y Fw(W)-108 b(e)603 b(will)i(pro)36 b(duce)602 b(the)h(pro)36 b(of)604 b(of)g(the)e(Theorem)i(2.1)g(at)f(the)g(end)g (of)g(this)g(section.)1088 b(The)2717 51237 y(follo)-36 b(wing)436 b(is)e(a)g(consequence)f(of)h(Theorem)g(2.1:)2717 54005 y Ff(Theorem)499 b(2.2)651 b Fg(L)-66 b(et)495 b Fm(H)598 b Fg(b)-66 b(e)496 b(the)f(self-adjoint)g(op)-66 b(er)g(ator)495 b(asso)-66 b(ciate)g(d)495 b(with)h Fw(\()p Fn(E)119 b Fm(;)221 b Fn(H)13 b Fw(\))496 b Fg(de\014ne)-66 b(d)494 b(as)j(in)2717 55665 y(\(2.9\))339 b(and)f(let)f Fm(T)10703 55864 y Fk(t)11468 55665 y Fw(=)368 b Fm(e)13451 55183 y Fi(\000)p Fk(tH)15419 55665 y Fm(;)443 b(t)369 b Fn(\025)g Fw(0)p Fg(.)555 b(Then)338 b Fn(f)p Fm(T)24755 55864 y Fk(t)25150 55665 y Fn(g)25814 55864 y Fk(t)p Fi(\025)p Fv(0)27750 55665 y Fg(is)f(a)i Fm(J)123 b Fg(-r)-66 b(e)g(al,)361 b(str)-66 b(ongly)338 b(c)-66 b(ontinuous,)362 b(symmetric)2717 57326 y(Markovian)464 b(semigr)-66 b(oup.)2717 61422 y(Pr)g(o)g(of:)898 b Fw(Clearly)596 b Fn(E)119 b Fw([)p Fn(\001)p Fw(])642 b Fn(\025)h Fw(0)p Fm(:)594 b Fw(Th)-36 b(us)594 b(Theorem)g(2.1)h(and)e(\(2.3\))i(imply)g(that)e Fn(f)p Fm(T)43084 61621 y Fk(t)43480 61422 y Fn(g)44144 61621 y Fk(t)p Fi(\025)p Fv(0)46335 61422 y Fw(is)i Fm(J)123 b Fw(-real,)2717 63082 y(strongly)512 b(con)-36 b(tin)g(uous,)529 b(sub-Mark)-36 b(o)g(vian.)810 b(It)510 b(follo)-36 b(ws)513 b(from)e(Theorem)g(2.1)h(\(a\))e(that)h Fm(T)46941 63281 y Fk(t)47336 63082 y Fm(\030)47904 63281 y Fv(0)48930 63082 y Fw(=)500 b Fm(\030)51010 63281 y Fv(0)2717 64742 y Fw(for)434 b(an)-36 b(y)434 b Fm(t)369 b Fn(\025)g Fw(0.)579 b(Th)-36 b(us)433 b Fn(f)p Fm(T)15713 64941 y Fk(t)16108 64742 y Fn(g)16772 64941 y Fk(t)p Fi(\025)p Fv(0)18803 64742 y Fw(is)h(Mark)-36 b(o)g(vian.)580 b Fe(\003)2717 69800 y Ff(Remark)499 b(2.1)650 b Fg(Consider)465 b(the)f(symmetric)g(emb)-66 b(e)g(dding:)22907 72789 y Fm(i)23351 72988 y Fv(0)24467 72789 y Fw(:)590 b Fn(M)812 b(\000)-221 b(!)812 b(H)24235 74781 y Fm(i)24679 74980 y Fv(0)25205 74781 y Fw(\()p Fm(A)p Fw(\))368 b(=)h(\001)30025 74233 y Fv(1)p Fk(=)p Fv(4)31492 74781 y Fm(A\030)33035 74980 y Fv(0)26801 78590 y Fw(7)p eop %%Page: 8 8 8 7 bop 2717 7306 a Fg(De\014ne)464 b(the)g(maps)i Fm(S)13166 7505 y Fk(t)14026 7306 y Fg(on)e Fn(M)h Fg(by)17929 10144 y Fm(S)18729 10343 y Fk(t)19493 10144 y Fw(:)369 b Fn(M)h(!)f(M)p Fm(;)1550 b(i)27835 10343 y Fv(0)28656 10144 y Fn(\016)295 b Fm(S)30415 10343 y Fk(t)31179 10144 y Fn(\021)369 b Fm(T)33343 10343 y Fk(t)34034 10144 y Fn(\016)295 b Fm(i)35437 10343 y Fv(0)35963 10144 y Fm(:)2717 12982 y Fg(It)450 b(fol)66 b(lows)451 b(fr)-66 b(om)449 b(The)-66 b(or)g(em)449 b(2.12)h(of)g([Cip1)o(])g(that)g Fn(f)p Fm(S)29205 13181 y Fk(t)29600 12982 y Fn(g)30264 13181 y Fk(t)p Fi(\025)p Fv(0)32312 12982 y Fg(is)g(a)g(we)-66 b(ak*)450 b(c)-66 b(ontinuous,)452 b(Markovian)2717 14643 y(semigr)-66 b(oup)333 b(on)h Fn(M)p Fg(.)554 b(Notic)-66 b(e)331 b(that)j Fm(S)20271 14842 y Fk(t)21000 14643 y Fg(pr)-66 b(eserves)333 b(p)-66 b(arity)333 b(:)532 b Fm(S)31817 14842 y Fk(t)32212 14643 y Fw(\()p Fm(\015)72 b Fw(\()p Fm(A)p Fw(\)\))368 b(=)h Fm(\015)72 b Fw(\()p Fm(S)39760 14842 y Fk(t)40154 14643 y Fw(\()p Fm(A)p Fw(\)\))p Fg(,)359 b(wher)-66 b(e)334 b Fm(A)369 b Fn(2)f(M)p Fg(.)2717 16303 y(This)465 b(c)-66 b(an)464 b(b)-66 b(e)464 b(che)-66 b(cke)g(d)463 b(fr)-66 b(om)464 b(the)h(fact)f Fm(U)23011 16502 y Fk(\015)23604 16303 y Fm(H)103 b(U)25807 15821 y Fi(\000)p Fv(1)25668 16632 y Fk(\015)27433 16303 y Fw(=)368 b Fm(H)567 b Fg(wher)-66 b(e)465 b Fm(H)567 b Fg(is)465 b(de\014ne)-66 b(d)464 b(in)g(The)-66 b(or)g(em)463 b(2.2.)4668 18941 y Fw(W)-108 b(e)548 b(no)-36 b(w)548 b(pro)36 b(duce)547 b(the)g(pro)36 b(of)548 b(of)g(Theorem)g(2.1.)922 b(Let)547 b(us)g(men)-36 b(tion)548 b(that)f(w)-36 b(e)548 b(mo)36 b(dify)549 b(the)2717 20602 y(pro)36 b(of)434 b(of)h(Theorem)e(3.1)i(of)f([P)-36 b(ar].)2717 23207 y Ff(Pro)42 b(of)519 b(of)g(Theorem)h(2.1)452 b Fw(\(a\))669 b(Notice)452 b(that)f Fm(U)27449 23406 y Fk(\015)28042 23207 y Fm(\030)28610 23406 y Fv(0)29536 23207 y Fw(=)400 b Fm(\030)31516 23406 y Fv(0)32041 23207 y Fw(,)457 b Fm(J)123 b(A\030)35244 23406 y Fv(0)36170 23207 y Fw(=)399 b(\001)38798 22355 y Fd(1)p 38798 22516 406 40 v 38798 23064 a(2)39392 23207 y Fm(A)40367 22725 y Fi(\003)40893 23207 y Fm(\030)41461 23406 y Fv(0)42439 23207 y Fw(for)452 b(an)-36 b(y)452 b Fm(A)400 b Fn(2)g(M)p Fm(:)2717 24868 y Fw(W)-108 b(e)434 b(obtain)f(that)8298 26630 y Fh(\000)8907 27706 y Fm(\033)9646 27912 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)12035 27706 y Fw(\()p Fm(x)p Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)17267 27912 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)19655 27706 y Fw(\()p Fm(x)20900 27157 y Fi(\003)21426 27706 y Fw(\)\))p Fm(U)23321 27905 y Fk(\015)23913 26630 y Fh(\001)24522 27706 y Fm(\030)25090 27905 y Fv(0)26723 27706 y Fw(=)1106 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Fk(t)p Fi(\000)p Fk(i=)p Fv(4)31063 42602 y Fw(\()p Fm(x)p Fw(\))p Fm(U)33836 42054 y Fi(\000)p Fv(1)33697 42931 y Fk(\015)35093 42602 y Fm(:)12758 b Fw(\(2.13\))2717 45440 y(Th)-36 b(us)433 b(a)h(direct)f(calculation)i(sho)-36 b(ws)434 b(that)f(for)h(an)-36 b(y)434 b Fm(\030)f Fn(2)369 b(H)13 b Fw(,)15578 48279 y Fn(k)16242 47203 y Fh(\000)16851 48279 y Fm(\033)17590 48485 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)19979 48279 y Fw(\()p Fm(x)p Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)25211 48485 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)27599 48279 y Fw(\()p Fm(x)28844 47730 y Fi(\003)29370 48279 y Fw(\)\))p Fm(U)31265 48478 y Fk(\015)31857 47203 y Fh(\001)32466 48279 y Fm(J)123 b(\030)64 b Fn(k)34604 47730 y Fv(2)16879 50271 y Fw(=)368 b Fn(k)p Fm(U)19806 50470 y Fk(\015)20399 50271 y Fm(J)21241 49195 y Fh(\000)22145 50271 y Fn(\000)295 b Fm(j)75 b Fw(\()p Fm(\033)25331 50478 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)27720 50271 y Fw(\()p Fm(x)p Fw(\)\))p Fm(U)30860 50470 y Fk(\015)31747 50271 y Fw(+)295 b Fm(\033)33793 50478 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)36181 50271 y Fw(\()p Fm(x)37426 49723 y Fi(\003)37952 50271 y Fw(\))38458 49195 y Fh(\001)39067 50271 y Fm(\030)64 b Fn(k)40363 49723 y Fv(2)16879 52264 y Fw(=)368 b Fn(k)18923 51188 y Fh(\000)19532 52264 y Fm(\033)20271 52470 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)22660 52264 y Fw(\()p Fm(x)23905 51715 y Fi(\003)24431 52264 y Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)28418 52470 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)30806 52264 y Fw(\()p Fm(x)p Fw(\)\))p Fm(U)33946 52463 y Fk(\015)34538 51188 y Fh(\001)35147 52264 y Fm(\030)64 b Fn(k)36443 51715 y Fv(2)36969 52264 y Fm(:)2717 55102 y Fw(Here)321 b(w)-36 b(e)320 b(ha)-36 b(v)g(e)321 b(used)f Fm(U)14241 54620 y Fi(\000)p Fv(1)14102 55430 y Fk(\015)15867 55102 y Fw(=)369 b Fm(U)18131 55301 y Fk(\015)19044 55102 y Fw(and)320 b Fm(U)22343 55301 y Fk(\015)23256 55102 y Fw(comm)-36 b(utes)320 b(with)h Fm(J)123 b Fw(.)540 b(Th)-36 b(us)320 b(w)-36 b(e)321 b(ha)-36 b(v)g(e)321 b Fn(E)42539 54620 y Fv(\(1\))43796 55102 y Fw([)p Fm(J)123 b(\030)64 b Fw(])370 b(=)e Fn(E)48562 54620 y Fv(\(2\))49820 55102 y Fw([)p Fm(\030)64 b Fw(])p Fm(:)2717 56762 y Fw(The)434 b(metho)36 b(d)433 b(used)g(in)g(the)g(ab)36 b(o)-36 b(v)g(e)434 b(also)h(implies)f(that)f Fn(E)31198 56280 y Fv(\(2\))32456 56762 y Fw([)p Fm(J)123 b(\030)64 b Fw(])369 b(=)g Fn(E)37222 56280 y Fv(\(1\))38479 56762 y Fw([)p Fm(\030)64 b Fw(])p Fm(:)4668 58423 y Fw(\(c\))433 b(By)i(the)e(expression)g(of)i Fn(E)119 b Fw(\()p Fm(\030)64 b(;)221 b(\021)48 b Fw(\))433 b(in)h(\(2.9\),)g Fn(E)119 b Fw(\()p Fm(\030)29768 58622 y Fv(+)30555 58423 y Fm(;)221 b(\030)31705 58622 y Fi(\000)32493 58423 y Fw(\))433 b(can)h(b)36 b(e)433 b(written)h(as)15361 61261 y Fn(E)119 b Fw(\()p Fm(\030)17255 61460 y Fv(+)18042 61261 y Fm(;)221 b(\030)19192 61460 y Fi(\000)19980 61261 y Fw(\))1107 b(=)f Fn(E)24531 60712 y Fv(\(1\))25789 61261 y Fw(\()p Fm(\030)26863 61460 y Fv(+)27650 61261 y Fm(;)221 b(\030)28800 61460 y Fi(\000)29588 61261 y Fw(\))295 b(+)g Fn(E)32516 60712 y Fv(\(2\))33773 61261 y Fw(\()p Fm(\030)34847 61460 y Fv(+)35634 61261 y Fm(;)221 b(\030)36784 61460 y Fi(\000)37572 61261 y Fw(\))21593 64360 y(=)1106 b(\(I)24686 63812 y Fv(\(1\))26239 64360 y Fw(+)295 b(I)36 b(I)28521 63812 y Fv(\(1\))29778 64360 y Fw(\))295 b(+)g(\(I)32861 63812 y Fv(\(2\))34414 64360 y Fw(+)728 b(I)36 b(I)37129 63812 y Fv(\(2\))38386 64360 y Fw(\))9320 b(\(2.14\))2717 67198 y(where)13534 71955 y(I)14003 71407 y Fv(\(1\))16368 71955 y Fw(=)18487 70147 y Fh(Z)20036 70481 y(\020)20830 71955 y Fn(h)p Fm(\033)22086 72162 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)24474 71955 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)26793 72154 y Fv(+)27580 71955 y Fm(;)221 b(\033)28901 72162 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)31290 71955 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)33609 72154 y Fi(\000)34396 71955 y Fn(i)19787 75070 y Fw(+)p Fn(h)p Fm(\033)22055 75276 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)24443 75070 y Fw(\()p Fm(x)25688 74521 y Fi(\003)26214 75070 y Fw(\))p Fm(\030)27288 75269 y Fi(\000)28075 75070 y Fm(;)g(\033)29396 75276 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)31785 75070 y Fw(\()p Fm(x)33030 74521 y Fi(\003)33556 75070 y Fw(\))p Fm(\030)34630 75269 y Fv(+)35417 75070 y Fn(i)35934 73595 y Fh(\021)36727 75070 y Fm(f)142 b Fw(\()p Fm(t)p Fw(\))221 b Fm(dt;)7492 b Fw(\(2.15\))26801 78590 y(8)p eop %%Page: 9 9 9 8 bop 11884 7786 a Fw(I)36 b(I)12858 7237 y Fv(\(1\))15223 7786 y Fw(=)1106 b Fn(\000)18595 5978 y Fh(Z)20146 6311 y(\020)20939 7786 y Fn(h)p Fm(\033)22195 7992 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)24583 7786 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)26902 7985 y Fv(+)27689 7786 y Fm(;)221 b(j)297 b Fw(\()p Fm(\033)30351 7992 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)32739 7786 y Fw(\()p Fm(x)33984 7237 y Fi(\003)34510 7786 y Fw(\)\))p Fm(U)36405 7985 y Fk(\015)36997 7786 y Fm(\030)37565 7985 y Fi(\000)38353 7786 y Fn(i)18642 10900 y Fw(+)p Fn(h)p Fm(j)e Fw(\()p Fm(\033)22249 11107 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)24638 10900 y Fw(\()p Fm(x)25883 10351 y Fi(\003)26409 10900 y Fw(\)\))p Fm(U)28304 11099 y Fk(\015)28896 10900 y Fm(\030)29464 11099 y Fv(+)30251 10900 y Fm(;)221 b(\033)31572 11107 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)33961 10900 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)36280 11099 y Fi(\000)37067 10900 y Fn(i)37584 9425 y Fh(\021)38377 10900 y Fm(f)142 b Fw(\()p Fm(t)p Fw(\))221 b Fm(dt:)2717 14288 y Fw(and)482 b(I)5764 13806 y Fv(\(2\))7503 14288 y Fw(and)g(I)36 b(I)11055 13806 y Fv(\(2\))12795 14288 y Fw(are)482 b(obtained)f(from)i(I)23995 13806 y Fv(\(1\))25734 14288 y Fw(and)f(I)36 b(I)29286 13806 y Fv(\(1\))31026 14288 y Fw(resp)g(ectiv)-36 b(ely)483 b(b)-36 b(y)481 b(replacing)i Fm(x)f Fw(b)-36 b(y)481 b Fm(x)49444 13806 y Fi(\003)50452 14288 y Fw(in)2717 15948 y(the)350 b(ab)36 b(o)-36 b(v)g(e.)552 b(Here)351 b(w)-36 b(e)351 b(ha)-36 b(v)g(e)351 b(used)f(the)g(facts)h(that)g Fm(U)28432 16147 y Fk(\015)29393 15948 y Fw(=)369 b Fm(U)31796 15466 y Fi(\000)p Fv(1)31657 16277 y Fk(\015)33054 15948 y Fm(;)351 b Fw(the)f(an)-36 b(tilinearit)g(y)352 b(of)f Fm(J)474 b Fw(and)350 b(\(2.13\))2717 17609 y(to)398 b(obtain)g(the)f(second)h(term)f(of)i(I)19635 17127 y Fv(\(1\))21291 17609 y Fw(in)e(\(2.15\).)568 b(As)398 b(a)g(consequence)g(of)g(Theorem)g(4\(7\))g(of)h([Ara],)2717 19269 y Fn(M)p Fm(\030)4880 19468 y Fv(+)6050 19269 y Fn(?)383 b(M)p Fm(\030)9629 19468 y Fi(\000)10416 19269 y Fw(,)444 b(whic)-36 b(h)441 b(implies)h(I)19842 18787 y Fv(\(1\))21482 19269 y Fw(=)382 b(0.)602 b(Similarly)443 b(I)30494 18787 y Fv(\(2\))32134 19269 y Fw(=)382 b(0.)603 b(See)441 b(also)h(pro)36 b(of)442 b(of)h(Prop)36 b(osition)2717 20930 y(5.3\(ii\))435 b(of)f([Cip1)q(].)4668 22590 y(Next,)637 b(w)-36 b(e)597 b(consider)e(I)36 b(I)16690 22108 y Fv(\(1\))17949 22590 y Fw(.)1065 b(Since)596 b Fm(\033)23673 22797 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)26061 22590 y Fw(\()p Fm(x)p Fw(\))27812 22108 y Fi(\003)28983 22590 y Fw(=)645 b Fm(\033)31379 22797 y Fk(t)p Fv(+)p Fk(i=)p Fv(4)33767 22590 y Fw(\()p Fm(x)35012 22108 y Fi(\003)35538 22590 y Fw(\),)637 b(it)596 b(follo)-36 b(ws)598 b(from)e(\(2.7\))h(and)2717 24250 y(\(2.13\))435 b(that)15036 27084 y Fn(h)p Fm(\033)16292 27291 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)18680 27084 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)20999 27283 y Fv(+)22007 27084 y Fm(;)444 b(j)296 b Fw(\()p Fm(\033)24891 27291 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)27279 27084 y Fw(\()p Fm(x)28524 26602 y Fi(\003)29050 27084 y Fw(\)\))p Fm(U)30945 27283 y Fk(\015)31537 27084 y Fm(\030)32105 27283 y Fi(\000)32892 27084 y Fn(i)18938 29796 y Fw(=)369 b Fn(h)221 b Fm(\030)21625 29995 y Fv(+)22633 29796 y Fm(;)443 b(\033)24176 30003 y Fk(t)p Fv(+)p Fk(i=)p Fv(4)26565 29796 y Fw(\()p Fm(x)27810 29314 y Fi(\003)28335 29796 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)30699 30003 y Fk(t)p Fv(+)p Fk(i=)p Fv(4)33087 29796 y Fw(\()p Fm(x)p Fw(\))34838 29314 y Fi(\003)35364 29796 y Fw(\))p Fm(U)36753 29995 y Fk(\015)37345 29796 y Fm(\030)37913 29995 y Fi(\000)38701 29796 y Fn(i)48212 28451 y Fw(\(2.16\))2717 32579 y(and)12790 36741 y Fn(h)p Fm(j)296 b Fw(\()p Fm(\033)15386 36947 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)17774 36741 y Fw(\()p Fm(x)19019 36259 y Fi(\003)19545 36741 y Fw(\)\))p Fm(U)21440 36940 y Fk(\015)22032 36741 y Fm(\030)22600 36940 y Fv(+)23609 36741 y Fm(;)443 b(\033)25152 36947 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)27540 36741 y Fw(\()p Fm(x)p Fw(\))p Fm(\030)29859 36940 y Fi(\000)30646 36741 y Fn(i)16692 39453 y Fw(=)369 b Fn(h\000)p Fm(U)20506 39652 y Fk(\015)21098 39453 y Fm(j)296 b Fw(\()p Fm(\033)23177 39659 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)25566 39453 y Fw(\()p Fm(x)26811 38971 y Fi(\003)27336 39453 y Fw(\)\))p Fm(\030)28916 39652 y Fv(+)29925 39453 y Fm(;)443 b Fn(\000)p Fm(U)32645 39652 y Fk(\015)33237 39453 y Fm(\033)33976 39659 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)36365 39453 y Fw(\()p Fm(x)p Fw(\))p Fm(U)38999 39652 y Fk(\015)39591 39453 y Fm(\030)40159 39652 y Fi(\000)40947 39453 y Fn(i)16692 42165 y Fw(=)369 b Fn(h)p Fm(\030)19158 42364 y Fv(+)20166 42165 y Fm(;)443 b(\033)21709 42372 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)24097 42165 y Fw(\()p Fm(x)p Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)27706 42372 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)30094 42165 y Fw(\()p Fm(x)31339 41683 y Fi(\003)31865 42165 y Fw(\))32371 41683 y Fi(\003)32897 42165 y Fw(\))p Fm(U)34286 42364 y Fk(\015)34878 42165 y Fm(\030)35446 42364 y Fi(\000)36233 42165 y Fn(i)p Fm(:)48212 39464 y Fw(\(2.17\))2717 45021 y(Substituting)432 b(\(2.16\))j(and)e(\(2.17\))h(in)-36 b(to)434 b(the)f(second)g(expression)h(of)g(\(2.15\),)h(w)-36 b(e)434 b(get)f(that)11882 48539 y(I)36 b(I)12857 47991 y Fv(\(1\))15221 48539 y Fw(=)1107 b Fn(\000)18594 46731 y Fh(Z)19923 48539 y Fn(h)221 b Fm(\030)21229 48738 y Fv(+)22016 48539 y Fm(;)g(\033)23337 48746 y Fk(t)p Fv(+)p Fk(i=)p Fv(4)25726 48539 y Fw(\()p Fm(x)26971 47991 y Fi(\003)27497 48539 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)29861 48746 y Fk(t)p Fv(+)p Fk(i=)p Fv(4)32249 48539 y Fw(\()p Fm(x)p Fw(\))34000 47991 y Fi(\003)34525 48539 y Fw(\))p Fm(U)35914 48738 y Fk(\015)36507 48539 y Fm(\030)37075 48738 y Fi(\000)37862 48539 y Fn(i)221 b Fm(f)142 b Fw(\()p Fm(t)p Fw(\))221 b Fm(dt)17340 51934 y Fn(\000)18594 50126 y Fh(Z)19923 51934 y Fn(h)p Fm(\030)21008 52133 y Fv(+)21795 51934 y Fm(;)g(\033)23116 52141 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)25505 51934 y Fw(\()p Fm(x)p Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)29114 52141 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)31502 51934 y Fw(\()p Fm(x)32747 51385 y Fi(\003)33272 51934 y Fw(\))33778 51385 y Fi(\003)34304 51934 y Fw(\))p Fm(U)35693 52133 y Fk(\015)36285 51934 y Fm(\030)36853 52133 y Fi(\000)37862 51934 y Fn(i)p Fm(f)142 b Fw(\()p Fm(t)p Fw(\))221 b Fm(dt:)2717 55378 y Fw(Notice)434 b(that)f(for)i(an)-36 b(y)433 b Fm(x)369 b Fn(2)g(M)18019 55577 y Fk(an)19579 55378 y Fw(and)433 b Fm(\030)g Fn(2)369 b(H)13 b Fw(,)433 b(the)g(map)22122 58306 y Fm(z)c Fn(7!)369 b Fm(j)75 b Fw(\()p Fm(\033)26710 58505 y Fk(z)27241 58306 y Fw(\()p Fm(x)p Fw(\))28992 57758 y Fi(\003)29517 58306 y Fw(\))p Fm(U)30906 58505 y Fk(\015)31499 58306 y Fm(\030)2717 61234 y Fw(is)296 b(analytic)i(on)d(a)i(domain)f(con)-36 b(taining)296 b(the)f(strip)g Fm(I)27567 61441 y Fv(1)p Fk(=)p Fv(2)29034 61234 y Fw(.)532 b(In)296 b(fact,)324 b(the)296 b(analyticit)-36 b(y)297 b(follo)-36 b(ws)298 b(from)f(the)2717 62895 y(facts)498 b(that)f Fn(h)p Fm(\021)48 b(;)221 b(j)11143 61819 y Fh(\000)11752 62895 y Fm(\033)12491 63094 y Fk(z)13021 62895 y Fw(\()p Fm(x)p Fw(\))14772 62413 y Fi(\003)15298 61819 y Fh(\001)15907 62895 y Fm(U)16790 63094 y Fk(\015)17382 62895 y Fm(\030)64 b Fn(i)478 b Fw(=)f Fn(h)p Fm(\033)21754 63094 y Fk(z)22284 62895 y Fw(\()p Fm(x)p Fw(\))24035 62413 y Fi(\003)24561 62895 y Fm(J)123 b(U)26286 63094 y Fk(\015)26879 62895 y Fm(\030)64 b(;)221 b(J)123 b(\021)48 b Fn(i)477 b Fw(=)g Fn(h)p Fm(J)123 b(U)34350 63094 y Fk(\015)34942 62895 y Fm(\030)64 b(;)221 b(\033)36895 63094 y Fk(z)37427 62895 y Fw(\()p Fm(x)p Fw(\))p Fm(J)123 b(\021)48 b Fn(i)496 b Fw(for)i(an)-36 b(y)498 b Fm(\021)48 b(;)221 b(\030)542 b Fn(2)477 b(H)13 b Fw(,)2717 64555 y(and)433 b(that)g(w)-36 b(eak)435 b(analyticit)-36 b(y)435 b(implies)f(strong)g(analyticit)-36 b(y)435 b([RS,)f(Theorem)g(VI.4].) 4668 66216 y(Using)499 b(the)f(Cauc)-36 b(h)g(y)498 b(in)-36 b(tegral)499 b(theorem,)514 b(the)498 b(assumption)g(in)g(the)g (theorem,)515 b(the)498 b(prop)36 b(ert)-36 b(y)2717 67876 y(\(c\))433 b(in)h(De\014nition)f(2.1)i(and)e Fm(\033)17673 68075 y Fk(t)18068 67876 y Fw(\()p Fm(x)p Fw(\))19819 67394 y Fi(\003)20714 67876 y Fw(=)368 b Fm(\033)22833 68075 y Fk(t)23229 67876 y Fw(\()p Fm(x)24474 67394 y Fi(\003)25000 67876 y Fw(\),)433 b(w)-36 b(e)434 b(obtain)g(that)12298 71395 y(I)36 b(I)13273 70846 y Fv(\(1\))14899 71395 y Fw(=)2214 b Fn(\000)19379 69586 y Fh(Z)20708 71395 y Fn(h)p Fm(\030)21793 71594 y Fv(+)22580 71395 y Fm(;)221 b(\033)23901 71594 y Fk(t)24297 71395 y Fw(\()p Fm(x)25542 70846 y Fi(\003)26067 71395 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)28431 71594 y Fk(t)28826 71395 y Fw(\()p Fm(x)30071 70846 y Fi(\003)30597 71395 y Fw(\)\))p Fm(U)32492 71594 y Fk(\015)33084 71395 y Fm(\030)33652 71594 y Fi(\000)34440 71395 y Fn(i)p Fm(f)142 b Fw(\()p Fm(t)294 b Fn(\000)h Fm(i=)p Fw(4\))221 b Fm(dt)18125 74789 y Fn(\000)19379 72981 y Fh(Z)20708 74789 y Fn(h)p Fm(\030)21793 74988 y Fv(+)22580 74789 y Fm(;)g(\033)23901 74988 y Fk(t)24297 74789 y Fw(\()p Fm(x)p Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)27906 74988 y Fk(t)28300 74789 y Fw(\()p Fm(x)p Fw(\)\))p Fm(U)31440 74988 y Fk(\015)32033 74789 y Fm(\030)32601 74988 y Fi(\000)33388 74789 y Fn(i)p Fm(f)142 b Fw(\()p Fm(t)294 b Fw(+)h Fm(i=)p Fw(4\))221 b Fm(dt:)26801 78590 y Fw(9)p eop %%Page: 10 10 10 9 bop 2717 7306 a Fw(Replacing)435 b Fm(x)e Fw(b)-36 b(y)434 b Fm(x)12451 6824 y Fi(\003)13410 7306 y Fw(in)g(the)f(ab)36 b(o)-36 b(v)g(e,)434 b(w)-36 b(e)434 b(obtain)g(the)f(expression)g(of)i (I)36 b(I)38136 6824 y Fv(\(2\))39394 7306 y Fw(.)578 b(Th)-36 b(us)433 b(w)-36 b(e)434 b(get)11751 10735 y(I)36 b(I)1108 b(=)e Fn(\000)17205 8927 y Fh(Z)18534 10735 y Fn(h)p Fm(\030)19619 10934 y Fv(+)20406 10735 y Fm(;)221 b Fw([)p Fm(\033)22088 10934 y Fk(t)22485 10735 y Fw(\()p Fm(x)p Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)26094 10934 y Fk(t)26488 10735 y Fw(\()p Fm(x)p Fw(\)\))295 b(+)g Fm(\033)31086 10934 y Fk(t)31481 10735 y Fw(\()p Fm(x)32726 10187 y Fi(\003)33252 10735 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)35616 10934 y Fk(t)36011 10735 y Fw(\()p Fm(x)37256 10187 y Fi(\003)37782 10735 y Fw(\)\)])p Fm(U)40038 10934 y Fk(\015)40630 10735 y Fm(\030)41198 10934 y Fi(\000)41985 10735 y Fn(i)17252 13318 y(\001)p Fw(\()p Fm(f)142 b Fw(\()p Fm(t)294 b Fn(\000)i Fm(i=)p Fw(4\))f(+)g Fm(f)142 b Fw(\()p Fm(t)295 b Fw(+)f Fm(i=)p Fw(4\)\))221 b Fm(dt:)2717 16231 y Fw(Recall)290 b(that)f Fm(U)10049 16430 y Fk(\015)10642 16231 y Fn(P)478 b(\032)369 b(P)398 b Fw(in)289 b(\(2.7\).)531 b(Since)289 b Fm(\033)23697 16430 y Fk(t)24092 16231 y Fw(\()p Fm(x)25337 15749 y Fv(#)26176 16231 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)28540 16430 y Fk(t)28935 16231 y Fw(\()p Fm(x)30180 15749 y Fv(#)31020 16231 y Fw(\)\))p Fm(U)32915 16430 y Fk(\015)33507 16231 y Fm(\030)34075 16430 y Fi(\000)35231 16231 y Fn(2)369 b(P)109 b Fw(,)752 b Fn(h)p Fm(\030)39717 16430 y Fv(+)40504 16231 y Fm(;)221 b(\033)41825 16430 y Fk(t)42221 16231 y Fw(\()p Fm(x)43466 15749 y Fv(#)44305 16231 y Fw(\))p Fm(j)75 b Fw(\()p Fm(\033)46669 16430 y Fk(t)47064 16231 y Fw(\()p Fm(x)48309 15749 y Fv(#)49148 16231 y Fw(\)\))p Fm(U)51043 16430 y Fk(\015)51635 16231 y Fm(\030)52203 16430 y Fi(\000)52991 16231 y Fn(i)2717 17891 y(\025)407 b Fw(0)456 b(for)g Fn(8)221 b Fm(t)407 b Fn(2)f Fo(R)p Fw(,)461 b(where)456 b Fm(x)16700 17409 y Fv(#)17995 17891 y Fw(is)g(either)f Fm(x)h Fw(or)g Fm(x)26578 17409 y Fi(\003)27104 17891 y Fw(.)644 b(By)456 b(the)f(prop)36 b(ert)-36 b(y)456 b(\(b\))e(in)i(De\014nition)f(2.1,)463 b(w)-36 b(e)2717 19552 y(conclude)433 b(that)g(I)36 b(I)435 b Fn(\024)369 b Fw(0.)578 b(This)434 b(pro)-36 b(v)g(ed)433 b(the)g(part)h(\(c\))f (of)h(the)f(theorem.)4668 21212 y(Clearly)296 b Fn(E)119 b Fw([)p Fn(\001)p Fw(])370 b Fn(\025)f Fw(0)p Fm(:)295 b Fw(Note)g(that)g Fn(E)119 b Fw(\()p Fm(\030)64 b(;)221 b(\030)22826 21411 y Fv(0)23352 21212 y Fw(\))369 b(=)g(0)295 b(for)g(all)g Fm(\030)434 b Fn(2)368 b(H)13 b Fw(.)532 b(Theorem)295 b(2.1)g(\(b\),\(c\))f(imply)i(that)2717 22872 y(the)433 b(form)h(\()p Fn(E)119 b Fm(;)221 b Fn(H)13 b Fw(\))434 b(satis\014es)f(the)g(conditions)h(in)f(\(2.2\).)579 b(Th)-36 b(us)433 b(\()p Fn(E)119 b Fm(;)221 b Fn(H)13 b Fw(\))434 b(is)g(a)g(Diric)-36 b(hlet)434 b(form.)1301 b Fe(\003)2717 27298 y Fp(3)2152 b(Mark)-60 b(o)g(vian)478 b(semigroups)e(on)i(CAR)g(algebras)g(with)f(quasi-)5945 29734 y(free)717 b(states)2717 32709 y Fw(In)615 b(this)g(section,)660 b(w)-36 b(e)616 b(apply)f(the)f(results)h(in)g(Section)f(2)i(to)f (construct)f(Diric)-36 b(hlet)615 b(forms)h(and)2717 34370 y(asso)36 b(ciated)435 b(Mark)-36 b(o)g(vian)435 b(semigroups)e(on)h(CAR)g(algebras)g(with)g(resp)36 b(ect)433 b(to)h(quasi-free)g(states.)4668 36030 y(W)-108 b(e)553 b(\014rst)e(review)j(the)e(notion)g(of)i(CAR)e(algebras.)936 b(F)-108 b(or)552 b(the)g(details)h(w)-36 b(e)553 b(refer)g(to)f (Section)2717 37691 y(5.2.2)j(of)f([BR)q(].)938 b(Let)553 b Fl(h)14587 37890 y Fv(0)15667 37691 y Fw(b)36 b(e)553 b(a)h(separable)f(pre-Hilb)36 b(ert)553 b(space)g(with)h(inner)e(pro)36 b(duct)553 b(\()p Fn(\001)p Fm(;)221 b Fn(\001)p Fw(\))554 b(and)2717 39351 y Fl(h)495 b Fw(the)f(completion)h(of)g Fl(h)15148 39550 y Fv(0)15674 39351 y Fw(.)762 b(Let)494 b Fn(A)p Fw(\()p Fl(h)p Fw(\))g(b)36 b(e)494 b(the)g Fm(C)27605 38869 y Fi(\003)28131 39351 y Fw(-algebra)h(generated)f(b) -36 b(y)495 b(the)f(iden)-36 b(tit)g(y)494 b Ff(1)h Fw(and)2717 41012 y(elemen)-36 b(ts)433 b Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fm(;)443 b(f)511 b Fn(2)590 b Fl(h)p Fw(,)434 b(satisfying)20756 43924 y(\()p Fm(a)p Fw(\))221 b Fm(f)511 b Fn(7!)369 b Fm(a)p Fw(\()p Fm(f)142 b Fw(\))221 b(is)651 b(an)-36 b(tilinear)p Fm(;)13373 b Fw(\(3.1\))20756 45917 y(\()p Fm(b)p Fw(\))221 b Fn(f)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(g)48 b Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)20756 47909 y Fw(\()p Fm(c)p Fw(\))221 b Fn(f)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(g)48 b Fw(\))28639 47361 y Fi(\003)29164 47909 y Fn(g)369 b Fw(=)g(\()p Fm(f)70 b(;)221 b(g)48 b Fw(\))p Ff(1)2717 50822 y Fw(for)434 b(all)h Fm(f)70 b(;)443 b(g)416 b Fn(2)369 b Fl(h)p Fm(;)434 b Fw(where)f Fn(f)p Fm(A;)221 b(B)67 b Fn(g)370 b Fw(:=)f Fm(AB)363 b Fw(+)295 b Fm(B)67 b(A:)434 b Fw(Notice)g(that)20641 53735 y Fn(k)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fn(k)368 b Fw(=)h Fn(k)p Fm(f)142 b Fn(k)886 b(8)p Fm(f)511 b Fn(2)368 b Fl(h)p Fm(;)15472 b Fw(\(3.2\))2717 56647 y(and)433 b(so)h Fn(A)p Fw(\()p Fl(h)9102 56846 y Fv(0)9628 56647 y Fw(\))369 b(=)f Fn(A)p Fw(\()p Fl(h)p Fw(\).)578 b(Let)433 b Fm(\015)663 b Fw(:)590 b Fn(A)p Fw(\()p Fl(h)p Fw(\))369 b Fn(!)g(A)p Fw(\()p Fl(h)p Fw(\))434 b(b)36 b(e)433 b(the)g Fn(\003)p Fw(-automorphism)g(de\014ned)f(b)-36 b(y)22087 59560 y Fm(\015)72 b Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\)\))368 b(=)h Fn(\000)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fm(;)16917 b Fw(\(3.3\))2717 62472 y(for)434 b(all)h Fm(f)511 b Fn(2)368 b Fl(h)p Fm(:)435 b Fw(Then)e(\()p Fn(A)p Fw(\()p Fl(h)p Fw(\))p Fm(;)221 b(\015)72 b Fw(\))433 b(is)h(a)g Fo(Z)22618 62671 y Fv(2)23143 62472 y Fw(-graded)f Fm(C)28866 61990 y Fi(\003)29392 62472 y Fw(-algebra.)4668 64133 y(Next,)412 b(w)-36 b(e)405 b(describ)36 b(e)405 b(quasi-free)g(states) h(on)f Fn(A)p Fw(\()p Fl(h)p Fw(\).)568 b(Let)405 b Fm(A)g Fw(b)36 b(e)405 b(a)g(b)36 b(ounded)404 b(and)g(non-negativ)-36 b(e)2717 65793 y(op)36 b(erator)471 b(on)f Fl(h)p Fw(.)688 b(Recall)472 b(that)e(a)g(v)-36 b(ector)471 b Fm(g)479 b Fn(2)431 b Fl(h)470 b Fw(is)h(called)g(an)f(analytic)h(v)-36 b(ector)471 b(for)g(an)f(op)36 b(erator)2717 67454 y Fm(B)501 b Fw(on)433 b Fl(h)h Fw(if)h(for)f(eac)-36 b(h)433 b Fm(n)370 b Fn(2)e Fo(N)p Fm(;)434 b(g)417 b Fn(2)368 b Fm(D)36 b Fw(\()p Fm(B)22369 66972 y Fk(n)22996 67454 y Fw(\))433 b(and)g(for)h(some)g Fm(t)369 b(>)f Fw(0)p Fm(;)22305 69663 y Fi(1)21816 70061 y Fh(X)21889 72851 y Fk(n)p Fv(=0)24089 70425 y Fn(k)p Fm(B)25808 69942 y Fk(n)26435 70425 y Fm(g)48 b Fn(k)p 24089 71018 3681 54 v 25360 72234 a Fm(n)p Fw(!)27902 71323 y Fm(t)28372 70775 y Fk(n)29367 71323 y Fm(<)369 b Fn(1)p Fm(:)2717 75269 y Fw(In)434 b(the)f(rest)g(of)h(this)f(pap)36 b(er,)434 b(w)-36 b(e)434 b(assume)f(that)g Fm(A)h Fw(satis\014es)g(the)f(follo) -36 b(wing)435 b(prop)36 b(erties:)26476 78590 y(10)p eop %%Page: 11 11 11 10 bop 2717 7306 a Ff(Assumption)501 b(3.1)651 b Fg(\(a\))466 b(Ther)-66 b(e)464 b(exists)g Fm(\013)378 b(>)369 b Fw(0)465 b Fg(such)g(that)g Fw(0)369 b Fm(<)g(A)g Fn(\024)g Fm(\013)8 b Ff(1)q Fm(:)2717 8966 y Fg(\(b\))442 b(The)f(inverse)e Fm(A)12869 8484 y Fi(\000)p Fv(1)14568 8966 y Fg(of)i Fm(A)g Fg(exists)g(as)g(a)h(\(unb)-66 b(ounde)g(d\))441 b(self-adjoint)e(and)i(p)-66 b(ositive)440 b(op)-66 b(er)g(ator)440 b(on)2717 10627 y Fl(h)p Fg(.)2717 12287 y(\(c\))574 b(F)-100 b(or)574 b(any)e Fm(z)629 b Fn(2)569 b Fo(C)p Fm(;)574 b(A)15640 11805 y Fk(z)16744 12287 y Fg(le)-66 b(aves)572 b Fl(h)21263 12486 y Fv(0)22363 12287 y Fg(invariant,)597 b(i.e.,)h Fm(A)32292 11805 y Fk(z)32823 12287 y Fl(h)33515 12486 y Fv(0)34611 12287 y Fn(\032)569 b Fl(h)36905 12486 y Fv(0)37431 12287 y Fm(:)574 b Fg(Mor)-66 b(e)g(over,)598 b Fm(z)629 b Fn(7!)570 b Fm(A)48720 11805 y Fk(z)49250 12287 y Fm(f)715 b Fg(is)2717 13948 y(entir)-66 b(e)463 b(analytic)h(for)h(any)f Fm(f)511 b Fn(2)369 b Fl(h)19000 14147 y Fv(0)19526 13948 y Fg(.)2717 15608 y(\(d\))466 b(A)-33 b(ny)464 b Fm(g)416 b Fn(2)369 b Fl(h)10654 15807 y Fv(0)11645 15608 y Fg(is)465 b(an)f(analytic)g(ve)-66 b(ctor)464 b(for)h Fm(A)26701 15126 y Fi(\000)p Fv(1)p Fk(=)p Fv(2)28899 15608 y Fg(.)2717 18376 y Fw(W)-108 b(e)626 b(remark)h(that)e(a)h(dense)g(subspace)f Fl(h)24131 18575 y Fv(0)25283 18376 y Fw(of)i Fl(h)f Fw(satisfying)i(Assumption)d (3.1)i(exists)g(b)-36 b(y)626 b(the)2717 20036 y(sp)36 b(ectral)434 b(theorem.)2717 23135 y Ff(Example)499 b(3.1)f(\(Ideal)h (F)-125 b(ermi)500 b(gases\))652 b Fg(L)-66 b(et)511 b Fl(h)i Fg(b)-66 b(e)512 b(the)h(sp)-66 b(ac)g(e)512 b Fm(L)36077 22653 y Fv(2)36602 23135 y Fw(\()p Fo(R)38067 22653 y Fk(d)38607 23135 y Fm(;)221 b(dx)p Fw(\))513 b Fg(and)g Fw(\001)g Fg(the)f(L)-66 b(apla-)2717 24796 y(cian)464 b(op)-66 b(er)g(ator)464 b(on)h Fl(h)p Fg(.)598 b(L)-66 b(et)464 b Fm(A)h Fg(b)-66 b(e)464 b(given)f(by)22567 27785 y Fm(A)369 b Fw(=)g Fm(e)25895 27236 y Fi(\000)p Fk(\014)46 b Fv(\()p Fi(\000)p Fv(\001)p Fi(\000)p Fk(\026)p Fc(1)o Fv(\))31325 27785 y Fm(;)2717 30773 y Fg(wher)-66 b(e)489 b Fm(\014)g(>)414 b Fw(0)p Fm(;)221 b(\026)415 b Fn(2)f Fo(R)p Fg(.)671 b(The)490 b(subsp)-66 b(ac)g(e)489 b Fl(h)23431 30972 y Fv(0)24447 30773 y Fg(is)g(given)f(by)h Fl(h)31688 30972 y Fv(0)32629 30773 y Fw(=)414 b Fn(f)p Fm(f)557 b Fn(2)414 b Fl(h)g Fw(:)39338 30423 y Fh(b)39099 30773 y Fm(f)557 b Fn(2)414 b Fm(C)42528 30972 y Fk(c)42990 30773 y Fw(\()p Fo(R)44455 30291 y Fk(d)44995 30773 y Fw(\))p Fn(g)p Fg(,)495 b(wher)-66 b(e)50992 30423 y Fh(b)50753 30773 y Fm(f)2717 32434 y Fg(denotes)465 b(the)f(F)-100 b(ourier)465 b(tr)-66 b(ansform)464 b(of)h Fm(f)142 b Fg(.)597 b(Then)464 b(Assumption)h(3.1)f(satis\014e)-66 b(d)465 b(with)f Fm(\013)378 b Fw(=)369 b Fm(e)47191 31952 y Fk(\014)46 b(\026)48385 32434 y Fg(.)4668 35533 y Fw(F)-108 b(or)420 b(giv)-36 b(en)422 b(b)36 b(ounded)419 b(op)36 b(erator)421 b Fm(A)f Fw(on)h Fl(h)g Fw(satisfying)h (Assumption)e(3.1,)425 b(the)420 b(gauge)h(in)-36 b(v)-72 b(arian)-36 b(t)2717 37194 y(quasi-free)434 b(state)g Fm(!)481 b Fw(on)434 b Fn(A)p Fw(\()p Fl(h)p Fw(\))f(is)h(de\014ned)e (b)-36 b(y)8886 40183 y Fm(!)48 b Fw(\()p Fm(a)p Fw(\()p Fm(f)12080 40382 y Fk(m)12967 40183 y Fw(\))13473 39634 y Fi(\003)14294 40183 y Fn(\001)295 b(\001)g(\001)h Fm(a)p Fw(\()p Fm(f)18117 40382 y Fv(1)18642 40183 y Fw(\))19148 39634 y Fi(\003)19674 40183 y Fm(a)p Fw(\()p Fm(g)21486 40382 y Fv(1)22011 40183 y Fw(\))f Fn(\001)g(\001)g(\001)g Fm(a)p Fw(\()p Fm(g)26616 40382 y Fk(n)27242 40183 y Fw(\)\))369 b(=)f Fm(\016)30579 40382 y Fk(nm)32259 40183 y Fw(det\(\()p Fm(g)35701 40382 y Fk(i)36076 40183 y Fm(;)221 b(A)p Fw(\()p Ff(1)296 b Fw(+)f Fm(A)p Fw(\))41970 39634 y Fi(\000)p Fv(1)43227 40183 y Fm(f)43868 40382 y Fk(j)44355 40183 y Fw(\)\))3495 b(\(3.4\))2717 43171 y(for)372 b(an)-36 b(y)372 b Fm(f)7678 43370 y Fv(1)8204 43171 y Fm(;)221 b Fn(\001)168 b(\001)g(\001)p Fm(;)221 b(f)11452 43370 y Fk(m)12342 43171 y Fm(;)g(g)13547 43370 y Fv(1)14073 43171 y Fm(;)g Fn(\001)168 b(\001)g(\001)p Fm(;)221 b(g)17303 43370 y Fk(n)18300 43171 y Fn(2)369 b Fl(h)p Fm(:)j Fw(Let)f Fm(\033)23987 43370 y Fk(t)24751 43171 y Fw(:)e Fn(A)p Fw(\()p Fl(h)p Fw(\))g Fn(!)g(A)p Fw(\()p Fl(h)p Fw(\))j(b)36 b(e)371 b(the)g Fn(\003)p Fw(-automorphisms)g(de\014ned)2717 44832 y(b)-36 b(y)21856 47821 y Fm(\033)22595 48020 y Fk(t)22991 47821 y Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\)\))368 b(=)g Fm(a)p Fw(\()p Fm(A)30393 47272 y Fk(it)31108 47821 y Fm(f)142 b Fw(\))16465 b(\(3.5\))2717 50809 y(for)434 b(an)-36 b(y)434 b Fm(f)511 b Fn(2)369 b Fl(h)p Fm(;)443 b(t)369 b Fn(2)f Fo(R)p Fm(:)434 b Fw(Then)g(one)f(can)h(c)-36 b(hec)g(k)433 b(KMS)g(conditions)20872 53798 y Fm(!)48 b Fw(\()p Fm(B)67 b(\033)24030 53997 y Fi(\000)p Fk(i)25138 53798 y Fw(\()p Fm(C)95 b Fw(\)\))368 b(=)h Fm(!)48 b Fw(\()p Fm(C)95 b(B)67 b Fw(\))2717 56787 y(for)434 b(an)-36 b(y)434 b(analytic)h(elemen)-36 b(ts)433 b Fm(B)67 b(;)221 b(C)465 b Fn(2)368 b(A)p Fw(\()p Fl(h)p Fw(\)[BR].)4668 58447 y(Let)349 b(\()p Fn(H)8542 58646 y Fk(!)9213 58447 y Fm(;)221 b(\031)10533 58646 y Fk(!)11206 58447 y Fm(;)g Fw(\012)12727 58646 y Fk(!)13400 58447 y Fw(\))349 b(b)36 b(e)349 b(the)f(GNS)g(represen)-36 b(tation)348 b(of)i(\()p Fn(A)p Fw(\()p Fl(h)p Fw(\))p Fm(;)221 b(!)48 b Fw(\).)550 b(De\014ne)348 b Fn(M)42639 58646 y Fk(!)43681 58447 y Fw(:=)369 b Fm(\031)46161 58646 y Fk(!)46833 58447 y Fw(\()p Fn(A)p Fw(\()p Fl(h)p Fw(\)\))50610 57965 y Fi(0)o(0)51175 58447 y Fw(,)2717 60108 y Fm(a)p Fw(\()p Fm(f)142 b Fw(\))5195 60307 y Fk(!)6535 60108 y Fw(:=)669 b Fm(\031)9315 60307 y Fk(!)9987 60108 y Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\)\),)217 b Fm(a)p Fw(\()p Fm(f)142 b Fw(\))16533 59626 y Fi(\003)16533 60436 y Fk(!)17872 60108 y Fw(:=)669 b Fm(\031)20652 60307 y Fk(!)21324 60108 y Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))24308 59626 y Fi(\003)24833 60108 y Fw(\),)217 b Fm(f)810 b Fn(2)668 b Fl(h)p Fw(,)871 b Fm(\033)31633 59626 y Fk(!)31585 60436 y(t)32974 60108 y Fw(:=)669 b Fm(\031)35754 60307 y Fk(!)36426 60108 y Fm(\033)37165 60307 y Fk(t)37560 60108 y Fm(\031)38346 59626 y Fi(\000)p Fv(1)38298 60436 y Fk(!)40213 60108 y Fw(and)609 b Fm(\015)43593 60307 y Fk(!)44933 60108 y Fw(:=)669 b Fm(\031)47713 60307 y Fk(!)48385 60108 y Fm(\015)72 b(\031)49918 59626 y Fi(\000)p Fv(1)49870 60436 y Fk(!)51175 60108 y Fm(:)2717 61768 y Fw(F)-108 b(rom)493 b(no)-36 b(w)493 b(on)h(w)-36 b(e)493 b(suppress)f Fm(!)541 b Fw(from)494 b(the)f(notations,)509 b(e.g.,)g Fn(M)471 b Fw(:=)g Fn(M)39344 61967 y Fk(!)40016 61768 y Fw(,)726 b Fn(H)483 b Fw(:=)471 b Fn(H)45674 61967 y Fk(!)46346 61768 y Fw(,)508 b Fm(a)p Fw(\()p Fm(f)142 b Fw(\))470 b(:=)2717 63429 y Fm(a)3400 63628 y Fk(!)4072 63429 y Fw(\()p Fm(f)142 b Fw(\),)217 b Fm(a)p Fw(\()p Fm(f)142 b Fw(\))8923 62947 y Fi(\003)9909 63429 y Fw(:=)462 b Fm(a)12427 63628 y Fk(!)13099 63429 y Fw(\()p Fm(f)142 b Fw(\))14894 62947 y Fi(\003)15419 63429 y Fw(,)718 b Fm(\033)17237 63628 y Fk(t)18094 63429 y Fw(:=)461 b Fm(\033)20715 62947 y Fk(!)20667 63757 y(t)21875 63429 y Fw(and)488 b Fm(\015)533 b Fw(:=)461 b Fm(\015)28176 63628 y Fk(!)29336 63429 y Fw(etc.)741 b(By)488 b(con)-36 b(tin)g(uit)g(y)-108 b(,)501 b Fm(\033)41397 63628 y Fk(t)42280 63429 y Fw(and)488 b Fm(\015)559 b Fw(extend)488 b(to)2717 65089 y Fn(M)p Fm(:)435 b Fw(W)-108 b(e)435 b(also)h(write)f Fm(\030)13852 65288 y Fv(0)14748 65089 y Fw(=)371 b(\012)17070 65288 y Fk(!)17742 65089 y Fm(:)435 b Fw(Notice)g(that)g Fm(!)482 b Fw(satis\014es)435 b(the)f Fm(\033)34505 65288 y Fk(t)34900 65089 y Fw(-KMS)g(conditions)h([BR].)g (Th)-36 b(us,)2717 66750 y(b)g(y)603 b(the)g(uniqueness)f(of)i(the)f (mo)36 b(dular)603 b(automorphism,)646 b Fm(\033)33142 66949 y Fk(t)34140 66750 y Fw(is)604 b(the)e(mo)36 b(dular)604 b(automorphism)2717 68410 y(on)519 b Fn(M)p Fw(.)836 b(See)519 b(Theorem)g(5.3.10)i(of)f([BR].)g(Let)f(\001)g(and)g Fm(J)642 b Fw(denote)518 b(the)h(mo)36 b(dular)519 b(op)36 b(erator)520 b(and)2717 70071 y(the)433 b(mo)36 b(dular)433 b(conjugation)h(asso)36 b(ciated)434 b(with)g(the)e(pair)i(\()p Fn(M)p Fm(;)443 b(\030)34709 70270 y Fv(0)35235 70071 y Fw(\))433 b(resp)36 b(ectiv)-36 b(ely)-108 b(.)579 b(Th)-36 b(us)432 b Fm(\033)47693 70270 y Fk(t)48089 70071 y Fw(\()p Fm(B)67 b Fw(\))368 b(=)2717 71731 y(\001)3801 71249 y Fk(it)4517 71731 y Fm(B)67 b Fw(\001)6656 71249 y Fi(\000)p Fk(it)8103 71731 y Fm(;)221 b Fn(8)p Fm(B)437 b Fn(2)368 b(M)p Fw(.)579 b(It)433 b(follo)-36 b(ws)436 b(from)e(\(3.3\))g(and)f(\(3.4\))h(that)19138 74720 y Fm(!)48 b Fw(\()p Fm(\015)72 b Fw(\()p Fm(B)67 b Fw(\)\))368 b(=)h Fm(!)48 b Fw(\()p Fm(B)67 b Fw(\))p Fm(;)885 b Fn(8)p Fm(B)436 b Fn(2)369 b(M)p Fm(:)26476 78590 y Fw(11)p eop %%Page: 12 12 12 11 bop 2717 7306 a Fw(Since)365 b Fm(!)414 b Fw(is)366 b Fm(\015)72 b Fw(-in)-36 b(v)-72 b(arian)-36 b(t,)379 b(that)365 b(is,)380 b Fn(h)p Fm(\030)20902 7505 y Fv(0)21428 7306 y Fm(;)221 b(\015)72 b Fw(\()p Fm(B)67 b Fw(\))p Fm(\030)25392 7505 y Fv(0)25918 7306 y Fn(i)368 b Fw(=)h Fn(h)p Fm(\030)29269 7505 y Fv(0)29794 7306 y Fm(;)221 b(B)67 b(\030)31999 7505 y Fv(0)32526 7306 y Fn(i)p Fm(;)443 b Fn(8)p Fm(B)436 b Fn(2)368 b(M)p Fm(;)f Fw(there)e(exists)h(a)g (unitary)2717 8966 y(op)36 b(erator)434 b Fm(U)8839 9165 y Fk(\015)9865 8966 y Fw(on)g Fn(H)446 b Fw(satisfying)435 b(the)e(prop)36 b(erties)433 b(in)h(\(2.7\).)4668 10627 y(F)-108 b(or)433 b(an)-36 b(y)434 b Fm(g)417 b Fn(2)368 b Fl(h)p Fm(;)434 b Fw(de\014ne)f(an)g(o)36 b(dd)434 b(elemen)-36 b(t)433 b Fm(B)67 b Fw(\()p Fm(g)48 b Fw(\))368 b Fn(2)h(M)32127 10826 y Fk(o)33070 10627 y Fw(b)-36 b(y)19576 14153 y Fm(B)67 b Fw(\()p Fm(g)48 b Fw(\))368 b(:=)25110 13254 y(1)p 24557 13847 1758 54 v 24557 14113 a Fn(p)p 25664 14113 651 54 v 1099 x Fw(2)26447 14153 y(\()p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))294 b(+)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))33286 13604 y Fi(\003)33810 14153 y Fw(\))p Fm(:)14185 b Fw(\(3.6\))2717 17823 y(Using)434 b(the)f(an)-36 b(tilinearit)g(y)434 b(of)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))432 b(and)h(\(3.6\),)i(w)-36 b(e)433 b(get)h(that)f(for)h(an)-36 b(y)434 b Fm(g)417 b Fn(2)368 b Fl(h)p Fm(;)18804 21349 y(a)p Fw(\()p Fm(g)48 b Fw(\))1106 b(=)25080 20450 y(1)p 24527 21044 1758 54 v 24527 21309 a Fn(p)p 25634 21309 651 54 v 1099 x Fw(2)26417 21349 y(\()p Fm(B)67 b Fw(\()p Fm(g)48 b Fw(\))294 b(+)h Fm(iB)67 b Fw(\()p Fm(ig)48 b Fw(\)\))p Fm(;)13107 b Fw(\(3.7\))18278 24782 y Fm(a)p Fw(\()p Fm(g)48 b Fw(\))20644 24233 y Fi(\003)22276 24782 y Fw(=)25080 23883 y(1)p 24527 24476 1758 54 v 24527 24742 a Fn(p)p 25634 24742 651 54 v 1099 x Fw(2)26417 24782 y(\()p Fm(B)67 b Fw(\()p Fm(g)48 b Fw(\))294 b Fn(\000)i Fm(iB)67 b Fw(\()p Fm(ig)48 b Fw(\)\))p Fm(:)2717 28453 y Fw(F)-108 b(rom)433 b(the)g(CAR)h(relations)h (\(3.1\),)f(w)-36 b(e)434 b(obtain)f(that)g(for)i(an)-36 b(y)433 b Fm(f)70 b(;)221 b(g)417 b Fn(2)369 b Fl(h)p Fm(;)19050 31978 y Fn(f)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))p Fm(;)221 b(B)67 b Fw(\()p Fm(g)48 b Fw(\))p Fn(g)1106 b Fw(=)30087 31080 y(1)p 29534 31673 1758 54 v 29534 31939 a Fn(p)p 30641 31939 651 54 v 1098 x Fw(2)31424 31978 y(\()p Fm(f)70 b(;)221 b(g)48 b Fw(\))p Ff(1)p Fm(;)13354 b Fw(\(3.8\))18524 35411 y Fn(f)p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))21666 34863 y Fi(\003)22191 35411 y Fm(;)221 b(B)67 b Fw(\()p Fm(g)48 b Fw(\))p Fn(g)1107 b Fw(=)30087 34513 y(1)p 29534 35106 1758 54 v 29534 35371 a Fn(p)p 30641 35371 651 54 v 1099 x Fw(2)31424 35411 y(\()p Fm(g)48 b(;)221 b(f)142 b Fw(\))p Ff(1)p Fm(:)2717 39183 y Fw(Denote)445 b(b)-36 b(y)444 b Fn(M)10602 39382 y Fk(f)98 b(in)12543 39183 y Fw(the)444 b(algebra)h(generated)f (b)-36 b(y)444 b Ff(1)h Fw(and)e Fm(a)p Fw(\()p Fm(f)142 b Fw(\))33319 38701 y Fv(#)34158 39183 y Fw(,)448 b Fm(f)529 b Fn(2)387 b Fl(h)38102 39382 y Fv(0)38628 39183 y Fw(,)447 b(where)e Fm(a)p Fw(\()p Fm(f)142 b Fw(\))45683 38701 y Fv(#)46966 39183 y Fw(is)444 b(either)2717 40843 y Fm(a)p Fw(\()p Fm(f)142 b Fw(\))455 b(or)h Fm(a)p Fw(\()p Fm(f)142 b Fw(\))9740 40361 y Fi(\003)10721 40843 y Fw(for)457 b Fm(f)549 b Fn(2)406 b Fl(h)15905 41042 y Fv(0)16431 40843 y Fw(.)646 b(Because)456 b(of)g(\(3.7\),)463 b Fn(M)29030 41042 y Fk(f)98 b(in)30982 40843 y Fw(is)456 b(equal)h(to)f(the)f(algebra)i(generated)e(b)-36 b(y)2717 42504 y Ff(1)485 b Fw(and)f Fm(B)67 b Fw(\()p Fm(f)142 b Fw(\),)497 b Fm(f)597 b Fn(2)455 b Fl(h)13508 42703 y Fv(0)14034 42504 y Fw(.)731 b(Since)484 b Fn(M)20168 42703 y Fk(f)98 b(in)22149 42504 y Fw(is)484 b(norm)g(dense)g(in)g Fm(\031)32859 42703 y Fk(!)33531 42504 y Fw(\()p Fn(A)p Fw(\()p Fl(h)p Fw(\)\),)497 b Fn(M)39761 42703 y Fk(f)98 b(in)41257 42504 y Fm(\030)41825 42703 y Fv(0)42835 42504 y Fw(is)485 b(also)g(dense)f(in)2717 44164 y Fn(H)13 b Fm(:)434 b Fw(F)-108 b(or)433 b(an)-36 b(y)434 b Fm(f)511 b Fn(2)368 b Fl(h)12532 44363 y Fv(0)13058 44164 y Fm(;)443 b(z)429 b Fn(2)368 b Fo(C)p Fm(;)435 b Fw(w)-36 b(e)433 b(write)21719 47153 y Fm(\033)22458 47352 y Fk(z)22989 47153 y Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\)\))589 b(:=)i Fm(a)p Fw(\()p Fm(A)31196 46604 y Fk(i)p 31516 46104 476 40 v(z)32046 47153 y Fm(f)142 b Fw(\))p Fm(;)15166 b Fw(\(3.9\))21719 49145 y Fm(\033)22458 49344 y Fk(z)22989 49145 y Fw(\()p Fm(a)p Fw(\()p Fm(f)142 b Fw(\))25973 48597 y Fi(\003)26498 49145 y Fw(\))590 b(:=)h Fm(a)p Fw(\()p Fm(A)31722 48597 y Fk(iz)32572 49145 y Fm(f)142 b Fw(\))33861 48597 y Fi(\003)34386 49145 y Fm(:)2717 52134 y Fw(In)561 b(fact,)595 b(using)561 b(\(3.4\))h(and)e(Assumption) h(3.1\(c\),)594 b(one)562 b(can)f(c)-36 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b(;)221 b(\030)64 b Fw(\))504 b Fg(in)f(\(3.10\))h(and)g (\(3.11\),)513 b(and)504 b(mo)-66 b(di\014es)2717 12287 y(Assumption)491 b(3.1)h(\(d\))g(appr)-66 b(opriately,)498 b(then)491 b(al)66 b(l)492 b(of)g(the)f(r)-66 b(esults)492 b(in)f(this)g(se)-66 b(ction)490 b(stil)66 b(l)492 b(hold)g(with)2717 13948 y(a)465 b(mo)-66 b(di\014e)g(d)463 b(sp)-66 b(e)g(ctr)g(al)464 b(gap)h(in)f(The)-66 b(or)g(em)464 b(3.2)h(\(b\),)f(i.e.,)f Fw(\(0)p Fm(;)221 b Fw(2)32735 13466 y Fv(1+2)p Fk(\025)35013 13948 y Fw(\))295 b Fn(\\)g Fm(\033)48 b Fw(\()p Fm(H)103 b Fw(\))368 b(=)g Fn(;)p Fg(.)4668 16563 y Fw(In)581 b(the)g(rest)h(of)g(this)f(section,)619 b(w)-36 b(e)582 b(pro)36 b(duce)581 b(the)g(pro)36 b(of)582 b(of)g(Theorem)g(3.1.)1023 b(The)581 b(pro)36 b(of)582 b(of)2717 18223 y(Theorem)434 b(3.2)g(will)h(b)36 b(e)434 b(p)36 b(ostp)g(oned)432 b(to)i(the)f(next)h(section.)2717 20514 y Ff(Pro)42 b(of)582 b(of)f(Theorem)i(3.1)1088 b Fw(Notice)507 b(that)f Fm(a)p Fw(\()p Fm(g)26981 20713 y Fk(j)27467 20514 y Fw(\))492 b Fn(2)h(M)31439 20713 y Fk(o)32294 20514 y Fn(\\)344 b(M)35119 20713 y Fk(an)36245 20514 y Fm(;)443 b Fn(8)p Fm(j)568 b Fn(2)492 b Fo(N)507 b Fw(\(see)f(b)36 b(elo)-36 b(w)507 b(of)g(the)2717 22174 y(equation)519 b(\(3.9\)\).)831 b(Since)518 b Fn(k)p Fm(\033)17368 22373 y Fk(t)p Fv(+)p Fk(is)19251 22174 y Fw(\()p Fm(a)p Fw(\()p Fm(g)21569 22373 y Fk(j)22055 22174 y Fw(\)\))p Fn(k)512 b Fw(=)g Fn(k)p Fm(A)27406 21692 y Fk(s)27897 22174 y Fm(g)28520 22373 y Fk(j)29007 22174 y Fn(k)518 b Fw(\(see)g(\(3.9\))g(and)g (\(3.2\)\),)539 b(the)518 b(assumption)2717 23835 y(\(2.12\))480 b(is)f(satis\014ed.)713 b(Th)-36 b(us)478 b(Theorem)h(2.1)g(impies)g (that)f(eac)-36 b(h)478 b Fn(E)35165 24034 y Fk(j)35652 23835 y Fm(;)444 b(j)520 b Fn(2)445 b Fo(N)p Fm(;)479 b Fw(is)g(a)g(Diric)-36 b(hlet)479 b(form.)2717 25495 y(Th)-36 b(us)425 b(if)h(one)g(can)f(sho)-36 b(w)426 b(that)f Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))425 b(is)h(dense)e(in)i Fn(H)13 b Fw(,)427 b(then)d(it)i(follo)-36 b(ws)427 b(from)f(Theorem)g (5.2\(ii\))g(of)2717 27155 y([Cip1)q(])378 b(that)f(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))378 b(is)g(a)g(Diric)-36 b(hlet)378 b(form)g(and)g(it)f(generates)h(a)g Fm(J)123 b Fw(-real,)389 b(strongly)379 b(con)-36 b(tin)g(uous,)2717 28816 y(symmetric)574 b(and)g(sub-Mark)-36 b(o)g(vian)573 b(semigroup)h Fm(T)27979 29015 y Fk(t)28982 28816 y Fw(=)607 b Fm(e)31204 28334 y Fi(\000)p Fk(tH)33172 28816 y Fm(;)443 b(t)608 b Fn(\025)f Fw(0.)1000 b(Using)574 b(the)f(calculations)2717 30476 y(used)433 b(in)g(the)g(pro)36 b(of)435 b(of)f(Theorem)g(2.1)g (\(a\),)g(w)-36 b(e)434 b(get)g(that)f(for)h(eac)-36 b(h)433 b Fm(j)444 b Fn(2)369 b Fo(N)p Fm(;)13677 31965 y Fh(\000)14286 33041 y Fm(\033)15025 33248 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)17413 33041 y Fw(\()p Fm(a)p Fw(\()p Fm(g)19731 33240 y Fk(j)20217 33041 y Fw(\))20723 32493 y Fv(#)21563 33041 y Fw(\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)25550 33248 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)27938 33041 y Fw(\(\()p Fm(a)p Fw(\()p Fm(g)30762 33240 y Fk(j)31248 33041 y Fw(\))31754 32493 y Fv(#)32593 33041 y Fw(\))33099 32493 y Fi(\003)33625 33041 y Fw(\)\))p Fm(U)35520 33240 y Fk(\015)36112 31965 y Fh(\001)36721 33041 y Fm(\030)37289 33240 y Fv(0)38184 33041 y Fw(=)369 b(0)p Fm(:)2717 35606 y Fw(Th)-36 b(us)433 b Fm(H)103 b(\030)7761 35805 y Fv(0)8655 35606 y Fw(=)369 b(0)434 b(and)f(so)h Fn(f)p Fm(T)16672 35805 y Fk(t)17067 35606 y Fn(g)17731 35805 y Fk(t)p Fi(\025)p Fv(0)19763 35606 y Fw(is)g(Mark)-36 b(o)g(vian.)4668 37267 y(Let)391 b(us)h(sho)-36 b(w)392 b(that)f Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))391 b(is)h(dense.)564 b(Since)392 b Fn(M)28136 37466 y Fk(f)98 b(in)29632 37267 y Fm(\030)30200 37466 y Fv(0)31118 37267 y Fw(is)392 b(dense)f(in)g Fn(H)13 b Fm(;)392 b Fw(it)g(is)g (su\016cien)-36 b(t)391 b(to)h(sho)-36 b(w)2717 38927 y(that)518 b Fn(M)7215 39126 y Fk(f)98 b(in)8711 38927 y Fm(\030)9279 39126 y Fv(0)10323 38927 y Fw(is)518 b(con)-36 b(tained)517 b(in)h Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))p Fm(:)518 b Fw(F)-108 b(or)517 b(an)-36 b(y)518 b Fm(B)580 b Fn(2)512 b(M)p Fm(;)518 b Fw(it)g(follo)-36 b(ws)520 b(from)e(\(2.11\))h(and)e(\(3.9\))2717 40588 y(that)16982 42077 y Fh(\000)17591 43153 y Fm(\033)18330 43359 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)20719 43153 y Fw(\()p Fm(a)p Fw(\()p Fm(g)23037 43352 y Fk(j)23523 43153 y Fw(\)\))295 b Fn(\000)g Fm(j)75 b Fw(\()p Fm(\033)28016 43359 y Fk(t)p Fi(\000)p Fk(i=)p Fv(4)30404 43153 y Fw(\()p Fm(a)p Fw(\()p Fm(g)32722 43352 y Fk(j)33208 43153 y Fw(\))33714 42604 y Fi(\003)34240 43153 y Fw(\)\))p Fm(U)36135 43352 y Fk(\015)36727 42077 y Fh(\001)37336 43153 y Fm(B)67 b(\030)38959 43352 y Fv(0)16982 45273 y Fw(=)369 b Fm(a)p Fw(\()p Fm(A)20527 44724 y Fk(it)p Fi(\000)p Fv(1)p Fk(=)p Fv(4)23385 45273 y Fm(g)24008 45472 y Fk(j)24495 45273 y Fw(\))p Fm(B)67 b(\030)26624 45472 y Fv(0)27445 45273 y Fn(\000)295 b Fm(\015)72 b Fw(\()p Fm(B)67 b Fw(\))p Fm(a)p Fw(\()p Fm(A)33751 44724 y Fk(it)p Fi(\000)p Fv(1)p Fk(=)p 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63272 y Fk(f)98 b(in)11365 63073 y Fw(b)36 b(e)455 b(an)-36 b(y)455 b(elemen)-36 b(t)454 b(of)i(the)e(form)h Fm(B)472 b Fw(=)405 b Fm(B)67 b Fw(\()p Fm(f)32396 63272 y Fv(1)32922 63073 y Fw(\))309 b Fn(\001)h(\001)f(\001)h Fm(B)67 b Fw(\()p Fm(f)37975 63272 y Fk(n)38601 63073 y Fw(\))405 b Fn(2)g(M)42398 63272 y Fk(f)98 b(in)43894 63073 y Fm(;)665 b(f)45561 63272 y Fv(1)46087 63073 y Fm(;)221 b Fn(\001)310 b(\001)f(\001)p Fm(;)221 b(f)49618 63272 y Fk(n)50650 63073 y Fn(2)2717 64734 y Fl(h)3409 64933 y Fv(0)3935 64734 y Fm(;)443 b(n)370 b Fn(2)e Fo(N)412 b Fw(\(see)f(\(3.6\))h(for)f(the)g(de\014nition)f(of)i Fm(B)67 b Fw(\()p Fm(f)27782 64933 y Fk(i)28158 64734 y Fw(\)\).)570 b(Consider)412 b(the)e(case)i(for)f Fm(n)370 b Fw(=)e(2)p Fm(m;)412 b Fw(that)e(is,)2717 66394 y Fm(B)436 b Fn(2)369 b(M)6991 66593 y Fk(e)7483 66394 y Fw(.)578 b(Using)434 b(the)f(CARs)h(\(3.1\))g(and)g(\(3.8\),)g(w)-36 b(e)434 b(obtain)f(that)5313 69916 y([)p Fm(a)p Fw(\()p Fm(A)7838 69368 y Fk(it)p Fi(\000)p 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Fw(for)g(an)-36 b(y)434 b Fm(h)368 b Fn(2)h Fl(h)p Fw(,)434 b(w)-36 b(e)433 b(get)g(from)h(\(3.13\),)2717 17529 y(\(3.14\))h(and)e(\(3.15\))h(that)22796 18486 y Fi(1)22307 18884 y Fh(X)22450 21684 y Fk(j)51 b Fv(=1)24447 20146 y Fn(E)25148 20345 y Fk(j)25635 20146 y Fw([)p Fm(B)67 b(\030)27619 20345 y Fv(0)28146 20146 y Fw(])369 b Fm(<)f Fn(1)p Fm(:)2717 23548 y Fw(Th)-36 b(us)340 b Fm(B)67 b(\030)7542 23747 y Fv(0)8437 23548 y Fn(2)369 b Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\).)547 b(The)341 b(result)f(same)h (as)g(that)g(in)f(the)h(ab)36 b(o)-36 b(v)g(e)341 b(holds)g(for)g(o)36 b(dd)340 b Fm(n)p Fw(.)548 b(Hence)341 b Fn(M)48946 23747 y Fk(f)98 b(in)50442 23548 y Fm(\030)51010 23747 y Fv(0)2717 25208 y Fw(is)434 b(con)-36 b(tained)433 b(in)h Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))433 b(and)g(so)h Fm(D)36 b Fw(\()p Fn(E)119 b Fw(\))433 b(is)h(dense.)4668 26869 y(Next)290 b(w)-36 b(e)290 b(will)h(sho)-36 b(w)290 b(that)f(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))289 b(is)h(indep)36 b(enden)-36 b(t)288 b(of)i(the)f(c)-36 b(hoice)290 b(of)g(the)g(CONS)f Fn(f)p Fm(g)47396 27068 y Fk(j)47883 26869 y Fn(g)369 b(\032)g Fl(h)51010 27068 y Fv(0)2717 28529 y Fw(for)489 b Fl(h)f Fw(and)f(the)g(normalized)h (admissible)h(function)e Fm(f)142 b Fw(.)741 b(Using)488 b(the)g(P)-36 b(arsev)-72 b(al)488 b(relations,)503 b(w)-36 b(e)488 b(get)2717 30190 y(that)433 b(for)h(all)h Fm(t)369 b Fn(2)f Fo(R)p Fm(;)444 b(k)45 b(;)221 b(k)15210 29708 y Fi(0)15890 30190 y Fn(2)368 b Fo(N)p Fm(;)19866 31777 y Fi(1)19377 32176 y Fh(X)19520 34975 y Fk(j)51 b Fv(=1)21296 33438 y Fw(\()p Fm(f)22443 33637 y Fk(k)23012 33438 y Fm(;)221 b(A)24569 32889 y Fk(it)p Fi(\000)p Fv(1)p Fk(=)p Fv(4)27428 33438 y Fm(g)28051 33637 y Fk(j)28538 33438 y Fw(\)\()p Fm(A)30525 32889 y Fk(it)p Fi(\000)p Fv(1)p Fk(=)p Fv(4)33383 33438 y Fm(g)34006 33637 y Fk(j)34493 33438 y Fm(;)g(f)35716 33637 y Fk(k)36230 33385 y Fb(0)36584 33438 y Fw(\))19377 36936 y(=)369 b(\()p Fm(f)21905 37135 y Fk(k)22474 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53062 y Fn(E)119 b Fw([)p Fm(B)67 b(\030)12901 53261 y Fv(0)13427 53062 y Fw(])369 b(=)16212 51402 y Fk(n)15538 51800 y Fh(X)15639 54629 y Fk(k)24 b Fv(=1)18400 51402 y Fk(n)17726 51800 y Fh(X)17678 54635 y Fk(k)18192 54383 y Fb(0)18490 54635 y Fv(=1)20046 52163 y Fw(\()p Fn(\000)p Fw(1\))22741 51681 y Fk(k)g Fv(+)p Fk(k)24501 51369 y Fb(0)p 20046 52757 4808 54 v 22125 53973 a Fw(2)24987 53062 y Fn(f)p Fw(\()p Fm(f)26798 53261 y Fk(k)27367 53062 y Fm(;)221 b(A)28924 52514 y Fi(\000)p Fv(1)p Fk(=)p Fv(2)31123 53062 y Fm(f)31764 53261 y Fk(k)32278 53009 y Fb(0)32631 53062 y Fw(\))295 b(+)g(\()p Fm(f)35886 53261 y Fk(k)36400 53009 y Fb(0)36753 53062 y Fm(;)221 b(A)38310 52514 y Fv(1)p Fk(=)p Fv(2)39778 53062 y Fm(f)40419 53261 y Fk(k)40988 53062 y Fw(\))p Fn(g)6054 b Fw(\(3.16\))12698 56506 y Fn(\001h)p Fm(B)67 b Fw(\()p Fm(f)15786 56705 y Fv(1)16312 56506 y Fw(\))295 b Fn(\001)g(\001)g(\001)19372 56170 y Fh(b)19105 56506 y Fm(B)67 b Fw(\()p Fm(f)21307 56705 y Fk(k)21876 56506 y Fw(\))295 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Fw(is)498 b(a)g(core)f(for)i(the)2717 60531 y(Diric)-36 b(hlet)580 b(op)36 b(erator)579 b Fm(H)681 b Fw(asso)36 b(ciated)580 b(to)g(the)e(form)i(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\))579 b(\(Theorem)g(3.2)h(\(a\)\).)1014 b(Th)-36 b(us)579 b Fm(H)2717 62192 y Fw(is)559 b(indep)36 b(enden)-36 b(t)557 b(of)i(normalized)g(admissible)g(function)g Fm(f)700 b Fw(and)558 b(CONS)h Fn(f)p Fm(g)41161 62391 y Fk(j)41647 62192 y Fn(g)583 b(\032)f Fl(h)45201 62391 y Fv(0)46286 62192 y Fw(and)558 b(so)h(is)2717 63852 y(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\).)1879 b Fe(\003)2717 68197 y Fp(4)2152 b(Decomp)60 b(osition)477 b(of)h(quasi-free)e(Hilb)60 b(ert)477 b(space)h(:)836 b(Pro)60 b(of)5945 70633 y(of)717 b(Theorem)f(3.2)2717 73608 y Fw(As)289 b(in)g([BKP)q(],)318 b(w)-36 b(e)290 b(will)g(decomp)36 b(ose)289 b(the)g(Hilb)36 b(ert)289 b(space)g Fn(H)381 b Fw(=)369 b Fn(H)34339 73807 y Fk(!)35300 73608 y Fw(in)-36 b(to)289 b(direct)g(sum)f(of)i Fn(H)46414 73126 y Fv(\()p Fk(m;n)p Fv(\))48865 73608 y Fm(;)221 b(m;)g(n)371 b Fn(2)2717 75269 y Fo(N)34 b Fn([)g(f)p Fw(0)p Fn(g)p Fm(;)308 b Fw(where)d Fn(H)12041 74787 y Fv(\()p Fk(m;n)p Fv(\))14799 75269 y Fw(is)h(the)f(Hilb)36 b(ert)306 b(space)g(of)h Fm(m)e Fw(quasi-particles)i(and)e Fm(n)h Fw(an)-36 b(ti)306 b(quasi-particles.)26476 78590 y(15)p eop %%Page: 16 16 16 15 bop 2717 7306 a Fw(W)-108 b(e)483 b(then)g(use)g(the)g(results)g (to)g(pro)-36 b(v)g(e)483 b(Theorem)h(3.2.)728 b(The)484 b(decomp)36 b(osition)483 b(metho)36 b(d)483 b(w)-36 b(e)484 b(use)f(is)2717 8966 y(essen)-36 b(tially)435 b(same)f(as)g(that)f(in)g([A)-145 b(W)q(].)579 b(See)433 b(also)i(Example)f(5.2.20)i(in)d([BR)q(].)4668 10627 y(Denote)h(b)-36 b(y)433 b Fm(B)501 b Fw(the)433 b(op)36 b(erator)434 b(giv)-36 b(en)434 b(b)-36 b(y)22494 13257 y Fm(B)436 b Fw(:=)369 b Fm(A)26635 12709 y Fi(\000)27500 12347 y Fd(1)p 27500 12508 406 40 v 27500 13056 a(2)28389 13257 y Fw(+)295 b Fm(A)30804 12347 y Fd(1)p 30804 12508 V 30804 13056 a(2)31398 13257 y Fm(:)17103 b Fw(\(4.1\))2717 15888 y(In)434 b(this)f(section,)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))13949 15406 y Fv(#)14787 15888 y Fm(;)443 b(g)417 b Fn(2)368 b Fl(h)18577 16087 y Fv(0)19537 15888 y Fw(stands)433 b(for)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))433 b(or)g Fm(a)p Fw(\()p Fm(g)48 b Fw(\))32339 15406 y Fi(\003)32864 15888 y Fw(.)578 b(And)433 b(w)-36 b(e)434 b(write)14508 18518 y Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(g)e Fw(\))18006 17969 y Fv(#)18844 18518 y Fw(\))1107 b(:=)g Fm(a)p Fw(\()p Fm(g)48 b Fw(\))25303 17969 y Fv(#)26436 18518 y Fn(\000)295 b Fm(j)75 b Fw(\()p Fm(\033)29622 18725 y Fi(\000)p Fk(i=)p Fv(2)31671 18518 y Fw(\(\()p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))35049 17969 y Fv(#)35887 18518 y Fw(\))36393 17969 y Fi(\003)36918 18518 y Fw(\)\))p Fm(U)38813 18717 y Fk(\015)48862 18518 y Fw(\(4.2\))20637 20510 y(=)1288 b Fm(a)p Fw(\()p Fm(g)48 b Fw(\))25303 19962 y Fv(#)26436 20510 y Fw(+)295 b Fm(U)28626 20709 y Fk(\015)29218 20510 y Fm(j)75 b Fw(\()p Fm(\033)31076 20717 y Fi(\000)p Fk(i=)p Fv(2)33125 20510 y Fw(\(\()p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))36503 19962 y Fv(#)37341 20510 y Fw(\))37847 19962 y Fi(\003)38372 20510 y Fw(\)\))p Fm(;)2717 23141 y Fw(as)434 b(a)g(b)36 b(ounded)432 b(op)36 b(erator)434 b(on)f Fn(H)13 b Fw(.)578 b(F)-108 b(or)433 b Fm(g)417 b Fn(2)368 b Fl(h)25144 23340 y Fv(0)25671 23141 y Fw(,)434 b Fm(C)463 b Fn(2)369 b(M)p Fw(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)15756 25771 y Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(g)e Fw(\))19254 25223 y Fv(#)20092 25771 y Fw(\))p Fm(C)95 b(\030)22192 25970 y Fv(0)23086 25771 y Fw(=)369 b Fm(a)p Fw(\()p Fm(g)48 b Fw(\))26833 25223 y Fv(#)27671 25771 y Fm(C)95 b(\030)29265 25970 y Fv(0)30086 25771 y Fn(\000)296 b Fm(\015)72 b Fw(\()p Fm(C)95 b Fw(\))p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))36566 25223 y Fv(#)37404 25771 y Fm(\030)37972 25970 y Fv(0)48862 25771 y Fw(\(4.3\))2717 28402 y(\(see)444 b(the)f(argumen)-36 b(t)442 b(of)j(\(2.11\)\).)608 b(Recall)445 b(\(2.13\),)i Fm(U)28785 27920 y Fi(\000)p Fv(1)28646 28730 y Fk(\015)30429 28402 y Fw(=)385 b Fm(U)32709 28601 y Fk(\015)33745 28402 y Fw(and)443 b(the)g(fact)h(that)f Fm(U)44820 28601 y Fk(\015)45856 28402 y Fw(comm)-36 b(utes)2717 30062 y(with)434 b Fm(J)556 b Fw(and)433 b(\001.)579 b(It)433 b(follo)-36 b(ws)436 b(from)e(\(3.9\))g(and)f (\(4.2\))h(that)f(for)h(an)-36 b(y)434 b Fm(g)417 b Fn(2)368 b Fl(h)39249 30261 y Fv(0)9202 32813 y Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)12578 32265 y Fi(\000)13443 31903 y Fd(1)p 13442 32064 V 13442 32612 a(2)14036 32813 y Fm(A)15011 32265 y Fi(\000)15876 31903 y Fd(1)p 15876 32064 V 15876 32612 a(4)16470 32813 y Fm(g)e Fw(\)\))1106 b(=)h Fm(a)p Fw(\()p Fm(B)23622 32265 y Fi(\000)24487 31903 y Fd(1)p 24486 32064 V 24486 32612 a(2)25080 32813 y Fm(A)26055 32265 y Fi(\000)26920 31903 y Fd(1)p 26920 32064 V 26920 32612 a(4)27514 32813 y Fm(g)48 b Fw(\))294 b Fn(\000)i Fm(j)75 b Fw(\()p Fm(\033)32172 33020 y Fi(\000)p Fk(i=)p Fv(2)34220 32813 y Fw(\()p Fm(a)p Fw(\()p Fm(B)36970 32265 y Fi(\000)37835 31903 y Fd(1)p 37834 32064 V 37834 32612 a(2)38428 32813 y Fm(A)39403 32265 y Fi(\000)40268 31903 y Fd(1)p 40268 32064 V 40268 32612 a(4)40862 32813 y Fm(g)48 b Fw(\))42039 32265 y Fi(\003)42564 32813 y Fw(\)\))p Fm(U)44459 33012 y Fk(\015)48862 32813 y Fw(\(4.4\))19259 35023 y(=)1107 b Fm(a)p Fw(\()p Fm(B)23622 34474 y Fi(\000)24487 34112 y Fd(1)p 24486 34273 V 24486 34821 a(2)25080 35023 y Fm(A)26055 34474 y Fi(\000)26920 34112 y Fd(1)p 26920 34273 V 26920 34821 a(4)27514 35023 y Fm(g)48 b Fw(\))294 b(+)h Fm(U)31175 35222 y Fk(\015)31768 35023 y Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(B)35131 34474 y Fi(\000)35996 34112 y Fd(1)p 35995 34273 V 35995 34821 a(2)36589 35023 y Fm(A)37697 34112 y Fd(1)p 37697 34273 V 37697 34821 a(4)38291 35023 y Fm(g)48 b Fw(\))39468 34474 y Fi(\003)39993 35023 y Fw(\))p Fm(;)10037 38783 y(\016)i Fw(\()p Fm(a)p Fw(\()p Fm(B)13413 38235 y Fi(\000)14278 37873 y Fd(1)p 14277 38034 V 14277 38582 a(2)14871 38783 y Fm(A)15979 37873 y Fd(1)p 15979 38034 V 15979 38582 a(4)16573 38783 y Fm(g)e Fw(\))17750 38235 y Fi(\003)18275 38783 y Fw(\))1107 b(=)f Fm(a)p Fw(\()p Fm(B)24250 38235 y Fi(\000)25115 37873 y Fd(1)p 25115 38034 V 25115 38582 a(2)25709 38783 y Fm(A)26817 37873 y Fd(1)p 26817 38034 V 26817 38582 a(4)27411 38783 y Fm(g)48 b Fw(\))28588 38235 y Fi(\003)29408 38783 y Fn(\000)295 b Fm(j)75 b Fw(\()p Fm(\033)32594 38990 y Fi(\000)p Fk(i=)p Fv(2)34643 38783 y Fw(\()p Fm(a)p Fw(\()p Fm(B)37393 38235 y Fi(\000)38258 37873 y Fd(1)p 38257 38034 V 38257 38582 a(2)38851 38783 y Fm(A)39959 37873 y Fd(1)p 39959 38034 V 39959 38582 a(4)40553 38783 y Fm(g)48 b Fw(\)\)\))p Fm(U)43625 38982 y Fk(\015)19888 40993 y Fw(=)1106 b Fm(a)p Fw(\()p Fm(B)24250 40444 y Fi(\000)25115 40082 y Fd(1)p 25115 40243 V 25115 40791 a(2)25709 40993 y Fm(A)26817 40082 y Fd(1)p 26817 40243 V 26817 40791 a(4)27411 40993 y Fm(g)48 b Fw(\))28588 40444 y Fi(\003)29408 40993 y Fw(+)295 b Fm(U)31598 41192 y Fk(\015)32190 40993 y Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(B)35553 40444 y Fi(\000)36418 40082 y Fd(1)p 36417 40243 V 36417 40791 a(2)37012 40993 y Fm(A)37987 40444 y Fi(\000)38852 40082 y Fd(1)p 38851 40243 V 38851 40791 a(4)39445 40993 y Fm(g)48 b Fw(\)\))p Fm(:)2717 43623 y Fw(Since)433 b(\()p Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(g)48 b Fw(\)\)\))11116 43141 y Fi(\003)12009 43623 y Fw(=)369 b Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))16875 43141 y Fi(\003)17399 43623 y Fw(\))433 b(and)g Fm(U)21750 43822 y Fk(\015)22712 43623 y Fw(=)369 b Fm(U)25115 43141 y Fi(\000)p Fv(1)24976 43951 y Fk(\015)26741 43623 y Fw(=)g Fm(U)29144 43141 y Fi(\003)29005 43951 y Fk(\015)29670 43623 y Fw(,)434 b(a)g(computation)f(sho)-36 b(ws)434 b(that)10699 46543 y(\()p Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)14581 45994 y Fi(\000)15446 45632 y Fd(1)p 15445 45793 V 15445 46341 a(2)16039 46543 y Fm(A)17014 45994 y Fi(\000)17879 45632 y Fd(1)p 17878 45793 V 17878 46341 a(4)18473 46543 y Fm(g)e Fw(\)\)\))20662 45994 y Fi(\003)22293 46543 y Fw(=)1107 b Fm(a)p Fw(\()p Fm(B)26656 45994 y Fi(\000)27521 45632 y Fd(1)p 27520 45793 V 27520 46341 a(2)28114 46543 y Fm(A)29089 45994 y Fi(\000)29954 45632 y Fd(1)p 29954 45793 V 29954 46341 a(4)30548 46543 y Fm(g)48 b Fw(\))31725 45994 y Fi(\003)32545 46543 y Fw(+)295 b Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(B)37215 45994 y Fi(\000)38080 45632 y Fd(1)p 38079 45793 V 38079 46341 a(2)38673 46543 y Fm(A)39781 45632 y Fd(1)p 39781 45793 V 39781 46341 a(4)40375 46543 y Fm(g)48 b Fw(\)\))p Fm(U)42941 46742 y Fk(\015)22293 48695 y Fw(=)1107 b Fm(a)p Fw(\()p Fm(B)26656 48147 y Fi(\000)27521 47785 y Fd(1)p 27520 47946 V 27520 48494 a(2)28114 48695 y Fm(A)29089 48147 y Fi(\000)29954 47785 y Fd(1)p 29954 47946 V 29954 48494 a(4)30548 48695 y Fm(g)48 b Fw(\))31725 48147 y Fi(\003)32545 48695 y Fn(\000)296 b Fm(U)34757 48894 y Fk(\015)35349 48695 y Fm(j)75 b Fw(\()p Fm(a)p Fw(\()p Fm(B)38712 48147 y Fi(\000)39577 47785 y Fd(1)p 39576 47946 V 39576 48494 a(2)40170 48695 y Fm(A)41278 47785 y Fd(1)p 41278 47946 V 41278 48494 a(4)41872 48695 y Fm(g)48 b Fw(\)\))22293 50848 y(=)1107 b Fm(a)p Fw(\()p Fm(B)26789 49938 y Fd(1)p 26788 50099 V 26788 50647 a(2)27382 50848 y Fm(A)28490 49938 y Fd(1)p 28490 50099 V 28490 50647 a(4)29084 50848 y Fm(g)48 b Fw(\))30261 50300 y Fi(\003)31082 50848 y Fn(\000)295 b Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)35786 50300 y Fi(\000)36651 49938 y Fd(1)p 36650 50099 V 36650 50647 a(2)37244 50848 y Fm(A)38352 49938 y Fd(3)p 38352 50099 V 38352 50647 a(4)38946 50848 y Fm(g)e Fw(\))40123 50300 y Fi(\003)40648 50848 y Fw(\))p Fm(:)7347 b Fw(\(4.5\))2717 53479 y(Here)464 b(w)-36 b(e)465 b(ha)-36 b(v)g(e)464 b(used)g(the)g(fact)h(that)e Fm(B)22564 52988 y Fi(\000)23429 52626 y Fd(1)p 23429 52787 V 23429 53336 a(2)24023 53479 y Fw(\()p Fm(A)25504 52988 y Fi(\000)26369 52626 y Fd(1)p 26369 52787 V 26369 53336 a(4)27279 53479 y Fw(+)316 b Fm(A)29715 52626 y Fd(3)p 29715 52787 V 29715 53336 a(4)30309 53479 y Fw(\))421 b(=)g Fm(B)33857 52626 y Fd(1)p 33857 52787 V 33857 53336 a(2)34451 53479 y Fm(A)35559 52626 y Fd(1)p 35559 52787 V 35559 53336 a(4)36153 53479 y Fm(:)464 b Fw(Using)h(the)e(metho)36 b(d)464 b(similar)2717 55139 y(to)434 b(that)f(used)g(in)g(the)g(ab)36 b(o)-36 b(v)g(e,)435 b(w)-36 b(e)434 b(get)f(that)12353 57769 y(\()p Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)16235 57221 y Fi(\000)17100 56859 y Fd(1)p 17099 57020 V 17099 57568 a(2)17693 57769 y Fm(A)18801 56859 y Fd(1)p 18801 57020 V 18801 57568 a(4)19395 57769 y Fm(g)e Fw(\))20572 57221 y Fi(\003)21097 57769 y Fw(\)\))22109 57221 y Fi(\003)23004 57769 y Fw(=)368 b Fm(a)p Fw(\()p Fm(B)26761 56859 y Fd(1)p 26761 57020 V 26761 57568 a(2)27355 57769 y Fm(A)28330 57221 y Fi(\000)29195 56859 y Fd(1)p 29195 57020 V 29195 57568 a(4)29789 57769 y Fm(g)48 b Fw(\))294 b Fn(\000)i Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)35965 57221 y Fi(\000)36830 56859 y Fd(1)p 36829 57020 V 36829 57568 a(2)37423 57769 y Fm(A)38398 57221 y Fi(\000)39263 56859 y Fd(3)p 39262 57020 V 39262 57568 a(4)39857 57769 y Fm(g)e Fw(\)\))p Fm(:)6961 b Fw(\(4.6\))2717 60400 y(F)-108 b(rom)433 b(notational)h(brevit)-36 b(y)-108 b(,)434 b(w)-36 b(e)434 b(write)g(that)f(for)h(an)-36 b(y)434 b Fm(g)417 b Fn(2)368 b Fl(h)32529 60599 y Fv(0)18923 63030 y Fm(D)20003 63229 y Fv(1)20529 63030 y Fw(\()p Fm(g)48 b Fw(\))1106 b(:=)h Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)29174 62482 y Fi(\000)30039 62120 y Fd(1)p 30038 62281 V 30038 62829 a(2)30632 63030 y Fm(A)31607 62482 y Fi(\000)32472 62120 y Fd(1)p 32471 62281 V 32471 62829 a(4)33066 63030 y Fm(g)e Fw(\)\))p Fm(;)13752 b Fw(\(4.7\))18923 65144 y Fm(D)20003 65343 y Fv(2)20529 65144 y Fw(\()p Fm(g)48 b Fw(\))1106 b(:=)h Fm(\016)50 b Fw(\()p Fm(a)p Fw(\()p Fm(B)29174 64595 y Fi(\000)30039 64233 y Fd(1)p 30038 64394 V 30038 64942 a(2)30632 65144 y Fm(A)31740 64233 y Fd(1)p 31740 64394 V 31740 64942 a(4)32334 65144 y Fm(g)e Fw(\))33511 64595 y Fi(\003)34036 65144 y Fw(\))p Fm(:)2717 67774 y Fw(Then)433 b(it)h(follo)-36 b(ws)436 b(from)e(\(4.5\))g(-)f(\(4.7\))h(that)16463 70525 y Fm(D)17543 70724 y Fv(1)18069 70525 y Fw(\()p Fm(g)48 b Fw(\))19752 69977 y Fi(\003)21384 70525 y Fw(=)1107 b Fm(a)p Fw(\()p Fm(B)25880 69615 y Fd(1)p 25879 69776 V 25879 70324 a(2)26473 70525 y Fm(A)27581 69615 y Fd(1)p 27581 69776 V 27581 70324 a(4)28175 70525 y Fm(g)48 b Fw(\))29352 69977 y Fi(\003)30173 70525 y Fn(\000)295 b Fm(D)32581 70724 y Fv(2)33107 70525 y Fw(\()p Fm(A)34721 69615 y Fd(1)p 34720 69776 V 34720 70324 a(2)35314 70525 y Fm(g)48 b Fw(\))p Fm(;)12010 b Fw(\(4.8\))16463 72639 y Fm(D)17543 72838 y Fv(2)18069 72639 y Fw(\()p Fm(g)48 b Fw(\))19752 72090 y Fi(\003)21384 72639 y Fw(=)1107 b Fm(a)p Fw(\()p Fm(B)25880 71728 y Fd(1)p 25879 71889 V 25879 72437 a(2)26473 72639 y Fm(A)27448 72090 y Fi(\000)28313 71728 y Fd(1)p 28313 71889 V 28313 72437 a(4)28907 72639 y Fm(g)48 b Fw(\))295 b Fn(\000)g Fm(D)32787 72838 y Fv(1)33313 72639 y Fw(\()p Fm(A)34794 72090 y Fi(\000)35659 71728 y Fd(1)p 35658 71889 V 35658 72437 a(2)36252 72639 y Fm(g)48 b Fw(\))p Fm(:)2717 75269 y Fw(W)-108 b(e)434 b(\014rst)f(collect)h(some)g(prop)36 b(erties)433 b(of)h Fm(D)23617 75468 y Fk(i)23993 75269 y Fw(\()p Fm(g)48 b Fw(\))433 b(for)h Fm(g)416 b Fn(2)369 b Fl(h)31082 75468 y Fv(0)32042 75269 y Fw(and)433 b Fm(i)368 b Fw(=)h(1)p Fm(;)443 b Fw(2)p Fm(:)26476 78590 y Fw(16)p eop %%Page: 17 17 17 16 bop 2717 7306 a Ff(Lemma)499 b(4.1)651 b Fm(D)11652 7505 y Fk(i)12028 7306 y Fw(\()p Fm(g)48 b Fw(\))p Fm(\030)14279 7505 y Fv(0)15173 7306 y Fw(=)368 b(0)465 b Fg(for)g(any)g Fm(g)416 b Fn(2)369 b Fl(h)25221 7505 y Fv(0)26212 7306 y Fg(and)465 b Fm(i)368 b Fw(=)h(1)p Fm(;)443 b Fw(2)p Fg(.)2717 11400 y(Pr)-66 b(o)g(of:)577 b Fw(This)434 b(follo)-36 b(ws)435 b(from)f(\(4.3\))h(and)e(\(4.7\).)579 b Fe(\003)2717 16567 y Ff(Lemma)499 b(4.2)651 b Fg(The)465 b(fol)66 b(lowing)464 b(r)-66 b(elations)464 b(hold)i(for)e(any)h Fm(g)48 b(;)443 b(h)368 b Fn(2)h Fl(h)35933 16766 y Fv(0)36459 16567 y Fg(:)3592 19167 y(\(a\))651 b Fn(f)p Fm(D)7713 19366 y Fk(i)8089 19167 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(D)11434 19366 y Fk(j)11920 19167 y Fw(\()p Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)665 b(i)369 b Fw(=)f(1)p Fm(;)221 b Fw(2)p Fm(;)445 b(j)f Fw(=)368 b(1)p Fm(;)221 b Fw(2)p Fm(;)3658 21879 y Fg(\(b\))651 b Fn(f)p Fm(D)7713 22078 y Fv(1)8239 21879 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(h)p Fw(\))p Fn(g)368 b Fw(=)g(0)p Fm(;)3658 24591 y Fg(\(c\))651 b Fn(f)p Fm(D)7713 24790 y Fv(2)8239 24591 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(h)p Fw(\))12948 24109 y Fi(\003)13472 24591 y Fn(g)370 b Fw(=)e(0)p Fm(;)3592 27302 y Fg(\(d\))651 b Fn(f)p Fm(D)7713 27501 y Fv(1)8239 27302 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(B)12881 26450 y Fd(1)p 12880 26611 406 40 v 12880 27160 a(2)13474 27302 y Fm(A)14582 26450 y Fd(1)p 14582 26611 V 14582 27160 a(4)15176 27302 y Fm(h)p Fw(\))16431 26820 y Fi(\003)16956 27302 y Fn(g)370 b Fw(=)e(\()p Fm(g)48 b(;)221 b(h)p Fw(\))p Fa(1)p Fm(;)3658 30014 y Fg(\(e\))651 b Fn(f)p Fm(D)7713 30213 y Fv(2)8239 30014 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)p Fw(\()p Fm(B)12881 29162 y Fd(1)p 12880 29323 V 12880 29871 a(2)13474 30014 y Fm(A)14449 29524 y Fi(\000)15314 29162 y Fd(1)p 15314 29323 V 15314 29871 a(4)15908 30014 y Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(\()p Fm(h;)221 b(g)48 b Fw(\))p Fa(1)p Fm(:)2717 34108 y Fg(Pr)-66 b(o)g(of:)728 b Fw(Notice)509 b(that)g Fm(a)p Fw(\()p Fm(h)p Fw(\))16363 33626 y Fv(#)17201 34108 y Fm(U)18084 34307 y Fk(\015)19174 34108 y Fw(=)497 b Fn(\000)p Fm(U)22599 34307 y Fk(\015)23192 34108 y Fm(a)p Fw(\()p Fm(h)p Fw(\))25636 33626 y Fv(#)26983 34108 y Fw(for)510 b Fm(h)497 b Fn(2)g Fl(h)32367 34307 y Fv(0)32893 34108 y Fw(.)805 b(The)509 b(pro)36 b(ofs)509 b(follo)-36 b(ws)511 b(from)e(direct)2717 35768 y(computations)433 b(using)h(the)f(de\014nitions)g(in)g(\(4.7\))h(and)f(the)g(CARs)i(in)e (\(3.1\).)579 b Fe(\003)2717 40936 y Ff(Prop)42 b(osition)500 b(4.1)651 b Fg(The)352 b(fol)66 b(lowing)352 b(c)-66 b(anonic)g(al)350 b(anti-c)-66 b(ommutation)349 b(r)-66 b(elations)351 b(\(CARs\))i(hold)f(for)2717 42596 y(any)465 b Fm(g)48 b(;)442 b(h)369 b Fn(2)g Fl(h)9747 42795 y Fv(0)10273 42596 y Fg(:)3592 45196 y(\(a\))651 b Fn(f)p Fm(D)7713 45395 y Fv(1)8239 45196 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(D)11584 45395 y Fv(1)12109 45196 y Fw(\()p Fm(h)p Fw(\))13870 44714 y Fi(\003)14395 45196 y Fn(g)370 b Fw(=)e(\()p Fm(g)48 b(;)221 b(h)p Fw(\))p Fa(1)p Fm(;)5969 46857 y Fn(f)p Fm(D)7713 47056 y Fv(1)8239 46857 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(D)11584 47056 y Fv(1)12109 46857 y Fw(\()p Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)1551 b Fn(f)p Fm(D)20589 47056 y Fv(1)21115 46857 y Fw(\()p Fm(g)48 b Fw(\))22798 46375 y Fi(\003)23323 46857 y Fm(;)221 b(D)24985 47056 y Fv(1)25511 46857 y Fw(\()p Fm(h)p Fw(\))27272 46375 y Fi(\003)27797 46857 y Fn(g)369 b Fw(=)g(0)p Fm(;)3658 49569 y Fg(\(b\))651 b Fn(f)p Fm(D)7713 49768 y Fv(2)8239 49569 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(D)11584 49768 y Fv(2)12109 49569 y Fw(\()p Fm(h)p Fw(\))13870 49087 y Fi(\003)14395 49569 y Fn(g)370 b Fw(=)e(\()p Fm(h;)221 b(g)48 b Fw(\))p Fa(1)p Fm(;)5969 51229 y Fn(f)p Fm(D)7713 51428 y Fv(2)8239 51229 y Fw(\()p Fm(g)g Fw(\))p Fm(;)221 b(D)11584 51428 y Fv(2)12109 51229 y Fw(\()p Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)1551 b Fn(f)p Fm(D)20589 51428 y Fv(2)21115 51229 y Fw(\()p Fm(g)48 b Fw(\))22798 50747 y Fi(\003)23323 51229 y Fm(;)221 b(D)24985 51428 y Fv(2)25511 51229 y Fw(\()p Fm(h)p Fw(\))27272 50747 y Fi(\003)27797 51229 y Fn(g)369 b Fw(=)g(0)p Fm(;)3658 53941 y Fg(\(c\))651 b Fn(f)p Fm(D)7713 54140 y Fv(1)8239 53941 y Fw(\()p Fm(g)48 b Fw(\))9922 54140 y Fk(;)10238 53941 y Fm(D)11318 54140 y Fv(2)11843 53941 y Fw(\()p Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)1551 b Fn(f)p Fm(D)20323 54140 y Fv(1)20849 53941 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(D)24194 54140 y Fv(2)24719 53941 y Fw(\()p Fm(h)p Fw(\))26480 53459 y Fi(\003)27005 53941 y Fn(g)370 b Fw(=)e(0)p Fm(;)5969 55601 y Fn(f)p Fm(D)7713 55800 y Fv(1)8239 55601 y Fw(\()p Fm(g)48 b Fw(\))9922 55119 y Fi(\003)10447 55601 y Fm(;)221 b(D)12109 55800 y Fv(2)12635 55601 y Fw(\()p Fm(h)p Fw(\))p Fn(g)369 b Fw(=)f(0)p Fm(;)1551 b Fn(f)p Fm(D)21115 55800 y Fv(1)21640 55601 y Fw(\()p Fm(g)48 b Fw(\))23323 55119 y Fi(\003)23848 55601 y Fm(;)221 b(D)25510 55800 y Fv(2)26037 55601 y Fw(\()p Fm(g)48 b Fw(\))27720 55119 y Fi(\003)28245 55601 y Fn(g)369 b Fw(=)g(0)p Fm(:)2717 59695 y Fg(Pr)-66 b(o)g(of:)616 b Fw(The)454 b(an)-36 b(ti-comm)g(utation)452 b(relations)i(follo)-36 b(w)455 b(from)f(\(4.7\),)459 b(\(4.8\),)g(\(3.1\))454 b(and)f(Lemma)g(4.2.)2717 61356 y Fe(\003)4668 65306 y Fw(No)-36 b(w)338 b(w)-36 b(e)339 b(are)e(ready)h(to)g(decomp)36 b(ose)338 b(the)f(Hilb)36 b(ert)338 b(space)g Fn(H)382 b Fw(=)368 b Fn(H)36765 65505 y Fk(!)37437 65306 y Fm(;)338 b Fw(called)h(quasi-free)f(Hilb)36 b(ert)2717 66967 y(space.)532 b(According)294 b(to)f(Lemma)h(4.1)h(and) e(the)g(CARs)h(in)f(Prop)36 b(osition)295 b(4.1,)322 b Fm(D)40425 67166 y Fk(i)40801 66967 y Fw(\()p Fm(g)48 b Fw(\))293 b(and)g Fm(D)46246 67166 y Fk(i)46622 66967 y Fw(\()p Fm(h)p Fw(\))48383 66485 y Fi(\003)48908 66967 y Fm(;)443 b(i)368 b Fw(=)2717 68627 y(1)p Fm(;)221 b Fw(2)p Fm(;)445 b(g)48 b(;)221 b(h)377 b Fn(2)g Fl(h)9739 68826 y Fv(0)10266 68627 y Fm(;)439 b Fw(can)f(b)36 b(e)439 b(though)-36 b(t)437 b(as)i(annihilation)g(and)g(creation)g(op)36 b(erators)439 b(resp)36 b(ectiv)-36 b(ely)-108 b(.)594 b(W)-108 b(e)2717 70288 y(remark)439 b(that)e Fm(h)377 b Fn(7!)g Fm(D)13899 70487 y Fv(1)14425 70288 y Fw(\()p Fm(h)p Fw(\))16186 69806 y Fi(\003)17149 70288 y Fw(is)439 b(linear,)h(but)d Fm(g)424 b Fn(7!)377 b Fm(D)28663 70487 y Fv(2)29189 70288 y Fw(\()p Fm(g)48 b Fw(\))30872 69806 y Fi(\003)31835 70288 y Fw(is)439 b(conjugate)g(linear.)592 b(With)438 b(an)g(abuse)2717 71948 y(of)551 b(terminology)-108 b(,)580 b(w)-36 b(e)550 b(call)g Fm(D)17514 72147 y Fv(1)18040 71948 y Fw(\()p Fm(h)p Fw(\))19801 71466 y Fi(\003)20876 71948 y Fw(and)f Fm(D)24601 72147 y Fv(2)25127 71948 y Fw(\()p Fm(h)p Fw(\))26888 71466 y Fi(\003)27963 71948 y Fw(the)g(creation)h(op)36 b(erators)550 b(for)g(quasi-particles)2717 73608 y(and)442 b(an)-36 b(ti)442 b(quasi-particles)g(resp)36 b(ectiv)-36 b(ely)443 b(for)g Fm(h)383 b Fn(2)g Fl(h)28672 73807 y Fv(0)29198 73608 y Fw(.)604 b(The)442 b(follo)-36 b(wing)444 b(is)e(the)g(decomp)36 b(osition)442 b(of)2717 75269 y Fn(H)13 b Fw(:)26476 78590 y(17)p eop %%Page: 18 18 18 17 bop 2717 7306 a Ff(Theorem)499 b(4.1)651 b Fg(The)465 b(fol)66 b(lowing)465 b(de)-66 b(c)g(omp)g(osition)462 b(holds:)22347 10820 y Fn(H)382 b Fw(=)26194 9159 y Fi(1)25661 9558 y Fh(M)25231 12347 y Fk(m;n)p Fv(=0)28320 10820 y Fn(H)29455 10271 y Fv(\()p Fk(m;n)p Fv(\))2717 14879 y Fg(wher)-66 b(e)554 b(for)f(e)-66 b(ach)553 b Fm(m;)221 b(n)535 b Fn(2)e Fo(N)361 b Fn([)g(f)p Fw(0)p Fn(g)p Fm(;)555 b Fn(H)22697 14397 y Fv(\()p Fk(m;n)p Fv(\))25702 14879 y Fg(is)f(the)f(closur)-66 b(e)554 b(of)g(the)f(subsp)-66 b(ac)g(e)554 b(sp)-66 b(anne)g(d)553 b(by)g(the)2717 16540 y(ve)-66 b(ctors)464 b(of)h(the)f(form)15895 20054 y Fw(\()16834 18393 y Fk(m)16401 18792 y Fh(Y)16433 21591 y Fk(j)51 b Fv(=1)18320 20054 y Fm(D)19400 20253 y Fv(1)19926 20054 y Fw(\()p Fm(g)21055 20253 y Fk(j)21541 20054 y Fw(\))22047 19505 y Fi(\003)22573 20054 y Fw(\)\()24148 18393 y Fk(n)23585 18792 y Fh(Y)23687 21620 y Fk(l)11 b Fv(=1)25503 20054 y Fm(D)26583 20253 y Fv(2)27109 20054 y Fw(\()p Fm(h)28364 20253 y Fk(l)28710 20054 y Fw(\))29216 19505 y Fi(\003)29742 20054 y Fw(\))p Fm(\030)30816 20253 y Fv(0)31342 20054 y Fm(;)443 b(g)32769 20253 y Fk(j)33255 20054 y Fm(;)g(h)34808 20253 y Fk(l)35524 20054 y Fn(2)368 b Fl(h)37470 20253 y Fv(0)37997 20054 y Fm(:)2717 23973 y Fg(In)457 b(the)h(c)-66 b(ase)458 b(in)f(which)h Fm(m)369 b Fw(=)f(0)221 b(\()p Fm(n)370 b Fw(=)f(0\))p Fm(;)458 b Fg(we)g(r)-66 b(eplac)g(e)457 b(the)h(op)-66 b(er)g(ator)457 b(in)g(the)h(\014rst\(se)-66 b(c)g(ond\))457 b(p)-66 b(ar)g(en-)2717 25634 y(thesis)465 b(in)f(the)h(ab)-66 b(ove)464 b(by)g(identity.)2717 29447 y(Pr)-66 b(o)g(of:)606 b Fw(Remark)449 b(that)e Fn(M)16083 29646 y Fk(f)98 b(in)18028 29447 y Fw(is)448 b(the)g(algebra)h(generated)e(b)-36 b(y)448 b(1)q Fm(;)443 b(B)67 b Fw(\()p Fm(f)142 b Fw(\))p Fm(;)443 b(f)535 b Fn(2)393 b Fl(h)42188 29646 y Fv(0)43163 29447 y Fw(is)448 b(dense)g(in)g Fn(M)p Fm(:)2717 31108 y Fw(It)480 b(follo)-36 b(ws)481 b(from)f(\(4.8\))g(and)f(\(3.6\))h (that)f(an)-36 b(y)480 b Fm(B)67 b Fw(\()p Fm(g)48 b Fw(\))p Fm(;)442 b(g)495 b Fn(2)447 b Fl(h)32537 31307 y Fv(0)33542 31108 y Fw(can)480 b(b)36 b(e)479 b(written)h(as)f(linear) h(sum)f(of)2717 32768 y(four)562 b Fm(D)6635 32967 y Fk(i)7010 32768 y Fw(\()p Fm(h)p Fw(\))8771 32286 y Fv(#)9610 32768 y Fm(;)221 b(h)587 b Fn(2)f Fl(h)13692 32967 y Fv(0)14219 32768 y Fm(;)443 b(i)586 b Fw(=)g(1)p Fm(;)221 b Fw(2)p Fm(:)563 b Fw(Th)-36 b(us)561 b(an)-36 b(y)562 b(\()26971 31772 y Fh(Q)28225 32122 y Fk(m)28225 33156 y(l)11 b Fv(=1)29996 32768 y Fm(B)67 b Fw(\()p Fm(g)32180 32967 y Fk(l)32526 32768 y Fw(\)\))p Fm(\030)34106 32967 y Fv(0)34632 32768 y Fm(;)443 b(g)36059 32967 y Fk(l)36992 32768 y Fn(2)586 b Fl(h)39156 32967 y Fv(0)39682 32768 y Fm(;)444 b(l)616 b Fw(=)586 b(1)p Fm(;)221 b(:::;)g(m)564 b Fw(can)d(b)36 b(e)2717 34429 y(expressed)433 b(as)h(a)g(\014nite)f (linear)h(com)-36 b(bination)434 b(of)g(the)f(v)-36 b(ectors)434 b(of)g(the)f(form)15582 37996 y(\()16699 36282 y Fk(p)16088 36735 y Fh(Y)16120 39534 y Fk(j)51 b Fv(=1)18006 37996 y Fm(D)19086 38195 y Fv(1)19612 37996 y Fw(\()p Fm(g)20741 38195 y Fk(j)21228 37996 y Fw(\))21734 37448 y Fv(#)22573 37996 y Fw(\)\()24207 36282 y Fk(q)23585 36735 y Fh(Y)23687 39563 y Fk(l)11 b Fv(=1)25503 37996 y Fm(D)26583 38195 y Fv(2)27109 37996 y Fw(\()p Fm(h)28364 38195 y Fk(l)28710 37996 y Fw(\))29216 37448 y Fv(#)30055 37996 y Fw(\))p Fm(\030)31129 38195 y Fv(0)31655 37996 y Fm(;)443 b(g)33082 38195 y Fk(j)33569 37996 y Fm(;)g(h)35122 38195 y Fk(l)35837 37996 y Fn(2)369 b Fl(h)37784 38195 y Fv(0)38310 37996 y Fm(;)2717 42017 y Fw(where)422 b Fm(D)36 b Fw(\()p Fm(g)48 b Fw(\))9262 41535 y Fv(#)10523 42017 y Fw(is)423 b(either)f Fm(D)36 b Fw(\()p Fm(g)48 b Fw(\))421 b(or)i Fm(D)36 b Fw(\()p Fm(g)48 b Fw(\))23092 41535 y Fi(\003)23617 42017 y Fw(.)574 b(Using)423 b(Lemma)f(4.1)h(and)f(the)g(CARs)g(in)h (Prop)36 b(osition)2717 43677 y(4.1,)463 b(the)455 b(ab)36 b(o)-36 b(v)g(e)457 b(v)-36 b(ector)456 b(can)g(b)36 b(e)456 b(expressed)f(as)h(a)h(\014nite)e(linear)h(com)-36 b(bination)456 b(of)h(the)e(v)-36 b(ectors)456 b(of)2717 45338 y(the)433 b(form)11234 47977 y(\()12203 46263 y Fk(p)12677 45950 y Fb(0)11740 46715 y Fh(Y)11772 49515 y Fk(j)51 b Fv(=1)13659 47977 y Fm(D)14739 48176 y Fv(1)15264 47977 y Fw(\()p Fm(g)16393 48176 y Fk(j)16880 47977 y Fw(\))17386 47429 y Fi(\003)17912 47977 y Fw(\)\()19397 46263 y Fk(q)19849 45950 y Fb(0)18924 46715 y Fh(Y)19026 49544 y Fk(l)11 b Fv(=1)20842 47977 y Fm(D)21922 48176 y Fv(2)22448 47977 y Fw(\()p Fm(h)23703 48176 y Fk(l)24049 47977 y Fw(\))24555 47429 y Fi(\003)25081 47977 y Fw(\))p Fm(\030)26155 48176 y Fv(0)26680 47977 y Fm(;)443 b(g)28107 48176 y Fk(j)28594 47977 y Fm(;)g(h)30147 48176 y Fk(l)30862 47977 y Fn(2)369 b Fl(h)32809 48176 y Fv(0)33335 47977 y Fm(;)443 b(p)34792 47429 y Fi(0)35103 47977 y Fm(;)221 b(q)36310 47429 y Fi(0)36989 47977 y Fn(2)369 b Fo(N)295 b Fn([)g(f)p Fw(0)p Fn(g)p Fm(:)2717 51419 y Fw(The)515 b(set)f(of)h(\014nite)f(linear)h(com)-36 b(binations)514 b(of)h(the)f(v)-36 b(ectors)515 b(of)g(the)f(ab)36 b(o)-36 b(v)g(e)515 b(form)g(is)g(dense)f(in)g Fn(H)13 b Fw(.)2717 53079 y(Th)-36 b(us)425 b(the)g(decomp)36 b(osition)425 b(follo)-36 b(ws)427 b(from)f(Lemma)f(4.1)h(and)f(the)g(CARs)h(in)f (Prop)36 b(osition)425 b(4.1.)577 b Fe(\003)4668 57030 y Fw(Recall)553 b(that)d Fl(h)12333 57229 y Fv(0)13410 57030 y Fw(is)i(a)f(dense)g(subspace)f(of)i(a)f(complex)h(Hilb)36 b(ert)551 b(space)h Fl(h)p Fm(:)f Fw(Let)g Fn(F)701 b Fw(=)568 b Fn(F)132 b Fw(\()p Fl(h)p Fw(\))2717 58691 y(b)36 b(e)475 b(the)g(an)-36 b(ti-symmetric)475 b(F)-108 b(o)36 b(c)-36 b(k)476 b(space)f(o)-36 b(v)g(er)476 b Fl(h)p Fm(;)g Fw(and)e Fm(a)p Fw(\()p Fm(g)48 b Fw(\))474 b(and)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))37168 58209 y Fi(\003)37693 58691 y Fm(;)443 b(g)487 b Fn(2)440 b Fl(h)p Fm(;)476 b Fw(the)f(annihilation)2717 60351 y(and)519 b(creation)g(op)36 b(erator)519 b(on)g Fn(F)651 b Fw(resp)36 b(ectiv)-36 b(ely)-108 b(.)836 b(Denote)519 b(b)-36 b(y)519 b(\012)h(the)e(v)-72 b(acuum)519 b(v)-36 b(ector)519 b(in)g Fn(F)132 b Fm(:)519 b Fw(Let)2717 62012 y Fm(C)488 b Fw(:)394 b Fl(h)f Fn(!)h Fl(h)448 b Fw(b)36 b(e)448 b(an)g(an)-36 b(ti-unitary)447 b(op)36 b(erator.)621 b(If)448 b Fl(h)h Fw(is)f(a)g Fm(L)31231 61529 y Fv(2)31757 62012 y Fw(-space,)j(one)d(ma)-36 b(y)448 b(consider)g(that)f Fm(C)543 b Fw(is)2717 63672 y(the)488 b(complex)h(conjugation.)744 b(Denote)488 b(b)-36 b(y)488 b(\000\()p Fm(C)95 b Fw(\))488 b(the)g(second)g(quan)-36 b(tization)489 b(of)g Fm(C)23 b(:)488 b Fw(See)g(Section)2717 65332 y(5.2.1)426 b(of)f([BR].)h(Let)e Fn(F)13950 65531 y Fv(1)14475 65332 y Fm(;)444 b Fw(\012)16219 65531 y Fv(1)16745 65332 y Fm(;)f(a)18232 65531 y Fv(1)18757 65332 y Fw(\()p Fm(g)48 b Fw(\))424 b(and)g Fm(a)24067 65531 y Fv(1)24592 65332 y Fw(\()p Fm(g)48 b Fw(\))26275 64850 y Fi(\003)26800 65332 y Fm(;)443 b(g)417 b Fn(2)368 b Fl(h)425 b Fw(b)36 b(e)424 b(the)g(iden)-36 b(tical)425 b(copies)g(of)g Fn(F)132 b Fm(;)443 b Fw(\012)p Fm(;)g(a)p Fw(\()p Fm(g)48 b Fw(\))2717 66993 y(and)369 b Fm(a)p Fw(\()p Fm(g)48 b Fw(\))7548 66511 y Fi(\003)8073 66993 y Fm(;)443 b(g)416 b Fn(2)369 b Fl(h)p Fw(,)383 b(resp)36 b(ectiv)-36 b(ely)-108 b(.)557 b(Notice)370 b(that)f(\000\()p Fm(C)95 b Fw(\))p Fm(a)p Fw(\()p Fm(g)48 b Fw(\))32060 66511 y Fv(#)32898 66993 y Fw(\000\()p Fm(C)95 b Fw(\))35749 66511 y Fi(\000)p Fv(1)37375 66993 y Fw(=)368 b Fm(a)p Fw(\()p Fm(C)95 b(g)48 b Fw(\))42147 66511 y Fv(#)42986 66993 y Fm(:)369 b Fw(W)-108 b(e)370 b(write)g(that)2717 68653 y Fn(F)3672 68852 y Fv(2)4700 68653 y Fw(=)501 b(\000\()p Fm(C)95 b Fw(\))p Fn(F)132 b Fw(\(=)501 b Fn(F)132 b Fw(\),)531 b(\012)15594 68852 y Fv(2)16622 68653 y Fw(=)502 b(\012)p Fm(;)443 b(a)20562 68852 y Fv(2)21088 68653 y Fw(\()p Fm(g)48 b Fw(\))501 b(=)g Fm(a)p Fw(\()p Fm(C)95 b(g)48 b Fw(\),)531 b(and)511 b Fm(a)32359 68852 y Fv(2)32885 68653 y Fw(\()p Fm(g)48 b Fw(\))34568 68171 y Fi(\003)35595 68653 y Fw(=)501 b Fm(a)p Fw(\()p Fm(C)95 b(g)48 b Fw(\))40500 68171 y Fi(\003)41025 68653 y Fm(;)443 b(g)549 b Fn(2)502 b Fl(h)p Fw(.)813 b(Then)512 b(the)2717 70314 y(follo)-36 b(wing)436 b(an)-36 b(ti-comm)g(utation)433 b(relations)h(hold:)578 b(for)434 b(an)-36 b(y)434 b Fm(g)48 b(;)443 b(h)368 b Fn(2)h Fl(h)21039 72944 y Fn(f)p Fm(a)22386 73143 y Fv(2)22912 72944 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)25860 73143 y Fv(2)26385 72944 y Fw(\()p Fm(h)p Fw(\))28146 72396 y Fi(\003)28671 72944 y Fn(g)369 b Fw(=)g(\()p Fm(h;)221 b(g)48 b Fw(\))p Ff(1)p Fm(;)13655 b Fw(\(4.9\))21039 74937 y Fn(f)p Fm(a)22386 75136 y Fv(2)22912 74937 y Fw(\()p Fm(g)48 b Fw(\))p Fm(;)221 b(a)25860 75136 y Fv(2)26385 74937 y Fw(\()p Fm(h)p Fw(\))p Fn(g)368 b Fw(=)h(0)p Fm(:)26476 78590 y Fw(18)p eop %%Page: 19 19 19 18 bop 2717 7306 a Ff(Prop)42 b(osition)500 b(4.2)651 b Fg(L)-66 b(et)464 b Fm(U)604 b Fg(b)-66 b(e)464 b(the)g(op)-66 b(er)g(ator)464 b(de\014ne)-66 b(d)464 b(by)22173 9866 y Fm(U)508 b Fw(:)369 b Fn(H)382 b(!)369 b(F)28450 10065 y Fv(1)29271 9866 y Fn(\012)296 b(F)31555 10065 y Fv(2)6187 13251 y Fw(\()7125 11591 y Fk(m)6693 11989 y Fh(Y)6725 14788 y Fk(j)51 b Fv(=1)8612 13251 y Fm(D)9692 13450 y Fv(1)10217 13251 y Fw(\()p Fm(g)11346 13450 y Fk(j)11833 13251 y Fw(\))12339 12702 y Fi(\003)12864 13251 y Fw(\)\()14439 11591 y Fk(n)13876 11989 y Fh(Y)13978 14818 y Fk(l)11 b Fv(=1)15795 13251 y Fm(D)16875 13450 y Fv(2)17401 13251 y Fw(\()p Fm(h)18656 13450 y Fk(l)19002 13251 y Fw(\))19508 12702 y Fi(\003)20033 13251 y Fw(\))p Fm(\030)21107 13450 y Fv(0)22002 13251 y Fn(7!)369 b Fw(\()24638 11591 y Fk(m)24205 11989 y Fh(Y)24237 14788 y Fk(j)51 b Fv(=1)25903 13251 y Fw(\()p Fm(a)27092 13450 y Fv(1)27617 13251 y Fw(\()p Fm(g)28746 13450 y Fk(j)29233 13251 y Fw(\))29739 12702 y Fi(\003)30559 13251 y Fn(\012)296 b Fa(1)p Fw(\)\)\()34754 11591 y Fk(n)34191 11989 y Fh(Y)34293 14818 y Fk(l)11 b Fv(=1)35888 13251 y Fw(\()p Fm(\022)331 b Fn(\012)295 b Fm(a)39342 13450 y Fv(2)39868 13251 y Fw(\()p Fm(h)41123 13450 y Fk(l)41469 13251 y Fw(\))41975 12702 y Fi(\003)42501 13251 y Fw(\)\)\012)44452 13450 y Fv(1)45273 13251 y Fn(\012)g Fw(\012)47540 13450 y Fv(2)2717 16736 y Fg(for)486 b Fm(g)5437 16935 y Fk(j)5924 16736 y Fm(;)443 b(h)7477 16935 y Fk(l)8232 16736 y Fn(2)409 b Fl(h)10219 16935 y Fv(0)10745 16736 y Fm(;)443 b(j)484 b Fw(=)408 b(1)p Fm(;)221 b(:::;)g(m;)446 b(l)438 b Fw(=)409 b(1)p Fm(;)221 b(:::;)g(n;)446 b(m;)221 b(n)409 b Fn(2)g Fo(N)311 b Fn([)g(f)p Fw(0)p Fn(g)p Fg(,)492 b(wher)-66 b(e)486 b Fm(\022)523 b Fg(is)486 b(an)g(op)-66 b(er)g(ator)486 b(which)2717 18397 y(anti-c)-66 b(ommutes)635 b(with)h Fm(a)p Fw(\()p Fm(g)48 b Fw(\))p Fm(;)857 b(a)p Fw(\()p Fm(h)p Fw(\))20645 17915 y Fi(\003)21807 18397 y Fg(for)636 b Fm(g)48 b(;)221 b(h)687 b Fn(2)f Fl(h)29007 18596 y Fv(0)30170 18397 y Fg(and)637 b(satis\014es)f Fm(\022)36 b Fw(\012)39500 18596 y Fv(1)40714 18397 y Fw(=)686 b(\012)43351 18596 y Fv(1)43877 18397 y Fg(.)1112 b(Then)636 b Fm(U)776 b Fg(is)2717 20057 y(unitary.)2717 23541 y(Pr)-66 b(o)g(of:)577 b Fw(Notice)434 b(that)f Fm(\022)14259 23059 y Fv(2)15154 23541 y Fw(=)369 b Ff(1)434 b Fw(and)f(for)h Fm(g)48 b(;)221 b(h)433 b Fw(in)h Fl(h)26877 23740 y Fv(0)5768 26102 y Fw(\()p Fm(a)6957 26301 y Fv(1)7482 26102 y Fw(\()p Fm(g)48 b Fw(\))9165 25553 y Fi(\003)9985 26102 y Fn(\012)296 b Ff(1)p Fw(\)\()p Fm(\022)331 b Fn(\012)295 b Fm(a)16021 26301 y Fv(2)16547 26102 y Fw(\()p Fm(h)p Fw(\))18308 25553 y Fi(\003)18833 26102 y Fw(\)\012)20278 26301 y Fv(1)21099 26102 y Fn(\012)g Fw(\012)23366 26301 y Fv(2)24262 26102 y Fw(=)368 b Fn(\000)p Fw(\()p Fm(\022)332 b Fn(\012)295 b Fm(a)30130 26301 y Fv(2)30656 26102 y Fw(\()p Fm(h)p Fw(\))32417 25553 y Fi(\003)32942 26102 y Fw(\)\()p Fm(a)34637 26301 y Fv(1)35162 26102 y Fw(\()p Fm(g)48 b Fw(\))36845 25553 y Fi(\003)37665 26102 y Fn(\012)295 b Ff(1)q Fw(\)\012)41186 26301 y Fv(1)42007 26102 y Fn(\012)g Fw(\012)44274 26301 y Fv(2)44800 26102 y Fm(:)3051 b Fw(\(4.10\))2717 28662 y(Since)534 b Fm(D)7294 28861 y Fv(1)7820 28662 y Fw(\()p Fm(g)48 b Fw(\))9503 28180 y Fv(#)10875 28662 y Fw(and)534 b Fm(a)14188 28861 y Fv(1)14713 28662 y Fw(\()p Fm(g)48 b Fw(\))16396 28180 y Fv(#)17235 28662 y Fw(,)559 b(and)534 b Fm(D)21865 28861 y Fv(2)22391 28662 y Fw(\()p Fm(g)48 b Fw(\))24074 28180 y Fv(#)25446 28662 y Fw(and)534 b Fm(a)28759 28861 y Fv(2)29284 28662 y Fw(\()p Fm(g)48 b Fw(\))30967 28180 y Fv(#)32340 28662 y Fw(for)535 b Fm(g)808 b Fn(2)540 b Fl(h)37977 28861 y Fv(0)39037 28662 y Fw(satisfy)c(the)d(same)i(an)-36 b(ti-)2717 30322 y(comm)g(utation) 565 b(relations)g(resp)36 b(ectiv)-36 b(ely)566 b(b)-36 b(y)564 b(Prop)36 b(osition)566 b(4.1)f(and)f(\(4.9\),)599 b(the)564 b(unitarit)-36 b(y)564 b(of)i Fm(U)2717 31983 y Fw(follo)-36 b(ws)436 b(from)e(Lemma)f(4.1,)i(\(4.10\))f(and)f Fm(a)24127 32182 y Fk(i)24503 31983 y Fw(\()p Fm(g)48 b Fw(\)\012)27125 32182 y Fk(i)27869 31983 y Fw(=)369 b(0,)434 b Fm(i)369 b Fw(=)f(1)p Fm(;)221 b Fw(2)435 b(for)f(an)-36 b(y)434 b Fm(g)416 b Fn(2)369 b Fl(h)42635 32182 y Fv(0)43161 31983 y Fw(.)579 b Fe(\003)4668 35934 y Fw(W)-108 b(e)471 b(next)g(turn)f(to)i(the)e(sp)36 b(ectral)471 b(analysis)i(of)f Fm(H)103 b Fw(,)480 b(where)471 b Fm(H)573 b Fw(is)472 b(the)e(generator)i(of)g(the)e(sym-)2717 37594 y(metric)434 b(Mark)-36 b(o)g(vian)434 b(semigroup)g Fn(f)p Fm(T)20878 37793 y Fk(t)21274 37594 y Fn(g)21938 37793 y Fk(t)p Fi(\025)p Fv(0)23969 37594 y Fw(asso)36 b(ciated)434 b(to)g(the)f(Diric)-36 b(hlet)434 b(form)g(\()p Fn(E)119 b Fm(;)221 b(D)36 b Fw(\()p Fn(E)119 b Fw(\)\).)4668 39255 y(Let)532 b(us)g(\014rst)f(describ)36 b(e)532 b(the)g(basic)g (idea)h(of)g(the)f(pro)36 b(of)533 b(of)g(Theorem)f(3.2.)875 b(Recall)534 b(that)e(the)2717 40915 y(v)-36 b(ectors)500 b(in)f Fn(M)10376 41114 y Fk(f)98 b(in)11872 40915 y Fm(\030)12440 41114 y Fv(0)13465 40915 y Fw(can)499 b(b)36 b(e)499 b(expressed)g(as)g(\014nite)f(linear)i(com)-36 b(bination)499 b(of)h(the)e(v)-36 b(ectors)500 b(of)g(the)2717 42575 y(form)11526 44915 y(\()12644 43200 y Fk(p)12032 43653 y Fh(Y)12064 46452 y Fk(j)51 b Fv(=1)13951 44915 y Fm(D)15031 45114 y Fv(1)15557 44915 y Fw(\()p Fm(f)16704 45114 y Fk(j)17190 44915 y Fw(\))17696 44366 y Fi(\003)18222 44915 y Fw(\)\()19856 43200 y Fk(q)19234 43653 y Fh(Y)19336 46482 y Fk(l)11 b Fv(=1)21152 44915 y Fm(D)22232 45114 y Fv(2)22758 44915 y Fw(\()p Fm(h)24013 45114 y Fk(l)24359 44915 y Fw(\))24865 44366 y Fi(\003)25391 44915 y Fw(\))p Fm(\030)26465 45114 y Fv(0)26991 44915 y Fm(;)443 b(f)28436 45114 y Fk(j)28923 44915 y Fm(;)g(h)30476 45114 y Fk(l)31191 44915 y Fn(2)369 b Fl(h)33138 45114 y Fv(0)33664 44915 y Fm(;)443 b(p;)221 b(q)417 b Fn(2)369 b Fo(N)295 b Fn([)g(f)p Fw(0)p Fn(g)p Fm(:)2717 48400 y Fw(Let)464 b Fn(f)p Fm(g)6365 48599 y Fk(j)6852 48400 y Fn(g)7516 47918 y Fi(1)7516 48729 y Fk(j)51 b Fv(=1)9627 48400 y Fn(\032)421 b Fl(h)11773 48599 y Fv(0)12763 48400 y Fw(b)36 b(e)465 b(a)f(CONS)g(for)h Fl(h)p Fw(.)671 b(In)464 b(the)g(pro)36 b(of)465 b(of)g(Theorem)f(3.1,) 473 b(w)-36 b(e)465 b(sho)-36 b(w)g(ed)464 b(that)g(the)2717 50061 y(form)376 b(\()p Fn(E)119 b Fm(;)221 b Fn(M)9233 50260 y Fk(f)98 b(in)10730 50061 y Fm(\030)11298 50260 y Fv(0)11824 50061 y Fw(\))376 b(is)g(indep)36 b(enden)-36 b(t)373 b(of)k(the)e(CONS)g Fn(f)p Fm(g)30185 50260 y Fk(j)30672 50061 y Fn(g)370 b(2)e Fl(h)33652 50260 y Fv(0)34554 50061 y Fw(for)376 b Fl(h)p Fw(.)560 b(Th)-36 b(us)375 b(one)h(can)g(sho)-36 b(w)376 b(that)2717 51721 y(for)434 b(an)-36 b(y)434 b Fm(\030)f Fn(2)369 b(M)11012 51920 y Fk(f)98 b(in)12508 51721 y Fm(\030)13076 51920 y Fv(0)9041 55278 y Fn(E)119 b Fw([)p Fm(\030)64 b Fw(])1108 b(=)14930 53617 y Fi(1)14441 54016 y Fh(X)14584 56815 y Fk(j)51 b Fv(=1)16581 53470 y Fh(Z)18131 54202 y(\000)18740 55278 y Fn(k)p Fm(D)20484 55477 y Fv(1)21010 55278 y Fw(\()p Fm(A)22491 54729 y Fk(it)23206 55278 y Fm(B)24261 54729 y Fv(1)p Fk(=)p Fv(2)25728 55278 y Fm(g)26351 55477 y Fk(j)26838 55278 y Fw(\))p Fm(\030)64 b Fn(k)28640 54729 y Fv(2)29461 55278 y Fw(+)295 b Fn(k)p Fm(D)32512 55477 y Fv(2)33038 55278 y Fw(\()p Fm(A)34519 54729 y Fk(it)35234 55278 y Fm(B)36289 54729 y Fv(1)p Fk(=)p Fv(2)37756 55278 y Fm(g)38379 55477 y Fk(j)38866 55278 y Fw(\))p Fm(\030)64 b Fn(k)40668 54729 y Fv(2)41194 54202 y Fh(\001)41803 55278 y Fm(f)142 b Fw(\()p Fm(t)p Fw(\))p Fm(dt)12323 59738 y Fw(=)14930 58077 y Fi(1)14441 58476 y Fh(X)14584 61275 y Fk(j)51 b Fv(=1)16581 58662 y Fh(\000)17190 59738 y Fn(k)p Fm(D)18934 59937 y Fv(1)19460 59738 y Fw(\()p Fm(B)21021 59189 y Fv(1)p Fk(=)p Fv(2)22488 59738 y Fm(g)23111 59937 y Fk(j)23597 59738 y Fw(\))p Fm(\030)64 b Fn(k)25399 59189 y Fv(2)26221 59738 y Fw(+)295 b Fn(k)p Fm(D)29272 59937 y Fv(2)29797 59738 y Fw(\()p Fm(B)31358 59189 y Fv(1)p Fk(=)p Fv(2)32825 59738 y Fm(g)33448 59937 y Fk(j)33935 59738 y Fw(\))p Fm(\030)64 b Fn(k)35737 59189 y Fv(2)36263 58662 y Fh(\001)12323 63318 y Fw(=)1106 b Fn(h)p Fm(\030)64 b(;)16466 62982 y Fh(b)16172 63318 y Fm(H)103 b(\030)64 b Fn(i)p Fm(;)29349 b Fw(\(4.11\))2717 65879 y(where)10507 67828 y Fh(b)10213 68164 y Fm(H)472 b Fw(=)13632 66503 y Fi(1)13143 66902 y Fh(X)13286 69701 y Fk(j)51 b Fv(=1)15062 68164 y Fn(f)p Fm(D)16806 68363 y Fv(1)17332 68164 y Fw(\()p Fm(B)18893 67615 y Fv(1)p Fk(=)p Fv(2)20360 68164 y Fm(g)20983 68363 y Fk(j)21469 68164 y Fw(\))21975 67615 y Fi(\003)22501 68164 y Fm(D)23581 68363 y Fv(1)24107 68164 y Fw(\()p Fm(B)25668 67615 y Fv(1)p Fk(=)p Fv(2)27134 68164 y Fm(g)27757 68363 y Fk(j)28244 68164 y Fw(\))295 b(+)g Fm(D)31432 68363 y Fv(2)31958 68164 y Fw(\()p Fm(B)33519 67615 y Fv(1)p Fk(=)p Fv(2)34985 68164 y Fm(g)35608 68363 y Fk(j)36095 68164 y Fw(\))36601 67615 y Fi(\003)37127 68164 y Fm(D)38207 68363 y Fv(2)38732 68164 y Fw(\()p Fm(B)40293 67615 y Fv(1)p Fk(=)p Fv(2)41760 68164 y Fm(g)42383 68363 y Fk(j)42870 68164 y Fw(\))p Fn(g)2717 71948 y Fw(as)388 b(a)h(bilinear)f(form)g(on)g Fn(M)16339 72147 y Fk(f)98 b(in)17835 71948 y Fm(\030)18403 72147 y Fv(0)19131 71948 y Fn(\002)202 b(M)21961 72147 y Fk(f)98 b(in)23458 71948 y Fm(\030)24026 72147 y Fv(0)24552 71948 y Fw(.)563 b(If)388 b(one)g(can)g(sho)-36 b(w)388 b(that)f Fn(M)38952 72147 y Fk(f)98 b(in)40449 71948 y Fm(\030)41017 72147 y Fv(0)41912 71948 y Fn(\032)369 b Fm(D)36 b Fw(\()p Fm(H)103 b Fw(\))p Fm(;)442 b(H)471 b Fw(=)50649 71612 y Fh(b)50356 71948 y Fm(H)2717 73608 y Fw(on)409 b Fn(M)6094 73807 y Fk(f)98 b(in)7591 73608 y Fm(\030)8159 73807 y Fv(0)8684 73608 y Fm(;)410 b Fw(and)e(that)h Fn(M)16348 73807 y Fk(f)98 b(in)17844 73608 y Fm(\030)18412 73807 y Fv(0)19347 73608 y Fw(is)410 b(a)f(core)g(for)h Fm(H)103 b Fw(,)413 b(then)c(one)g(exp)36 b(ects)409 b(that)f(the)h(sp)36 b(ectrum)408 b(of)i Fm(H)2717 75269 y Fw(can)434 b(b)36 b(e)433 b(analyzed)h(completely)-108 b(.)579 b(The)434 b(follo)-36 b(wing)436 b(is)d(one)h(of)g(main)g (results)f(in)h(this)f(section:)26476 78590 y(19)p eop %%Page: 20 20 20 19 bop 2717 7306 a Ff(Theorem)499 b(4.2)1980 b Fg(\(a\))465 b(L)-66 b(et)17712 6970 y Fw(~)17374 7306 y Fm(H)472 b Fw(:)369 b Fn(M)21249 7505 y Fk(f)98 b(in)22745 7306 y Fm(\030)23313 7505 y Fv(0)24208 7306 y Fn(!)370 b(H)477 b Fg(b)-66 b(e)464 b(the)h(op)-66 b(er)g(ator)464 b(de\014ne)-66 b(d)464 b(by)11882 10955 y Fw(~)11545 11291 y Fm(H)12725 10215 y Fh(\000)13988 9631 y Fk(m)13556 10029 y Fh(Y)13566 12819 y Fk(p)p Fv(=1)15474 11291 y Fm(D)16554 11490 y Fv(1)17080 11291 y Fw(\()p Fm(f)18227 11490 y Fk(p)18756 11291 y Fw(\))19262 10743 y Fi(\003)19788 10215 y Fh(\001\000)21790 9631 y Fk(n)21227 10029 y Fh(Y)21248 12819 y Fk(q)32 b Fv(=1)23145 11291 y Fm(D)24225 11490 y Fv(2)24751 11291 y Fw(\()p Fm(h)26006 11490 y Fk(q)26513 11291 y Fw(\))27019 10743 y Fi(\003)27545 10215 y Fh(\001)28153 11291 y Fm(\030)28721 11490 y Fv(0)48212 11291 y Fw(\(4.12\))9427 15975 y(=)12088 14314 y Fk(m)11545 14713 y Fh(X)11647 17542 y Fk(k)24 b Fv(=1)13685 14899 y Fh(\000)14516 14314 y Fk(k)g Fi(\000)p Fv(1)14525 14713 y Fh(Y)14535 17502 y Fk(p)p Fv(=1)16453 15975 y Fm(D)17533 16174 y Fv(1)18058 15975 y Fw(\()p Fm(f)19205 16174 y Fk(p)19734 15975 y Fw(\))20240 15426 y Fi(\003)20766 14899 y Fh(\001)21375 15975 y Fm(D)22455 16174 y Fv(1)22981 15975 y Fw(\()p Fm(B)67 b(f)25183 16174 y Fk(k)25751 15975 y Fw(\))26257 15426 y Fi(\003)26783 14899 y Fh(\000)28658 14314 y Fk(m)28225 14713 y Fh(Y)27613 17542 y Fk(p)p Fv(=)p Fk(k)24 b Fv(+1)30756 15975 y Fm(D)31836 16174 y Fv(1)32362 15975 y Fw(\()p Fm(f)33509 16174 y Fk(p)34038 15975 y Fw(\))34544 15426 y Fi(\003)35069 14899 y Fh(\001\000)37072 14314 y Fk(n)36508 14713 y Fh(Y)36530 17502 y Fk(q)32 b Fv(=1)38427 15975 y Fm(D)39507 16174 y Fv(2)40033 15975 y Fw(\()p Fm(h)41288 16174 y Fk(q)41795 15975 y Fw(\))42301 15426 y Fi(\003)42826 14899 y Fh(\001)43435 15975 y Fm(\030)44003 16174 y Fv(0)11545 20698 y Fw(+)13452 19037 y Fk(n)12778 19436 y Fh(X)12880 22265 y Fk(k)24 b Fv(=1)14918 19622 y Fh(\000)16181 19037 y Fk(m)15749 19436 y Fh(Y)15759 22225 y Fk(p)p Fv(=1)17667 20698 y Fm(D)18747 20897 y Fv(1)19273 20698 y Fw(\()p Fm(f)20420 20897 y Fk(p)20949 20698 y Fw(\))21455 20149 y Fi(\003)21981 19622 y Fh(\001)o(\000)23420 19037 y Fk(k)g Fi(\000)p Fv(1)23429 19436 y Fh(Y)23450 22225 y Fk(q)32 b Fv(=1)25357 20698 y Fm(D)26437 20897 y Fv(2)26963 20698 y Fw(\()p Fm(h)28218 20897 y Fk(q)28724 20698 y Fw(\))29230 20149 y Fi(\003)29756 19622 y Fh(\001)30365 20698 y Fm(D)31445 20897 y Fv(2)31970 20698 y Fw(\()p Fm(B)67 b(h)34280 20897 y Fk(k)34849 20698 y Fw(\))35355 20149 y Fi(\003)35881 19622 y Fh(\000)37875 19037 y Fk(n)37312 19436 y Fh(Y)36711 22265 y Fk(q)32 b Fv(=)p Fk(k)24 b Fv(+1)39832 20698 y Fm(D)40912 20897 y Fv(2)41438 20698 y Fw(\()p 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y Fm(H)637 b Fg(is)533 b(essential)66 b(ly)534 b(self-adjoint)e(and)i(the)f(self-adjoint)g (extension)f(denote)-66 b(d)533 b(by)45640 32307 y Fw(~)45303 32643 y Fm(H)636 b Fg(again)533 b(is)2717 34304 y(e)-66 b(qual)465 b(to)g(the)f(Dirichlet)g(op)-66 b(er)g(ator)464 b Fm(H)103 b Fg(.)2717 38399 y(Pr)-66 b(o)g(of:)1300 b Fw(\(a\))433 b(Let)g(\()p Fn(E)13072 38598 y Fv(1)13598 38399 y Fm(;)221 b Fn(M)15775 38598 y Fk(f)98 b(in)17272 38399 y Fm(\030)17840 38598 y Fv(0)18366 38399 y Fw(\))433 b(b)36 b(e)434 b(the)f(form)h(giv)-36 b(en)434 b(b)-36 b(y)19712 42384 y Fn(E)20413 42583 y Fv(1)20939 42384 y Fw([)p Fm(\030)64 b Fw(])370 b(=)24533 40724 y Fi(1)24044 41122 y Fh(X)24186 43922 y Fk(j)51 b Fv(=1)26184 42384 y Fn(k)p Fm(D)27928 42583 y Fv(1)28454 42384 y Fw(\()p Fm(B)30148 41474 y Fd(1)p 30148 41635 406 40 v 30148 42183 a(2)30742 42384 y Fm(g)31365 42583 y Fk(j)31851 42384 y Fw(\))p Fm(\030)64 b Fn(k)33653 41836 y Fv(2)34179 42384 y Fm(;)2717 46883 y Fw(where)481 b Fn(f)p Fm(g)7809 47082 y Fk(j)8296 46883 y Fn(g)p Fm(;)443 b(g)10387 47082 y Fk(j)11323 46883 y Fn(2)449 b Fl(h)13350 47082 y Fv(0)14357 46883 y Fw(is)481 b(a)g(CONS)f(for)h Fl(h)p Fw(.)721 b(W)-108 b(e)480 b(write)i(that)33640 46547 y(~)33303 46883 y Fm(H)551 b Fw(=)36731 46547 y(~)36393 46883 y Fm(H)37471 47082 y Fv(1)38324 46883 y Fw(+)40000 46547 y(~)39663 46883 y Fm(H)40741 47082 y Fv(2)41266 46883 y Fw(,)493 b(where)481 b(the)f(image)2717 48544 y(under)6762 48208 y(~)6425 48544 y Fm(H)7503 48743 y Fv(1)8029 48544 y Fw(\(resp.)11774 48208 y(~)11437 48544 y Fm(H)12515 48743 y Fv(2)13040 48544 y Fw(\))456 b(is)h(de\014ned)e(b)-36 b(y)456 b(the)g(\014rst)g(\(resp.)646 b(second\))456 b(v)-36 b(ector)457 b(in)f(the)g(righ)-36 b(t)457 b(hand)e(side)2717 50204 y(in)434 b(\(4.12\).)579 b(The)433 b(CARs)i(in)e(Prop)36 b(osition)434 b(4.1)h(\(a\))e(and)g(Lemma)h(4.1)g(imply)g(that)11882 54150 y Fn(E)12583 54349 y Fv(1)13109 54150 y Fw([\()14409 52489 y Fk(m)13976 52888 y Fh(Y)13987 55677 y Fk(p)p Fv(=1)15895 54150 y Fm(D)16975 54349 y Fv(1)17501 54150 y Fw(\()p Fm(f)18648 54349 y Fk(p)19176 54150 y Fw(\))19682 53601 y Fi(\003)20208 54150 y Fw(\))p Fm(\030)21282 54349 y Fv(0)21808 54150 y Fw(])11882 58600 y(=)13752 56939 y Fi(1)13263 57338 y Fh(X)13405 60137 y Fk(j)51 b Fv(=1)15946 56939 y Fk(m)15403 57338 y Fh(X)15524 60127 y Fk(p)p Fv(=1)18086 56939 y Fk(m)17543 57338 y Fh(X)17675 60127 y Fk(q)32 b Fv(=1)19462 58600 y Fw(\()p Fn(\000)p Fw(1\))22157 58051 y Fk(p)p Fv(+)p Fk(q)23869 58600 y Fw(\()p Fm(B)25563 57689 y Fd(1)p 25563 57850 V 25563 58399 a(2)26157 58600 y Fm(f)26798 58799 y Fk(p)27327 58600 y Fm(;)221 b(g)28532 58799 y Fk(j)29020 58600 y Fw(\)\()p Fm(g)30655 58799 y Fk(j)31141 58600 y Fm(;)g(B)32911 57689 y Fd(1)p 32912 57850 V 32912 58399 a(2)33506 58600 y Fm(f)34147 58799 y Fk(q)34654 58600 y Fw(\))p Fm(G)p Fw(\()p Fm(f)37333 58799 y Fv(1)37859 58600 y Fm(;)g(:::;)g(f)40747 58799 y Fk(m)41637 58600 y Fw(;)g Fm(p;)g(q)48 b Fw(\))2717 62804 y(where)9421 65793 y Fm(G)p Fw(\()p Fm(f)11594 65992 y Fv(1)12120 65793 y Fm(;)221 b(:::;)g(f)15008 65992 y Fk(m)15898 65793 y Fw(;)g Fm(p;)g(q)48 b Fw(\))6941 68986 y(:=)1107 b Fn(h)9938 67511 y Fh(\020)10975 67271 y Fk(p)p Fi(\000)p Fv(1)10965 67724 y Fh(Y)10952 70513 y Fk(\034)102 b Fv(=1)12896 68986 y Fm(D)13976 69185 y Fv(1)14502 68986 y Fw(\()p Fm(f)15649 69185 y Fk(\034)16224 68986 y Fw(\))16730 68437 y Fi(\003)17255 67511 y Fh(\021)q(\020)20111 67325 y Fk(m)19678 67724 y Fh(Y)19063 70513 y Fk(\034)g Fv(=)p Fk(p)p Fv(+1)22212 68986 y Fm(D)23292 69185 y Fv(1)23818 68986 y Fw(\()p Fm(f)24965 69185 y Fk(\034)25540 68986 y Fw(\))26046 68437 y Fi(\003)26572 67511 y Fh(\021)27365 68986 y Fm(\030)27933 69185 y Fv(0)28459 68986 y Fm(;)29041 67511 y Fh(\020)30090 67271 y Fk(q)32 b Fi(\000)p Fv(1)30069 67724 y Fh(Y)30056 70513 y Fk(\034)102 b Fv(=1)32000 68986 y Fm(D)33080 69185 y Fv(1)33606 68986 y Fw(\()p Fm(f)34753 69185 y Fk(\034)35328 68986 y Fw(\))35834 68437 y Fi(\003)36359 67511 y Fh(\021)q(\020)39204 67325 y Fk(m)38772 67724 y Fh(Y)38167 70513 y Fk(\034)g Fv(=)p Fk(q)32 b Fv(+1)41295 68986 y Fm(D)42375 69185 y Fv(1)42900 68986 y Fw(\()p Fm(f)44047 69185 y Fk(\034)44622 68986 y Fw(\))45128 68437 y Fi(\003)45654 67511 y Fh(\021)46447 68986 y Fm(\030)47015 69185 y Fv(0)47541 68986 y Fn(i)p Fm(:)26476 78590 y Fw(20)p eop %%Page: 21 21 21 20 bop 2717 7306 a Fw(Using)434 b(the)f(P)-36 b(arserv)-72 b(al)434 b(relations)g(and)g(the)f(fact)h(that)9034 9327 y Fk(m)8490 9726 y Fh(X)8612 12515 y Fk(p)p Fv(=1)10409 10988 y Fw(\()p Fn(\000)p Fw(1\))13104 10439 y Fk(p)p Fv(+)p Fk(q)14817 10988 y Fw(\()p Fm(f)15964 11187 y Fk(p)16493 10988 y Fm(;)221 b(B)67 b(f)18771 11187 y Fk(q)19279 10988 y Fw(\))p Fm(G)p Fw(\()p Fm(f)21958 11187 y Fv(1)22483 10988 y Fm(;)221 b(:::;)g(f)25371 11187 y Fk(m)26261 10988 y Fw(;)g Fm(p;)g(q)48 b Fw(\))8490 15681 y(=)369 b(\()p Fn(\000)p Fw(1\))12566 15133 y Fk(q)32 b Fi(\000)p Fv(1)14276 15681 y Fn(h)p Fm(D)15873 15880 y Fv(1)16398 15681 y Fw(\()p Fm(B)67 b(f)18600 15880 y Fk(q)19107 15681 y Fw(\))19613 14605 y Fh(\000)20876 14021 y Fk(m)20443 14419 y Fh(Y)20454 17209 y Fk(p)p Fv(=1)22362 15681 y Fm(D)23442 15880 y Fv(1)23968 15681 y Fw(\()p Fm(f)25115 15880 y Fk(p)25644 15681 y Fw(\))26150 15133 y Fi(\003)26675 14605 y Fh(\001)27284 15681 y Fm(\030)27852 15880 y Fv(0)28378 15681 y Fm(;)28960 14207 y Fh(\020)30009 13966 y Fk(q)32 b Fi(\000)p Fv(1)29988 14419 y Fh(Y)29975 17209 y Fk(\034)102 b Fv(=1)31919 15681 y Fm(D)32999 15880 y Fv(1)33525 15681 y Fw(\()p Fm(f)34672 15880 y Fk(\034)35247 15681 y Fw(\))35753 15133 y Fi(\003)36278 14207 y Fh(\021)q(\020)39123 14021 y Fk(m)38691 14419 y Fh(Y)38086 17209 y Fk(\034)g Fv(=)p Fk(q)32 b Fv(+1)41214 15681 y Fm(D)42294 15880 y Fv(1)42819 15681 y Fw(\()p Fm(f)43966 15880 y Fk(\034)44541 15681 y Fw(\))45047 15133 y Fi(\003)45573 14207 y Fh(\021)46366 15681 y Fm(\030)46934 15880 y Fv(0)47460 15681 y Fn(i)8490 20375 y Fw(=)369 b Fn(h)10388 19299 y Fh(\000)11650 18714 y Fk(m)11218 19113 y Fh(Y)11228 21902 y Fk(p)p Fv(=1)13137 20375 y Fm(D)14217 20574 y Fv(1)14742 20375 y Fw(\()p Fm(f)15889 20574 y Fk(p)16418 20375 y Fw(\))16924 19826 y Fi(\003)17450 19299 y Fh(\001)18059 20375 y Fm(\030)18627 20574 y Fv(0)19153 20375 y Fm(;)19735 18900 y Fh(\020)20784 18660 y Fk(q)32 b Fi(\000)p Fv(1)20762 19113 y Fh(Y)20750 21902 y Fk(\034)102 b Fv(=1)22693 20375 y Fm(D)23773 20574 y Fv(1)24299 20375 y Fw(\()p Fm(f)25446 20574 y Fk(\034)26021 20375 y Fw(\))26527 19826 y Fi(\003)27053 18900 y Fh(\021)27846 20375 y Fm(D)28926 20574 y Fv(1)29452 20375 y Fw(\()p Fm(B)67 b(f)31654 20574 y Fk(q)32161 20375 y Fw(\))32667 19826 y Fi(\003)33193 18900 y Fh(\020)35244 18714 y Fk(m)34812 19113 y Fh(Y)34208 21902 y Fk(\034)102 b Fv(=)p Fk(q)32 b Fv(+1)37335 20375 y Fm(D)38415 20574 y Fv(1)38940 20375 y Fw(\()p Fm(f)40087 20574 y Fk(\034)40663 20375 y Fw(\))41169 19826 y Fi(\003)41694 18900 y Fh(\021)42487 20375 y Fm(\030)43055 20574 y Fv(0)43581 20375 y Fn(i)p Fm(;)2717 24378 y Fw(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)9470 27802 y Fn(E)10171 28001 y Fv(1)10697 27802 y Fw([\()11997 26141 y Fk(m)11564 26540 y Fh(Y)11575 29329 y Fk(p)p Fv(=1)13483 27802 y Fm(D)14563 28001 y Fv(1)15089 27802 y Fw(\()p Fm(f)16236 28001 y Fk(p)16765 27802 y Fw(\))17271 27253 y Fi(\003)17796 27802 y Fw(\))p Fm(\030)18870 28001 y Fv(0)19396 27802 y Fw(])9470 32495 y(=)11394 30835 y Fk(m)10851 31233 y Fh(X)10983 34023 y Fk(q)32 b Fv(=1)12770 32495 y Fn(h)13287 31419 y Fh(\000)14549 30835 y Fk(m)14117 31233 y Fh(Y)14127 34023 y Fk(p)p Fv(=1)16035 32495 y Fm(D)17115 32694 y Fv(1)17641 32495 y Fw(\()p Fm(f)18788 32694 y Fk(p)19317 32495 y Fw(\))19823 31947 y Fi(\003)20349 31419 y Fh(\001)20957 32495 y Fm(\030)21525 32694 y Fv(0)22051 32495 y Fm(;)22633 31021 y Fh(\020)23683 30781 y Fk(q)g Fi(\000)p Fv(1)23661 31233 y Fh(Y)23649 34023 y Fk(\034)102 b Fv(=1)25592 32495 y Fm(D)26672 32694 y Fv(1)27198 32495 y Fw(\()p Fm(f)28345 32694 y Fk(\034)28920 32495 y Fw(\))29426 31947 y Fi(\003)29952 31021 y Fh(\021)30745 32495 y Fm(D)31825 32694 y Fv(1)32351 32495 y Fw(\()p Fm(B)67 b(f)34553 32694 y Fk(q)35060 32495 y Fw(\))35566 31947 y Fi(\003)36092 31021 y Fh(\020)38143 30835 y Fk(m)37711 31233 y Fh(Y)37106 34023 y Fk(\034)102 b Fv(=)p Fk(q)32 b Fv(+1)40234 32495 y Fm(D)41314 32694 y Fv(1)41839 32495 y Fw(\()p Fm(f)42986 32694 y Fk(\034)43561 32495 y Fw(\))44067 31947 y Fi(\003)44593 31021 y Fh(\021)45386 32495 y Fm(\030)45954 32694 y Fv(0)46480 32495 y Fn(i)9470 36945 y Fw(=)369 b Fn(h)11368 35869 y Fh(\000)12630 35285 y Fk(m)12198 35683 y Fh(Y)12209 38473 y Fk(p)p Fv(=1)14117 36945 y Fm(D)15197 37144 y Fv(1)15722 36945 y Fw(\()p Fm(f)16869 37144 y Fk(p)17398 36945 y Fw(\))17904 36397 y Fi(\003)18430 35869 y Fh(\001)19039 36945 y Fm(\030)19607 37144 y Fv(0)20133 36945 y Fm(;)21052 36609 y Fw(~)20715 36945 y Fm(H)21793 37144 y Fv(1)22319 35869 y Fh(\000)23581 35285 y Fk(m)23149 35683 y Fh(Y)23160 38473 y Fk(p)p Fv(=1)25068 36945 y Fm(D)26148 37144 y Fv(1)26673 36945 y Fw(\()p Fm(f)27820 37144 y Fk(p)28349 36945 y Fw(\))28855 36397 y Fi(\003)29381 35869 y Fh(\001)29990 36945 y Fm(\030)30558 37144 y Fv(0)31084 36945 y Fn(i)p Fm(:)2717 41244 y Fw(Notice)482 b(that)10049 40908 y(~)9712 41244 y Fm(H)10790 41443 y Fv(2)11765 41244 y Fw(=)13564 40908 y(~)13226 41244 y Fm(H)430 b Fn(\000)16432 40908 y Fw(~)16095 41244 y Fm(H)17173 41443 y Fv(1)18179 41244 y Fw(comm)-36 b(utes)480 b(with)h Fm(D)28429 41443 y Fv(1)28955 41244 y Fw(\()p Fm(f)142 b Fw(\))30750 40762 y Fi(\003)31756 41244 y Fw(for)482 b(an)-36 b(y)481 b Fm(f)591 b Fn(2)450 b Fl(h)39555 41443 y Fv(0)40562 41244 y Fw(b)-36 b(y)480 b(\(4.12\).)722 b(Th)-36 b(us)480 b(b)-36 b(y)2717 42904 y(the)433 b(p)36 b(olarization)435 b(iden)-36 b(tit)g(y)-108 b(,)433 b(w)-36 b(e)434 b(pro)-36 b(v)g(ed)433 b(that)21910 45703 y Fn(E)22611 45902 y Fv(1)23137 45703 y Fw(\()p Fm(\021)48 b(;)221 b(\030)64 b Fw(\))369 b(=)f Fn(h)p Fm(\021)48 b(;)29928 45367 y Fw(~)29591 45703 y Fm(H)30669 45902 y Fv(1)31194 45703 y Fm(\030)64 b Fn(i)2717 48501 y Fw(for)434 b(an)-36 b(y)434 b Fm(\021)48 b(;)443 b(\030)433 b Fn(2)368 b(M)12505 48700 y Fk(f)98 b(in)14002 48501 y Fm(\030)14570 48700 y Fv(0)15096 48501 y Fw(.)578 b(The)434 b(metho)36 b(d)433 b(similar)h(to)g(that)f(used)g(in)g(the)g(ab)36 b(o)-36 b(v)g(e)435 b(implies)f(that)21910 51300 y Fn(E)22611 51499 y Fv(2)23137 51300 y Fw(\()p Fm(\021)48 b(;)221 b(\030)64 b Fw(\))369 b(=)f Fn(h)p Fm(\021)48 b(;)29928 50964 y Fw(~)29591 51300 y Fm(H)30669 51499 y Fv(2)31194 51300 y Fm(\030)64 b Fn(i)2717 54098 y Fw(for)434 b(an)-36 b(y)434 b Fm(\021)48 b(;)443 b(\030)433 b Fn(2)368 b(M)12505 54297 y Fk(f)98 b(in)14002 54098 y Fm(\030)14570 54297 y Fv(0)15096 54098 y Fw(.)578 b(This)434 b(pro)-36 b(v)g(ed)433 b(the)g(part)g(\(a\))h(of)g(the)f(theorem.)4668 55759 y(\(b\))g(By)h(Prop)36 b(osition)434 b(4.2,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)16467 58557 y Fm(U)17826 58221 y Fw(~)17489 58557 y Fm(H)102 b(U)19691 58009 y Fi(\000)p Fv(1)21318 58557 y Fw(=)368 b Fm(d)p Fw(\000)24187 58756 y Fv(1)24713 58557 y Fw(\()p Fm(B)67 b Fw(\))295 b Fn(\012)g Ff(1)g Fw(+)g Ff(1)h Fn(\012)f Fm(d)p Fw(\000)34612 58756 y Fv(2)35137 58557 y Fw(\()p Fm(B)67 b Fw(\))p Fm(;)10647 b Fw(\(4.13\))2717 61356 y(where)458 b(eac)-36 b(h)457 b Fm(i;)443 b(i)410 b Fw(=)g(1)p Fm(;)221 b Fw(2)p Fm(;)459 b(d)p Fw(\000)17164 61555 y Fk(i)17539 61356 y Fw(\()p Fm(B)67 b Fw(\))458 b(is)g(the)f(second)g(quan)-36 b(tization)458 b(of)h Fm(B)525 b Fw(on)457 b Fn(F)41225 61555 y Fk(i)41601 61356 y Fm(:)h Fw(W)-108 b(e)457 b(remark)i(that) 2717 63016 y Fm(d)p Fw(\000)4206 63215 y Fv(2)4732 63016 y Fw(\()p Fm(B)67 b Fw(\))530 b(is)i(an)-36 b(ti-unitary)530 b(equiv)-72 b(alen)-36 b(t)532 b(to)g Fm(d)p Fw(\000)25511 63215 y Fv(1)26036 63016 y Fw(\()p Fm(B)67 b Fw(\).)871 b(By)532 b(Assumption)e(3.1)i(\(d\),)555 b(an)-36 b(y)532 b Fm(f)677 b Fn(2)534 b Fl(h)49604 63215 y Fv(0)50662 63016 y Fw(is)2717 64676 y(an)446 b(analytic)h(v)-36 b(ector)447 b(for)f Fm(B)67 b Fw(,)450 b(and)445 b(so)i(it)f(is)g(easy) 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Fv(0)32797 67997 y Fw(b)-36 b(y)447 b(\(4.12\))g(and)41267 67661 y(~)40929 67997 y Fm(H)494 b Fw(=)391 b Fm(H)549 b Fw(on)447 b Fn(M)48946 68196 y Fk(f)98 b(in)50442 67997 y Fm(\030)51010 68196 y Fv(0)2717 69658 y Fw(b)-36 b(y)485 b(the)f(part)h(\(a\))g(of)h(the)e(theorem,)498 b(it)485 b(follo)-36 b(ws)487 b(from)e(Corollary)i(2)e(of)h(Theorem)f(X.)g(39)h (of)g([RS])2717 71318 y(that)5873 70982 y(~)5535 71318 y Fm(H)536 b Fw(and)433 b Fm(H)536 b Fw(are)434 b(essen)-36 b(tially)435 b(self-adjoin)-36 b(t)434 b(on)f Fn(M)29781 71517 y Fk(f)98 b(in)31278 71318 y Fm(\030)31846 71517 y Fv(0)32372 71318 y Fw(,)433 b(and)h(so)37630 70982 y(~)37292 71318 y Fm(H)472 b Fw(=)368 b Fm(H)30 b(:)434 b Fe(\003)4668 75269 y Fw(Finally)h(w)-36 b(e)434 b(can)f(pro)36 b(duce)433 b(the)g(pro)36 b(of)434 b(of)h(Theorem)e(3.2.)26476 78590 y(21)p eop %%Page: 22 22 22 21 bop 2717 7306 a Ff(Pro)42 b(of)498 b(of)g(Theorem)h(3.2)1735 b Fw(\(a\))434 b(follo)-36 b(ws)435 b(from)f(Theorem)g(4.2.)2717 8966 y(\(b\))f(Recall)i(0)369 b Fm(<)g(A)g Fn(\024)g Fm(\013)8 b Ff(1)q Fw(.)578 b(Let)18913 12686 y Fm(m)369 b Fw(=)21801 10813 y Fh(\032)23350 11884 y Fm(\013)24185 11402 y Fi(\000)p Fv(1)p Fk(=)p Fv(2)26680 11884 y Fw(+)295 b Fm(\013)28822 11402 y Fv(1)p Fk(=)p Fv(2)31407 11884 y Fm(\013)378 b(<)368 b Fw(1)26495 13489 y(2)4251 b Fm(\013)378 b Fn(\025)369 b Fw(1)2717 16690 y(Since)433 b Fm(B)j Fw(=)369 b Fm(A)9893 16199 y Fi(\000)10758 15837 y Fd(1)p 10758 15998 406 40 v 10758 16547 a(2)11647 16690 y Fw(+)295 b Fm(A)14062 15837 y Fd(1)p 14062 15998 V 14062 16547 a(2)14656 16690 y Fw(,)23167 18350 y(inf)314 b Fm(\033)48 b Fw(\()p Fm(B)67 b Fw(\))369 b Fn(\025)g Fm(m:)2717 20730 y Fw(It)414 b(follo)-36 b(ws)415 b(from)f(the)f(ab)36 b(o)-36 b(v)g(e)414 b(lo)-36 b(w)g(er)414 b(b)36 b(ound)412 b(and)h(\(4.13\))h(that)f(zero)h(is)g(a)f(simple)h(eigen)-36 b(v)-72 b(alue)414 b(with)2717 22391 y(eigen)-36 b(v)g(ector)434 b Fm(\030)10041 22590 y Fv(0)11001 22391 y Fw(and)19698 24051 y(inf)92 b(\()p Fm(\033)48 b Fw(\()p Fm(H)103 b Fw(\))294 b Fn(\000)i(f)p Fw(0)p Fn(g)p Fw(\))369 b Fn(\025)g Fm(m)g Fn(\025)g Fw(2)p Fm(:)2717 26431 y Fw(This)434 b(completes)g(the)f(pro)36 b(of)434 b(of)g(the)f(theorem.)1301 b Fe(\003)4668 28091 y Ff(Ac)-42 b(kno)g(wledgemen)g(t)454 b Fw(:)612 b(This)451 b(w)-36 b(ork)451 b(w)-36 b(as)451 b(supp)36 b(orted)449 b(in)i(part)f(b)-36 b(y)450 b(Korea)h(Researc)-36 b(h)451 b(F)-108 b(oun-)2717 29752 y(dation)434 b (\(KRF-2001-005-D00010\),)i(Korean)e(Ministry)g(of)g(Education.)2717 34189 y Fp(References)2717 37165 y Fw([AFL])651 b(L.)435 b(Accaradi,)i(A.)f(F)-108 b(rigerio,)436 b(and)f(J.)h(T.)g(Lewis,)h (Quan)-36 b(tum)434 b(sto)36 b(c)-36 b(hastic)436 b(pro)36 b(cesses,)436 b Fg(Publ.)5391 38825 y(R)-66 b(es.)464 b(Inst.)f(Math.)i(Sci.)432 b Ff(18)p Fw(,)i(97-133\(1982\).)2717 41593 y([AH-K])651 b(S.)403 b(Alb)36 b(ev)-36 b(erio)405 b(and)e(R.)h(H\034egh-Krohn,)410 b(Diric)-36 b(hlet)404 b(forms)g(and)f(Mark)-36 b(o)g(vian)405 b(semigroups)5391 43253 y(on)433 b Fm(C)8223 42771 y Fi(\003)8749 43253 y Fw(-algebras,)h Fg(Comm.)464 b(Math.)g(Phys.)p Ff(56)p Fw(,)435 b(173-187)g(\(1977\).)2717 46021 y([A)-145 b(W])652 b(H.)367 b(Araki)h(and)f(W.)h(Wyss,)381 b(Represen)-36 b(tations)367 b(of)g(canonical)h(an)-36 b(ticomm)g(utation)367 b(relations,)5391 47681 y Fg(Helv.)463 b(Phys.)j(A)-66 b(cta.)431 b Ff(37)p Fw(,)k(136-159)g(\(1964\).)2717 50448 y([Acc])651 b(L.)434 b(Accardi,)g(T)-108 b(opics)434 b(in)f(quan)-36 b(tum)433 b(probabilit)-36 b(y)-108 b(,)434 b Fg(Phys.R)-66 b(ep.)432 b Ff(77)p Fw(,)i(169-192)i(\(1981\).)2717 53216 y([Ara])651 b(H.)597 b(Araki,)637 b(Some)596 b(prop)36 b(erties)596 b(of)g(mo)36 b(dular)596 b(conjugation)h(op)36 b(erator)596 b(of)h(v)-36 b(on)596 b(Neumann)5391 54876 y(algebras)540 b(ans)g(noncomm)-36 b(utativ)g(e)539 b(Radon-Nik)-36 b(o)36 b(dym)540 b(theorem)g(with)f(c)-36 b(hain)540 b(rule,)566 b Fg(Paci\014c)5391 56537 y(J.)464 b(Math.)433 b Ff(50)h Fw(\(2\),)g(309-354)h(\(1974\).)2717 59304 y([BK])651 b(C.)438 b(Bahn)f(and)f(C.)i(K.)f(Ko,)i(Construction)e(of)g (un)-36 b(b)36 b(ounded)435 b(Diric)-36 b(hlet)437 b(forms)h(on)f (standard)5391 60964 y(forms)d(of)g(v)-36 b(on)434 b(Neumann)f (Algebras,)h Fg(J.)464 b(Kor)-66 b(e)g(an)464 b(Math.)g(So)-66 b(c.)432 b Ff(39)i Fw(\(6\))g(931-951)h(\(2002\).)2717 63732 y([BKP])651 b(C.)393 b(Bahn,)400 b(C.)392 b(K.)g(Ko)h(amd)e(Y.)h (M.)g(P)-36 b(ark,)401 b(Diric)-36 b(hlet)392 b(forms)h(and)e (symmetric)h(Mark)-36 b(o)g(vian)5391 65392 y(semigroups)451 b(on)g(CCR)g(Algebras)h(with)f(quasi-free)g(states,)456 b Fg(J.)480 b(Math.)h(Phys.)p Fw(,)456 b Ff(44)p Fw(,)g(723-753)5391 67053 y(\(2003\).)2717 69820 y([BR])651 b(O.)453 b(Bratteli)h(and)e(D.) h(W.)h(Robinson,)k Fg(Op)-66 b(er)g(ator)482 b(algebr)-66 b(as)482 b(and)h(quantum)f(statistic)-66 b(al)482 b(me-)5391 71481 y(chanics)p Fw(,)433 b(Springer-V)-108 b(erlag,)433 b(New)h(Y)-108 b(ork-Heidelb)36 b(erg-Berlin,)434 b(v)-36 b(ol)434 b(I)g(1979,)h(v)-36 b(ol.)435 b(I)36 b(I)434 b(\(1981\).)26476 78590 y(22)p eop %%Page: 23 23 23 22 bop 2717 7306 a Fw([CFL])651 b(F.)480 b(Cipriani,)493 b(F.)481 b(F)-108 b(agnola)481 b(and)f(J.)h(M.)f(Lindsa)-36 b(y)-108 b(,)492 b(Sp)36 b(ectral)480 b(Analysis)i(and)d(F)-108 b(eller)481 b(Prop-)5391 8966 y(erties)603 b(for)g(Quan)-36 b(tum)601 b(Ornstein-Uhlen)-36 b(b)36 b(ec)-36 b(k)601 b(Semigroups,)646 b Fg(Comm.)619 b(Math.)h(Phys.)603 b Ff(210)p Fw(,)5391 10627 y(85-105)435 b(\(2000\).)2717 13394 y([Cip1])652 b(F.)547 b(Cipriani,)576 b(Diric)-36 b(hlet)548 b(forms)g(and)f(Mark)-36 b(o)g(vian)548 b(semigroups)f(on)g (standard)g(forms)g(of)5391 15055 y(v)-36 b(on)434 b(Neumann)e (algebras,)j Fg(J.)464 b(F)-100 b(unct.)465 b(A)-33 b(nal.)432 b Ff(147)p Fw(,)i(259-300)h(\(1997\).)2717 17822 y([Cip2])652 b(F.)451 b(Cipriani,)456 b(P)-36 b(erron)451 b(theory)g(for)g(p)36 b(ositiv)-36 b(e)452 b(maps)f(and)g(semigroups)g(on)g(v)-36 b(on)451 b(Neumann)5391 19482 y(algebras,)434 b Fg(CMS)465 b(Conf.)e(Pr)-66 b(o)g(c.)433 b Ff(29)p Fw(,)h(115-123)h(\(2000\).)2717 22250 y([DL1])651 b(E.)646 b(B.)f(Da)-36 b(vies)647 b(and)d(J.M.)i (Lindsa)-36 b(y)-108 b(,)697 b(Non)645 b(comm)-36 b(utativ)g(e)646 b(symmetric)f(Mark)-36 b(o)g(v)646 b(semi-)5391 23910 y(groups,)433 b Fg(Math.)465 b(Zeit.)432 b Ff(210)p Fw(,)i(379-411)h (\(1992\).)2717 26678 y([DL2])651 b(E.)462 b(B.)f(Da)-36 b(vies)462 b(and)f(J.M.)h(Lindsa)-36 b(y)-108 b(,)467 b(Sup)36 b(erderiv)-72 b(ations)460 b(and)h(symmetric)g(Mark)-36 b(o)g(v)462 b(semi-)5391 28338 y(groups,)433 b Fg(Commun.)464 b(Math.)g(Phys.)434 b Ff(157)p Fw(,)h(359-370)g(\(1993\).)2717 31106 y([Da)-36 b(v])652 b(E.)518 b(B.)h(Da)-36 b(vies,)541 b Fg(Quantum)i(the)-66 b(ory)542 b(of)g(op)-66 b(en)542 b(systems)p Fw(,)e(Academic)518 b(Press,)540 b(London-New)5391 32766 y(Y)-108 b(ork-San)433 b(F)-108 b(rancisco,)433 b(\(1976\).)2717 35533 y([FNW])651 b(M.)578 b(F)-108 b(anes,)612 b(B.)578 b(Nac)-36 b(h)g(tergaele,)614 b(and)577 b(R.)g(F.)h(W)-108 b(erner,)613 b(Finitely)577 b(correlated)h(states)f (on)5391 37194 y(quan)-36 b(tum)432 b(spin)i(c)-36 b(hains,)433 b Fg(Commun.)464 b(Math.)g(Phys.)434 b Ff(144)p Fw(,)h(443-490)g (\(1992\).)2717 39961 y([GIS])650 b(D.)470 b(Guido,)477 b(T.)469 b(Isola)h(and)e(S.)h(Scarlatti,)478 b(Non-comm)-36 b(utativ)g(e)468 b(Diric)-36 b(hlet)469 b(forms)h(on)e(semi-)5391 41622 y(\014nite)433 b(v)-36 b(on)433 b(Neumann)g(algebras,)i Fg(J.)464 b(F)-100 b(unct.)465 b(A)-33 b(nal.)432 b Ff(135)p Fw(,)i(50-75)h(\(1996\))2717 44389 y([GL1])651 b(S.)513 b(Goldstein)g(and)g(J.)i(M.)e(Lindsa)-36 b(y)-108 b(,)534 b(Beuring-Den)-36 b(y)513 b(conditions)g(for)i(KMS-symmetric)5391 46050 y(dynamical)434 b(semigroups,)g Fg(C.)464 b(R.)g(A)-66 b(c)g(ad.)464 b(Sci.)f(Paris.)i(Ser.)e(I)434 b Ff(317)p Fw(,)g(1053-1057)i(\(1993\).)2717 48817 y([GL2])651 b(S.)605 b(Goldstein)h(and)f(J.)i(M.)f(Lindsa)-36 b(y)-108 b(,)648 b(KMS-symmetric)606 b(Mark)-36 b(o)g(v)606 b(semigroups,)650 b Fg(Math.)5391 50477 y(Zeit.)432 b Ff(219)p Fw(,)i(590-608)h (\(1995\).)2717 53245 y([GL3])651 b(S.)695 b(Goldstein)g(and)g(J.)h(M.) f(Lindsa)-36 b(y)-108 b(,)761 b(Mark)-36 b(o)g(v)696 b(semigroups)g(KMS-symmetric)f(for)h(a)5391 54905 y(w)-36 b(eigh)g(t,)434 b Fg(Math.)464 b(A)-33 b(nn.)432 b Ff(313)p Fw(,)i(39-67)h(\(1999\))2717 57673 y([KP])651 b(C.)645 b(K.)h(Ko)f(and)f(Y.)h(M.)f(P)-36 b(ark,)699 b(Construction)644 b(of)h(a)g(F)-108 b(amily)645 b(of)h(Quan)-36 b(tum)643 b(Ornstein-)5391 59333 y(Uhlen)-36 b(b)36 b(ec)-36 b(k)433 b(Semigroups,)g(preprin)-36 b(t)432 b(preparation.)2717 62100 y([MZ1])651 b(A.)425 b(W.)h(Ma)72 b(jewski)427 b(and)d(B.)h(Zegarlinski,)j(Quan)-36 b(tum)423 b(sto)36 b(c)-36 b(hastic)426 b(dynamics)f(I:)g(Spin)f(sys-)5391 63761 y(tems)433 b(on)h(a)f(lattice,)i Fg(MPEJ)f Ff(1)p Fw(,)g(P)-36 b(ap)36 b(er)433 b(2)h(\(1995\).)2717 66528 y([MZ2])651 b(A.)632 b(W.)h(Ma)72 b(jewski)634 b(and)d(B.)i (Zegarlinski,)683 b(Quan)-36 b(tum)630 b(sto)36 b(c)-36 b(hastic)633 b(dynamics)f(I)36 b(I,)633 b Fg(R)-66 b(ev.)5391 68189 y(Math.)464 b(Phys.)434 b Ff(8)g Fw(\(5\),)f(689-713)j(\(1996\).) 2717 70956 y([Mat])651 b(T.)524 b(Matsui,)547 b(Mark)-36 b(o)g(v)524 b(semigroups)g(on)f(UHF)h(algebras,)547 b Fg(R)-66 b(ev.)546 b(Mo)-66 b(d.)547 b(Phys.)524 b Ff(5)p Fw(,)547 b(587-600)5391 72617 y(\(1993\).)26476 78590 y(23)p eop %%Page: 24 24 24 23 bop 2717 7306 a Fw([Nac])652 b(B.)445 b(Nac)-36 b(h)g(tergaele,)449 b(The)c(sp)36 b(ectral)445 b(gap)g(for)g(some)h (spin)e(c)-36 b(hains)445 b(with)g(discrete)g(symmetry)5391 8966 y(breaking,)434 b Fg(Commun.)464 b(Math.)g(Phys.)434 b Ff(175)p Fw(,)h(565-606)g(\(1996\).)2717 11734 y([P)-36 b(ar])651 b(Y.)412 b(M.)h(P)-36 b(ark,)417 b(Construction)411 b(of)i(Diric)-36 b(hlet)413 b(forms)f(on)g(standard)f(forms)i(of)g(v) -36 b(on)412 b(Neumann)5391 13394 y(algebras,)389 b Fg(In\014nite)410 b(Dimensional)i(A)-33 b(nalysis,)422 b(Quantum)413 b(Pr)-66 b(ob)g(ability)411 b(and)i(R)-66 b(elate)g(d)412 b(T)-100 b(opics)p Fw(,)5391 15055 y(V)-108 b(ol.)434 b(3,)g(No.)g(1,)g(1-14)h (\(2000\).)2717 17822 y([P)-36 b(art])651 b(K.)1151 b(R.)h(P)-36 b(arthasarath)g(y)-108 b(,)1330 b Fg(A)-33 b(n)1123 b(intr)-66 b(o)g(duction)1121 b(to)k(quantum)f(sto)-66 b(chastic)1123 b(c)-66 b(alculus)p Fw(,)5391 19482 y(Birkh\177)-650 b(auser,)434 b(Basel)g(\(1992\))2717 22250 y([RS])651 b(M.Reed)585 b(and)f(B.Simon,)622 b Fg(Metho)-66 b(d)603 b(of)h(mo)-66 b(dern)603 b(mathmatic)-66 b(al)602 b(physics)i(I,)f(II)p Fw(,)584 b(Academic)5391 23910 y(press)433 b(\(1980\).)26476 78590 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303281937898--