Content-Type: multipart/mixed; boundary="-------------9912300445719" This is a multi-part message in MIME format. ---------------9912300445719 Content-Type: text/plain; name="99-489.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-489.comments" 1991 Mathematical Subject Classification: Primary 47A56, 47Nxx; Secondary 47N50, 47A40. ---------------9912300445719 Content-Type: text/plain; name="99-489.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-489.keywords" Operator matrix, operator pencil, Herglotz function, resonance, unphysical sheet, Riesz basis ---------------9912300445719 Content-Type: application/postscript; name="Crimlanl.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Crimlanl.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: crimlanl.dvi %%CreationDate: Thu Dec 30 13:27:48 1999 %%Pages: 14 %%PageOrder: Ascend 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y(and)f(some)g(relations)h(b)q(et)o(w)o(een)h(their)g (sp)q(ectra.)19 b(Finally)m(,)10 b(in)j(Section)g(4)g(w)o(e)g(presen)o (t)h(a)f(simple)228 2446 y(example.)p eop %%Page: 3 3 3 2 bop 565 147 a Fl(F)m(actorization)15 b(Theorem)e(for)g(the)g(T)m (ransfer)g(F)m(unction)320 b(3)278 246 y Fz(A)12 b(detailed)h(exp)q (osition)g(of)f(the)h(material)e(presen)o(ted)k(here)f(including)e(pro) q(ofs)g(in)h(the)g(case)228 296 y(of)h(essen)o(tially)h(more)f(general) h(sp)q(ectral)h(situations)f(will)f(b)q(e)h(giv)o(en)g(in)f(the)i (extended)g(pap)q(er)228 346 y([HMM].)389 437 y(1.)22 b FA(Anal)m(ytic)16 b(continua)m(tion)h(of)f(the)g(transfer)g(function) g Fq(M)1542 443 y Fs(1)278 512 y Fz(F)m(or)f(the)h(sak)o(e)g(of)f (simplicit)o(y)e(w)o(e)j(assume)f(in)g(this)h(pap)q(er)g(that)g(the)g (sp)q(ectrum)g Fq(\033)q Fz(\()p Fq(A)1638 518 y Fs(0)1657 512 y Fz(\))g(of)228 562 y(the)j(en)o(try)g Fq(A)448 568 y Fs(0)484 562 y Fz(is)g(absolutely)e(con)o(tin)o(uous)h 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Fy(;)p Fz(,)e(if)f Fq(k)k Fy(6)p Fz(=)f Fq(l)q Fz(,)f(and)228 1404 y Fr(S)262 1447 y Fo(k)290 1435 y Fq(\016)308 1441 y Fo(k)340 1435 y Fz(=)12 b(\001)419 1441 y Fs(0)437 1435 y Fz(.)17 b(The)10 b(p)q(oin)o(ts)g Fq(\026)693 1441 y Fo(k)723 1435 y Fz(are)g(arbitrarily)f(c)o(hosen)i (p)q(oin)o(ts)f(of)f Fq(\016)1295 1441 y Fo(k)1316 1435 y Fz(.)16 b(The)11 b(n)o(um)o(b)q(er)e(V)m(ar)1638 1441 y Fo(\022)1656 1435 y Fz(\()p Fq(B)r Fz(\))228 1485 y(is)h(called)f(w)o (eigh)o(ted)h(v)n(ariation)e(of)i(the)g(op)q(erators)h Fq(B)1040 1491 y Fo(ij)1079 1485 y Fz(with)f(resp)q(ect)i(to)e(the)g (sp)q(ectral)h(measure)228 1534 y Fq(E)259 1540 y Fs(0)277 1534 y Fz(.)278 1584 y(Notice)g(that)g(in)f(con)o(trast)h(to)g([MM1)o (,)g(MM2)o(],)g(where)h(the)f(v)n(ariation)e(\(1.1\))h(w)o(as)h (considered)228 1634 y(in)16 b(case)i(of)e Fq(\022)h Fz(=)g(0,)f(w)o(e)h(no)o(w)f(will)f(mainly)f(consider)k Fq(\022)f Fz(=)g(1.)26 b(Of)16 b(course,)i(in)o(tro)q(ducing)f(the)228 1684 y(v)n(ariation)c(V)m(ar)467 1690 y Fo(\022)486 1684 y Fz(\()p Fq(B)r Fz(\))i(for)f Fq(\022)g Fy(6)p Fz(=)f(0)h(only)g(mak)o (es)f(sense)j(when)f(the)g(en)o(try)g Fq(A)1388 1690 y Fs(0)1421 1684 y Fz(is)f(an)h(un)o(b)q(ounded)228 1734 y(op)q(erator.)278 1784 y(W)m(e)e(supp)q(ose)j(that)e(the)g(function)g Fq(K)865 1790 y Fo(B)894 1784 y Fz(\()p Fq(\026)p Fz(\))g(is)g (di\013eren)o(tiable)g(in)g Fq(\026)e Fy(2)f Fz(\001)1416 1790 y Fs(0)1448 1784 y Fz(in)j(the)h(op)q(erator)228 1833 y(norm)e(top)q(ology)m(.)k(The)e(deriv)n(ativ)o(e)f Fq(K)836 1818 y Fx(0)833 1845 y Fo(B)862 1833 y Fz(\()p Fq(\026)p Fz(\))g(is)h(non-negativ)o(e,)e Fq(K)1270 1818 y Fx(0)1267 1845 y Fo(B)1296 1833 y Fz(\()p Fq(\026)p Fz(\))g Fy(\025)f Fz(0,)i(since)h Fq(K)1594 1839 y Fo(B)1623 1833 y Fz(\()p Fq(\026)p Fz(\))f(is)228 1883 y(a)f(non-decreasing)i (function.)j(Ob)o(viously)m(,)635 1972 y(V)m(ar)700 1978 y Fo(\022)719 1972 y Fz(\()p Fq(B)r Fz(\))12 b(=)841 1915 y Fr(Z)830 2034 y Fs(\001)857 2038 y Fk(0)890 1972 y Fq(d\026)7 b Fz(\(1)i(+)h Fy(j)p Fq(\026)p Fy(j)p Fz(\))1097 1954 y Fx(\000)p Fo(\022)1141 1972 y Fy(k)p Fq(K)1200 1954 y Fx(0)1197 1982 y Fo(B)1225 1972 y Fz(\()p Fq(\026)p Fz(\))p Fy(k)p Fq(:)278 2098 y Fz(F)m(urther,)h(w)o(e)f(supp)q(ose)h (that)f(the)g(function)g Fq(K)996 2083 y Fx(0)993 2109 y Fo(B)1022 2098 y Fz(\()p Fq(\026)p Fz(\))g(is)g(con)o(tin)o(uous)f (within)g(the)i(in)o(terv)n(al)e(\001)1704 2104 y Fs(0)228 2147 y Fz(and,)14 b(moreo)o(v)o(er,)g(that)h(it)g(admits)e(analytic)h (con)o(tin)o(uation)g(from)f(this)i(in)o(terv)n(al)f(to)h(a)f(simply) 228 2197 y(connected)h(domain)c(situated,)j(sa)o(y)m(,)e(in)h Fp(C)893 2182 y Fs(+)924 2197 y Fz(.)18 b(Let)c(this)f(domain)e(b)q(e)j (called)g Fq(D)1464 2182 y Fs(+)1492 2197 y Fz(.)k(W)m(e)13 b(assume)228 2247 y(that)e(the)i(b)q(oundary)e(of)g(the)h(domain)e Fq(D)864 2232 y Fs(+)903 2247 y Fz(includes)i(the)h(en)o(tire)f(sp)q (ectral)h(in)o(terv)n(al)d(\001)1580 2253 y Fs(0)1598 2247 y Fz(.)18 b(Since)228 2297 y Fq(K)266 2282 y Fx(0)263 2308 y Fo(B)292 2297 y 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Fx(0)1570 2458 y Fo(B)1599 2446 y Fz(\()p Fq(\026)p Fz(\))g(for)p eop %%Page: 4 4 4 3 bop 228 147 a Fl(4)377 b(V.)12 b(Hardt,)h(R.)g(Mennic)o(k)o(en,)h (A.)e(K.)h(Moto)o(vilo)o(v)228 246 y Fq(\026)e Fy(2)h Fz(\001)339 252 y Fs(0)368 246 y Fz(implies)d([)p Fq(K)556 231 y Fx(0)553 258 y Fo(B)581 246 y Fz(\()p Fq(\026)p Fz(\)])650 231 y Fx(\003)681 246 y Fz(=)i Fq(K)762 231 y Fx(0)759 258 y Fo(B)788 246 y Fz(\()s(\026)-24 b Fq(\026)p Fz(\))p Fq(;)11 b(\026)h Fy(2)f Fq(D)979 231 y Fx(\006)1014 246 y Fq(:)g Fz(Also,)g(w)o(e)g(shall)f(supp)q(ose)i(that)f(the)h Fq(K)1639 231 y Fx(0)1636 258 y Fo(B)1665 246 y Fz(\()p Fq(\026)p Fz(\))228 296 y(satis\014es)j(the)f(follo)o(wing)d(condition) j(at)f(the)i(end)f(p)q(oin)o(t)g Fq(\013)1146 302 y Fs(0)1178 296 y Fz(of)f(the)h(sp)q(ectral)h(in)o(terv)n(al)e(\001)1637 302 y Fs(0)1656 296 y Fz(:)662 365 y Fy(k)p Fq(K)721 348 y Fx(0)718 375 y Fo(B)746 365 y Fz(\()p Fq(\026)p Fz(\))p Fy(k)f(\024)g Fq(C)s Fy(j)p Fq(\026)c Fy(\000)i Fq(\013)1027 371 y Fs(0)1045 365 y Fy(j)1057 348 y Fo(\015)1078 365 y Fq(;)48 b(\026)11 b Fy(2)g Fq(D)1248 348 y Fx(\006)1277 365 y Fq(;)228 434 y Fz(with)i(some)g Fq(C)i(>)c Fz(0)j(and)g Fq(\015)g Fy(2)d Fz(\()p Fy(\000)p Fz(1)p Fq(;)c Fz(0].)278 483 y(Let)17 b(\000)381 489 y Fo(l)410 483 y Fz(\()p Fq(l)h Fz(=)e Fy(\006)p Fz(1\))h(b)q(e)h(a)e(recti\014able)i(Jordan)e (curv)o(e)i(in)e Fq(D)1223 468 y Fo(l)1253 483 y Fz(resulting)h(from)e (con)o(tin)o(uous)228 533 y(deformation)c(of)i(the)i(in)o(terv)n(al)d (\001)762 539 y Fs(0)781 533 y Fz(,)h(the)h(\014nite)g(end)g(p)q(oin)o (t)f(of)g(this)h(in)o(terv)n(al)f(b)q(eing)g(\014xed.)19 b(As)228 583 y(men)o(tioned)13 b(ab)q(o)o(v)o(e,)h(in)g(the)h(follo)o (wing)d(w)o(e)i(deal)h(with)f(the)h(v)n(ariation)d(V)m(ar)1413 589 y Fs(1)1432 583 y Fz(\()p Fq(B)r Fz(\).)20 b(W)m(e)14 b(extend)228 633 y(the)19 b(de\014nition)f(of)g(this)h(v)n(ariation)e (also)h(to)g(the)h(curv)o(e)h(\000)1171 639 y Fo(l)1202 633 y Fz(b)o(y)e(in)o(tro)q(ducing)g(the)i(mo)q(di\014ed)228 683 y(v)n(ariation)589 762 y(V)m(ar)654 768 y Fs(1)673 762 y Fz(\()p Fq(B)r(;)7 b Fz(\000)767 768 y Fo(l)780 762 y Fz(\))18 b(:=)876 706 y Fr(Z)872 825 y Fs(\000)893 829 y Fj(l)925 762 y Fy(j)p Fq(d\026)p Fy(j)7 b Fz(\(1)h(+)h Fy(j)p Fq(\026)p Fy(j)p Fz(\))1154 745 y Fx(\000)p Fs(1)1198 762 y Fy(k)p Fq(K)1257 745 y Fx(0)1254 773 y Fo(B)1282 762 y Fz(\()p Fq(\026)p Fz(\))p Fy(k)277 b Fz(\(1.2\))228 888 y(where)18 b Fy(j)p Fq(d\026)p Fy(j)d Fz(denotes)k(the)e(Leb)q (esgue)i(measure)e(on)f(\000)1099 894 y Fo(l)1112 888 y Fz(.)27 b(W)m(e)17 b(supp)q(ose)h(that)f(the)g(op)q(erators)228 938 y Fq(B)259 944 y Fo(ij)307 938 y Fz(are)i(suc)o(h)h(that)f(there)h (exists)f(a)g(con)o(tour)g(\(exist)g(con)o(tours\))h(\000)1335 944 y Fo(l)1366 938 y Fz(on)f(whic)o(h)f(the)i(v)n(alue)228 987 y(V)m(ar)292 993 y Fs(1)311 987 y Fz(\()p Fq(B)r(;)7 b Fz(\000)405 993 y Fo(l)418 987 y Fz(\))16 b(is)g(\014nite,)g(i.e.,)e (V)m(ar)759 993 y Fs(1)778 987 y Fz(\()p Fq(B)r(;)7 b Fz(\000)872 993 y Fo(l)885 987 y Fz(\))15 b Fq(<)g Fy(1)p Fz(.)23 b(The)16 b(con)o(tours)h(\000)1323 993 y Fo(l)1351 987 y Fz(satisfying)e(the)i(condi-)228 1037 y(tion)d(V)m(ar)379 1043 y Fs(1)398 1037 y Fz(\()p Fq(B)r(;)7 b Fz(\000)492 1043 y Fo(l)505 1037 y Fz(\))13 b Fq(<)h Fy(1)h Fz(are)g(said)g(to)g(b) q(e)h Fq(K)939 1043 y Fo(B)968 1037 y Fz(-b)q(ounded)f(con)o(tours.)23 b(Surely)m(,)14 b(in)h(the)h(case)g(of)228 1087 y(un)o(b)q(ounded)h Fq(A)476 1093 y Fs(0)512 1087 y Fz(the)g(condition)f(of)g(b)q (oundedness)j(of)d(V)m(ar)1183 1093 y Fs(1)1201 1087 y Fz(\()p Fq(B)r(;)7 b Fz(\000)1295 1093 y Fo(l)1308 1087 y Fz(\))17 b(is)f(m)o(uc)o(h)g(w)o(eak)o(er)h(than)228 1137 y(the)d(condition)f(of)h(b)q(oundedness)i(of)d(V)m(ar)884 1143 y Fs(0)903 1137 y Fz(\()p Fq(B)r(;)7 b Fz(\000)997 1143 y Fo(l)1010 1137 y Fz(\))14 b(used)h(in)e([MM1,)g(MM2]\).)228 1212 y Ft(Lemma)i(1.1.)21 b Fi(The)12 b(analytic)g(c)n(ontinuation)h (of)f(the)g(tr)n(ansfer)f(function)h Fq(M)1430 1218 y Fs(1)1449 1212 y Fz(\()p Fq(z)r Fz(\))p Fi(,)h Fq(z)h Fy(2)d Fp(C)e Fy(n)s Fz(\001)1689 1218 y Fs(0)1709 1212 y Fi(,)228 1262 y(thr)n(ough)16 b(the)f(sp)n(e)n(ctr)n(al)g(interval)g Fz(\001)784 1268 y Fs(0)817 1262 y Fi(into)h(the)g(sub)n(domain)g Fq(D)q Fz(\(\000)1256 1268 y Fo(l)1269 1262 y Fz(\))d Fy(\032)g Fq(D)1378 1247 y Fo(l)1407 1262 y Fz(\()p Fq(l)h Fz(=)f Fy(\006)p Fz(1\))j Fi(b)n(ounde)n(d)228 1312 y(by)f(the)g(set)g Fz(\001)451 1318 y Fs(0)484 1312 y Fi(and)g(a)g Fq(K)635 1318 y Fo(B)664 1312 y Fi(-b)n(ounde)n(d)h(c)n(ontour)f Fz(\000)1014 1318 y Fo(l)1042 1312 y Fi(is)f(given)i(by)679 1384 y Fq(M)719 1390 y Fs(1)738 1384 y Fz(\()p Fq(z)r(;)7 b Fz(\000)820 1390 y Fo(l)832 1384 y Fz(\))27 b(:=)955 1373 y Fr(e)945 1384 y Fq(A)976 1390 y Fs(1)1004 1384 y Fy(\000)10 b Fq(z)h Fz(+)e Fq(V)1141 1390 y Fs(1)1160 1384 y Fz(\()p Fq(z)r(;)e Fz(\000)1242 1390 y Fo(l)1255 1384 y Fz(\))366 b(\(1.3\))228 1453 y Fi(wher)n(e)676 1532 y Fq(V)700 1538 y Fs(1)719 1532 y Fz(\()p Fq(z)r(;)7 b Fz(\000)801 1538 y Fo(l)813 1532 y Fz(\))27 b(:=)926 1476 y Fr(Z)921 1595 y Fs(\000)942 1599 y Fj(l)974 1532 y Fq(d\026)7 b(K)1066 1515 y Fx(0)1063 1543 y Fo(B)1092 1532 y Fz(\()p Fq(\026)p Fz(\))1199 1504 y Fq(z)p 1161 1523 97 2 v 1161 1561 a(z)k Fy(\000)f Fq(\026)1263 1532 y(:)362 b Fz(\(1.4\))228 1662 y Fi(F)m(or)14 b Fq(z)g Fy(2)d Fq(D)412 1647 y Fo(l)435 1662 y Fy(\\)e Fq(D)q Fz(\(\000)549 1668 y Fo(l)562 1662 y Fz(\))15 b Fi(the)g(function)g Fq(M)865 1668 y Fs(1)884 1662 y Fz(\()p Fq(z)r(;)7 b Fz(\000)966 1668 y Fo(l)979 1662 y Fz(\))15 b Fi(may)g(b)n(e)g(written) e(as)652 1731 y Fq(M)692 1737 y Fs(1)711 1731 y Fz(\()p Fq(z)r(;)7 b Fz(\000)793 1737 y Fo(l)806 1731 y Fz(\))26 b(=)h Fq(M)947 1737 y Fs(1)966 1731 y Fz(\()p Fq(z)r Fz(\))10 b(+)f(2)p Fq(\031)q Fz(i)e Fq(l)q(z)r(K)1207 1713 y Fx(0)1204 1741 y Fo(B)1233 1731 y Fz(\()p Fq(z)r Fz(\))p Fq(:)339 b Fz(\(1.5\))228 1809 y Fi(Pr)n(o)n(of.)20 b Fz(Ob)o(viously)m(,)12 b(the)i(function)f(\(1.4\))g(is)h(w)o(ell)e (de\014ned)j(due)f(to)g(the)g Fq(K)1396 1815 y Fo(B)1425 1809 y Fz({b)q(oundedness)h(of)228 1859 y(the)j(con)o(tour)g(\000)484 1865 y Fo(l)514 1859 y Fz(and)f(since)i(for)e(all)f Fq(z)k Fy(2)d Fp(C)k Fy(n)11 b Fz(\000)1016 1865 y Fo(l)1046 1859 y Fz(there)19 b(exist)f(a)f Fq(c)p Fz(\()p Fq(z)r Fz(\))h Fq(>)g Fz(0)g(suc)o(h)g(that)g(the)228 1917 y(estimate)398 1882 y Fr(\014)398 1906 y(\014)412 1917 y Fz(\()p Fq(z)c Fy(\000)e Fq(\026)p Fz(\))546 1902 y Fx(\000)p Fs(1)591 1882 y Fr(\014)591 1906 y(\014)623 1917 y Fq(<)18 b(c)p Fz(\()p Fq(z)r Fz(\))751 1883 y Fr(\000)771 1917 y Fz(1)11 b(+)h Fy(j)p Fq(\026)p Fy(j)896 1883 y Fr(\001)915 1892 y Fx(\000)p Fs(1)977 1917 y Fz(\()p Fq(\026)18 b Fy(2)g Fz(\000)1108 1923 y Fo(l)1121 1883 y Fr(\001)1158 1917 y Fz(holds.)29 b(Th)o(us,)19 b(the)f(pro)q(of)g(of)f(this)228 1967 y(lemma)e(is)j(reduced)i(to)e(the)h(observ)n(ation)f(that)h(the)g (function)f Fq(M)1305 1973 y Fs(1)1323 1967 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1405 1973 y Fo(l)1418 1967 y Fz(\))18 b(is)g(holomorphic)228 2021 y(for)d Fq(z)h Fy(2)e Fp(C)20 b Fy(n)10 b Fz(\000)468 2027 y Fo(l)496 2021 y Fz(and)15 b(coincides)i(with)e Fq(M)892 2027 y Fs(1)911 2021 y Fz(\()p Fq(z)r Fz(\))h(for)f Fq(z)h Fy(2)e Fp(C)20 b Fy(n)p 1194 1984 107 2 v 10 w Fq(D)q Fz(\(\000)1271 2027 y Fo(l)1284 2021 y Fz(\).)j(The)16 b(equation)f(\(1.5\))g(is)228 2070 y(obtained)e(from)g(\(1.4\))g(using)g(the)i(Residue)f(Theorem.)p 1692 2070 2 29 v 1694 2044 25 2 v 1694 2070 V 1719 2070 2 29 v 278 2147 a(The)h(form)o(ula)d(\(1.5\))i(sho)o(ws)h(that)f(in)h (general)g(the)g(transfer)g(function)g Fq(M)1451 2153 y Fs(1)1470 2147 y Fz(\()p Fq(z)r Fz(\))g(has)f(a)h(Rie-)228 2197 y(mann)e(surface)j(with)f(at)g(least)g(t)o(w)o(o)g(sheets.)23 b(The)15 b(sheet)i(of)d(the)i(complex)e(plane)g(where)j(the)228 2247 y(transfer)f(function)f Fq(M)587 2253 y Fs(1)606 2247 y Fz(\()p Fq(z)r Fz(\))h(together)h(with)e(the)h(resolv)o(en)o(t)g Ft(R)p Fz(\()p Fq(z)r Fz(\))e(=)h(\()p Ft(H)c Fy(\000)f Fq(z)r Fz(\))1479 2232 y Fx(\000)p Fs(1)1539 2247 y Fz(is)16 b(initially)228 2297 y(considered)g(is)g(said)f(to)g(b)q(e)h(the)f(ph)o (ysical)g(sheet.)24 b(The)15 b(remaining)f(sheets)j(of)d(the)i(Riemann) 228 2347 y(surface)g(of)f Fq(M)458 2353 y Fs(1)477 2347 y Fz(\()p Fq(z)r Fz(\))h(are)f(said)g(to)h(b)q(e)g(unph)o(ysical)f (sheets)i(\(see,)f(e.)7 b(g.,)15 b([RS)o(]\).)22 b(In)16 b(the)g(presen)o(t)228 2396 y(w)o(ork)h(w)o(e)g(deal)g(with)f(the)i (unph)o(ysical)e(sheets)j(neigh)o(b)q(oring)e(the)g(ph)o(ysical)g(one,) g(i.)6 b(e.,)17 b(with)228 2446 y(the)d(sheets)i(connected)g(through)e (the)g(in)o(terv)n(al)f(\001)1029 2452 y Fs(0)1061 2446 y Fz(immediately)e(to)i(the)i(ph)o(ysical)e(sheet.)p eop %%Page: 5 5 5 4 bop 565 147 a Fl(F)m(actorization)15 b(Theorem)e(for)g(the)g(T)m (ransfer)g(F)m(unction)320 b(5)228 246 y Ft(Remark)15 b(1.1.)21 b Fz(F)m(or)e Fq(z)j Fy(2)e Fp(C)h Fy(n)13 b Fz(\000)775 252 y Fo(l)787 246 y Fz(,)20 b(the)g(equation)e(\(1.4\))g (de\014nes)j(v)n(alues)e(of)f(the)h(function)228 296 y Fq(V)252 302 y Fs(1)271 296 y Fz(\()p Fy(\001)p Fq(;)7 b Fz(\000)344 302 y Fo(l)355 296 y Fz(\))18 b(in)f(the)h(space)g(of)f (b)q(ounded)h(op)q(erators)g(in)f Fy(H)1128 302 y Fs(1)1146 296 y Fz(.)28 b(The)18 b(in)o(v)o(erse)g(transfer)g(function)228 321 y Fr(\002)245 354 y Fq(M)285 360 y Fs(1)304 354 y Fz(\()p Fq(z)r Fz(\))357 321 y Fr(\003)375 329 y Fx(\000)p Fs(1)433 354 y Fz(coincides)c(with)g(the)g(righ)o(t)g(lo)o(w)o(er)f (blo)q(c)o(k)h(comp)q(onen)o(t)f Ft(R)1341 360 y Fs(11)1376 354 y Fz(\()p Fq(z)r Fz(\))h(of)f(the)i(resolv)o(en)o(t)228 407 y Ft(R)p Fz(\()p Fq(z)r Fz(\))d(=)f(\()p Ft(H)5 b Fy(\000)g Fq(z)r Fz(\))504 392 y Fx(\000)p Fs(1)562 407 y Fz(and,)12 b(th)o(us,)g(it)f(is)h(holomorphic)e(in)h Fp(C)g Fy(n)5 b Fq(\033)q Fz(\()p Ft(H)q Fz(\))14 b Fy(\033)e Fp(C)f Fy(n)5 b Fp(R)p Fz(.)14 b(Since)f Fq(M)1593 413 y Fs(1)1612 407 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1694 413 y Fo(l)1706 407 y Fz(\))228 461 y(coincides)14 b(with)g Fq(M)538 467 y Fs(1)557 461 y Fz(\()p Fq(z)r Fz(\))g(for)f(all)g Fq(z)h Fy(2)d Fp(C)18 b Fy(n)p 886 425 107 2 v 9 w Fq(D)q Fz(\(\000)963 467 y Fo(l)976 461 y Fz(\),)c(one)g(concludes)h(that)f([) p Fq(M)1423 467 y Fs(1)1441 461 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1523 467 y Fo(l)1535 461 y Fz(\)])1563 446 y Fx(\000)p Fs(1)1621 461 y Fz(exists)228 515 y(as)13 b(a)f(b)q(ounded)i(op)q(erator)f(and)g(is)f(holomorphic)f(in)h Fq(z)j Fz(at)e(least)g(for)g Fq(z)g Fy(2)e Fp(C)16 b Fy(n)1429 481 y Fr(\000)1448 515 y Fq(\033)q Fz(\()p Ft(H)p Fz(\))8 b Fy([)p 1585 479 V 7 w Fq(D)q Fz(\(\000)1662 521 y Fo(l)1675 515 y Fz(\))1692 481 y Fr(\001)1711 515 y Fz(.)745 601 y(2.)22 b FA(The)16 b(basic)f(equa)m(tion)278 676 y Fz(Let)e(\000)f(b)q(e)i(a)e Fq(K)513 682 y Fo(B)542 676 y Fz(-b)q(ounded)h(con)o(tour.)18 b(If)12 b Fq(Y)22 b Fz(stands)14 b(for)e(an)h(arbitrary)f(b)q(ounded)i(op)q(erator)228 725 y(in)k Fy(H)316 731 y Fs(1)354 725 y Fz(suc)o(h)i(that)f(the)g(sp)q (ectrum)g(of)g Fq(Y)28 b Fz(is)19 b(separated)h(from)d(the)j(set)g (\000)e(then,)j(follo)o(wing)228 775 y(to)14 b([MM2)o(,)f(M1,)g(M2],)g (one)h(can)g(de\014ne)h(the)g(op)q(erator)624 864 y Fq(V)648 870 y Fs(1)667 864 y Fz(\()p Fq(Y)r(;)7 b Fz(\000\))26 b(:=)865 808 y Fr(Z)866 927 y Fs(\000)913 864 y Fq(d\026)7 b(K)1005 847 y Fx(0)1002 874 y Fo(B)1031 864 y Fz(\()p Fq(\026)p Fz(\))g Fq(Y)i Fz(\()p Fq(Y)19 b Fy(\000)9 b Fq(\026)p Fz(\))1269 847 y Fx(\000)p Fs(1)1314 864 y Fq(:)311 b Fz(\(2.1\))228 984 y(This)12 b(op)q(erator)i(is)e(b)q (ounded,)h Fq(V)732 990 y Fs(1)751 984 y Fz(\()p Fq(Y)r(;)7 b Fz(\000\))k Fy(2)h Ft(B)p Fz(\()p Fy(H)990 990 y Fs(1)1008 984 y Fq(;)7 b Fy(H)1062 990 y Fs(1)1081 984 y Fz(\),)12 b(and)h(its)f(norm)f(admits)h(the)h(estimate)461 1052 y Fy(k)p Fq(V)506 1058 y Fs(1)525 1052 y Fz(\()p Fq(Y)r(;)7 b Fz(\000\))p Fy(k)25 b(\024)h Fz(V)m(ar)796 1058 y Fs(1)815 1052 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))14 b Fy(k)p Fq(Y)9 b Fy(k)21 b Fz(sup)1035 1087 y Fo(\026)p Fx(2)p Fs(\000)1098 1052 y Fz(\(1)9 b(+)g Fy(j)p Fq(\026)p Fy(j)p Fz(\))p Fy(k)p Fz(\()p Fq(Y)18 b Fy(\000)10 b Fq(\026)p Fz(\))1412 1035 y Fx(\000)p Fs(1)1456 1052 y Fy(k)p Fq(:)148 b Fz(\(2.2\))228 1150 y(In)14 b(what)f(follo)o(ws)g(w)o(e)h(consider)h(the)f(equation)g (\(cf.)k([MM2)o(,)c(M1)o(,)g(M2]\))788 1223 y Fq(Y)35 b Fz(=)914 1213 y Fr(e)904 1223 y Fq(A)935 1229 y Fs(1)963 1223 y Fz(+)10 b Fq(V)1029 1229 y Fs(1)1048 1223 y Fz(\()p Fq(Y)r(;)d Fz(\000\))p Fq(:)474 b Fz(\(2.3\))228 1291 y(This)14 b(equation)g(p)q(ossesses)i(the)f(follo)o(wing)d(c)o (haracteristic)j(prop)q(ert)o(y:)20 b(If)13 b(an)h(op)q(erator)h Fq(H)1662 1297 y Fs(1)1694 1291 y Fz(is)228 1341 y(a)e(solution)g(of)h (\(2.3\))f(and)h Fq(u)671 1347 y Fs(1)703 1341 y Fz(is)g(an)f(eigen)o (v)o(ector)i(of)e Fq(H)1100 1347 y Fs(1)1118 1341 y Fz(,)h Fq(H)1179 1347 y Fs(1)1197 1341 y Fq(u)1221 1347 y Fs(1)1251 1341 y Fz(=)e Fq(z)r(u)1340 1347 y Fs(1)1358 1341 y Fz(,)h(then)332 1430 y Fq(z)r(u)377 1436 y Fs(1)437 1430 y Fz(=)520 1420 y Fr(e)510 1430 y Fq(A)541 1436 y Fs(1)560 1430 y Fq(u)584 1436 y Fs(1)612 1430 y Fz(+)c Fq(V)677 1436 y Fs(1)696 1430 y Fz(\()p 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Fz(\000\))p Fq(u)1416 1576 y Fs(1)1434 1570 y Fq(:)228 1687 y Fz(This)k(means)g(that)h(an)o(y)g(eigen)o(v)n(alue)f Fq(z)j Fz(of)d(suc)o(h)h(an)g(op)q(erator)g Fq(H)1231 1693 y Fs(1)1261 1687 y Fz(is)g(automatically)d(an)i(eigen-)228 1737 y(v)n(alue)16 b(for)g(the)i(analytically)c(con)o(tin)o(ued)j (transfer)h(function)f Fq(M)1260 1743 y Fs(1)1279 1737 y Fz(\()p Fq(z)r(;)7 b Fz(\000\))16 b(and)h Fq(u)1501 1743 y Fs(1)1536 1737 y Fz(is)g(a)f(corre-)228 1787 y(sp)q(onding)j (eigen)o(v)o(ector.)36 b(Th)o(us,)20 b(ha)o(ving)f(found)g(the)h (solution\(s\))f(of)g(the)h(equation)f(\(2.3\))228 1837 y(one)d(obtains)g(an)g(e\013ectiv)o(e)i(means)e(of)f(studying)h(the)h (sp)q(ectral)h(prop)q(erties)g(of)d(the)i(transfer)228 1887 y(function)e Fq(M)432 1893 y Fs(1)451 1887 y Fz(\()p Fq(z)r(;)7 b Fz(\000\),)15 b(referring)h(to)g(w)o(ell-kno)o(wn)e(facts) i(of)f(op)q(erator)h(theory)g([GK,)f(K].)23 b(It)15 b(is)228 1936 y(con)o(v)o(enien)o(t)f(to)g(rewrite)h(the)f(equation)g(\(2.3\))f (in)g(the)i(form)787 2009 y Fq(X)29 b Fz(=)d Fq(V)932 2015 y Fs(1)950 2009 y Fz(\()976 1999 y Fr(e)966 2009 y Fq(A)997 2015 y Fs(1)1026 2009 y Fz(+)9 b Fq(X)q(;)e Fz(\000\))474 b(\(2.4\))228 2089 y(where)15 b Fq(X)g Fz(:=)c Fq(Y)19 b Fy(\000)546 2079 y Fr(e)536 2089 y Fq(A)567 2095 y Fs(1)586 2089 y Fz(.)278 2140 y(Let)14 b(the)g(sp)q(ectrum)h(of)e(the)i(op)q(erator)901 2129 y Fr(e)891 2140 y Fq(A)922 2146 y Fs(1)955 2140 y Fz(b)q(e)f(separated) h(from)d(\000,)i(i.)6 b(e.,)701 2211 y Fq(d)723 2217 y Fs(0)741 2211 y Fz(\(\000\))26 b(:=)f(dist)p Fy(f)p Fq(\033)q Fz(\()1033 2200 y Fr(e)1023 2211 y Fq(A)1054 2217 y Fs(1)1073 2211 y Fz(\))p Fq(;)7 b Fz(\000)p Fy(g)k Fq(>)h Fz(0)7 b Fq(:)387 b Fz(\(2.5\))228 2290 y(Then,)13 b(since)457 2280 y Fr(e)447 2290 y Fq(A)478 2296 y Fs(1)509 2290 y Fz(is)g(selfadjoin)o(t)e(and)i(b)q(ounded,)g(it)f(is)g(ob)o (vious)g(that)h(the)g(follo)o(wing)d(quan)o(tit)o(y)621 2380 y(V)m(ar)694 2387 y Fh(e)686 2394 y Fo(A)711 2398 y Fk(1)729 2380 y Fz(\()p Fq(B)r(;)d Fz(\000\))26 b(:=)934 2323 y Fr(Z)935 2442 y Fs(\000)983 2380 y Fy(j)p Fq(d\026)p Fy(j)1113 2352 y(k)p Fq(K)1172 2336 y Fx(0)1169 2363 y Fo(B)1197 2352 y Fz(\()p Fq(\026)p Fz(\))p Fy(k)p 1064 2370 260 2 v 1064 2415 a Fz(dist)p Fy(f)p Fq(\026;)7 b(\033)q Fz(\()1247 2405 y Fr(e)1237 2415 y Fq(A)1268 2421 y Fs(1)1287 2415 y Fz(\))p Fy(g)1637 2380 y Fz(\(2.6\))p eop %%Page: 6 6 6 5 bop 228 147 a Fl(6)377 b(V.)12 b(Hardt,)h(R.)g(Mennic)o(k)o(en,)h (A.)e(K.)h(Moto)o(vilo)o(v)228 246 y Fz(is)h(\014nite,)330 324 y(V)m(ar)402 331 y Fh(e)395 338 y Fo(A)420 342 y Fk(1)438 324 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))25 b Fy(\024)h Fz(V)m(ar)696 330 y Fs(1)714 324 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))15 b(sup)838 359 y Fo(\026)p Fx(2)p Fs(\000)902 324 y Fz(\(1)9 b(+)g Fy(j)p Fq(\026)p Fy(j)p Fz(\))1054 291 y Fr(\002)1071 324 y Fz(dist)p Fy(f)p Fq(\026;)e(\033)q Fz(\()1253 314 y Fr(e)1244 324 y Fq(A)1275 330 y Fs(1)1293 324 y Fz(\))p Fy(g)1330 291 y Fr(\003)1347 299 y Fx(\000)p Fs(1)1417 324 y Fq(<)26 b Fy(1)7 b Fq(:)101 b Fz(\(2.7\))228 424 y(It)13 b(is)g(more)g(con)o(v)o(enien)o(t)g(to)h(mak)o(e)d(the)j (subsequen)o(t)h(estimations)d(in)h(terms)g(of)g(the)h(v)n(ariation)228 474 y(V)m(ar)300 481 y Fh(e)292 488 y Fo(A)317 492 y Fk(1)336 474 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))14 b(rather)h(then)f(in)g (terms)f(of)g(the)i(v)n(ariation)d(V)m(ar)1201 480 y Fs(1)1220 474 y Fz(\()p Fq(B)r(;)7 b Fz(\000\).)228 564 y Ft(Theorem)15 b(2.1.)21 b Fi(L)n(et)607 554 y Fr(e)597 564 y Fq(A)628 570 y Fs(1)662 564 y Fi(b)n(e)15 b(a)g(b)n(ounde)n(d)h (op)n(er)n(ator,)e(the)h(c)n(ontour)g Fz(\000)g Fi(b)n(e)g Fq(K)1430 570 y Fo(B)1459 564 y Fi(-b)n(ounde)n(d)h(and)301 653 y Fz(V)m(ar)374 660 y Fh(e)366 667 y Fo(A)391 671 y Fk(1)409 653 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))12 b Fq(<)g Fz(1)7 b Fq(;)91 b Fz(V)m(ar)778 660 y Fh(e)770 667 y Fo(A)795 671 y Fk(1)814 653 y Fz(\()p Fq(B)r(;)7 b Fz(\000\))p Fy(k)955 643 y Fr(e)945 653 y Fq(A)976 659 y Fs(1)994 653 y Fy(k)12 b Fq(<)1076 625 y Fz(1)p 1076 644 21 2 v 1076 682 a(4)1108 653 y Fq(d)1130 659 y Fs(0)1148 653 y Fz(\(\000\))7 b([1)i Fy(\000)h Fz(V)m(ar)1369 660 y Fh(e)1361 667 y Fo(A)1386 671 y Fk(1)1405 653 y Fz(\()p Fq(B)r(;)d Fz(\000\)])1527 636 y Fs(2)1552 653 y Fq(:)73 b Fz(\(2.8\))228 737 y Fi(L)n(et)270 818 y Fq(r)289 824 y Fs(min)346 818 y Fz(\(\000\))42 b(:=)536 790 y(1)p 536 808 V 536 846 a(2)569 818 y Fq(d)591 824 y Fs(0)609 818 y Fz(\(\000\))7 b([1)i Fy(\000)g Fz(V)m(ar)830 825 y Fh(e)822 832 y Fo(A)847 836 y Fk(1)865 818 y Fz(\()p Fq(B)r(;)e Fz(\000\)])531 937 y Fy(\000)563 871 y Fr(r)p 605 871 947 2 v 610 909 a Fz(1)p 610 927 21 2 v 610 965 a(4)643 937 y Fq(d)665 922 y Fs(2)665 948 y(0)683 937 y Fz(\(\000\))g([1)i Fy(\000)g Fz(V)m(ar)903 944 y Fh(e)896 951 y Fo(A)921 955 y Fk(1)939 937 y Fz(\()p Fq(B)r(;)e Fz(\000\)])1061 925 y Fs(2)1089 937 y Fy(\000)i Fq(d)1152 943 y Fs(0)1170 937 y Fz(\(\000\))e(V)m(ar)1308 944 y Fh(e)1300 951 y Fo(A)1325 955 y Fk(1)1343 937 y Fz(\()p Fq(B)r(;)g Fz(\000\))g Fy(k)1491 926 y Fr(e)1481 937 y Fq(A)1512 943 y Fs(1)1531 937 y Fy(k)1637 877 y Fz(\(2.9\))228 1025 y Fi(and)458 1105 y Fq(r)477 1111 y Fs(max)541 1105 y Fz(\(\000\))26 b(:=)g Fq(d)717 1111 y Fs(0)736 1105 y Fz(\(\000\))9 b Fy(\000)845 1054 y Fr(q)p 886 1054 594 2 v 886 1105 a Fz(V)m(ar)958 1112 y Fh(e)951 1119 y Fo(A)976 1123 y Fk(1)994 1105 y Fz(\()p Fq(B)r(;)e Fz(\000\))g Fq(d)1133 1111 y Fs(0)1151 1105 y Fz(\(\000\))g([)p Fq(d)1250 1111 y Fs(0)1268 1105 y Fz(\(\000\))j(+)f Fy(k)1408 1094 y Fr(e)1398 1105 y Fq(A)1429 1111 y Fs(1)1448 1105 y Fy(k)p Fz(])o Fq(:)124 b Fz(\(2.10\))228 1185 y Fi(Then)15 b(the)g(e)n(quation)g Fz(\(2.4\))g Fi(is)f(uniquely)i(solvable)e(in)h (any)h(close)n(d)f(b)n(al)r(l)613 1257 y Fy(S)638 1263 y Fs(1)657 1257 y Fz(\()p Fq(r)q Fz(\))26 b(:=)805 1223 y Fr(\010)830 1257 y Fq(X)15 b Fy(2)c Ft(B)p Fz(\()p Fy(H)1003 1263 y Fs(1)1022 1257 y Fq(;)c Fy(H)1076 1263 y Fs(1)1094 1257 y Fz(\))19 b(:)e Fy(k)p Fq(X)s Fy(k)12 b(\024)g Fq(r)1313 1223 y Fr(\011)228 1329 y Fi(wher)n(e)737 1401 y Fq(r)756 1407 y Fs(min)813 1401 y Fz(\(\000\))26 b Fy(\024)h Fq(r)g(<)g(r)1080 1407 y Fs(max)1143 1401 y Fz(\(\000\))p Fq(:)403 b Fz(\(2.11\))228 1475 y Fi(The)14 b(solution)h Fq(X)j Fi(of)d(the)f(e)n(quation)h Fz(\(2.4\))f Fi(is)g(the)h(same)g(for)f(any)h Fq(r)g Fi(satisfying)f Fz(\(2.11\))g Fi(and)h(in)228 1525 y(fact)f(it)h(b)n(elongs)g(to)g(the) g(smal)r(lest)f(b)n(al)r(l)g Fy(S)873 1531 y Fs(1)892 1525 y Fz(\()p Fq(r)927 1531 y Fs(min)984 1525 y Fz(\))p Fi(,)h Fy(k)p Fq(X)s Fy(k)d(\024)f Fq(r)1181 1531 y Fs(min)1238 1525 y Fz(\(\000\))p Fi(.)228 1602 y(Pr)n(o)n(of.)20 b Fz(One)c(can)g(pro)o(v)o(e)f(this)g(theorem)g(making)e(use)j(of)f (Banac)o(h's)h(Fixed)f(P)o(oin)o(t)g(Theorem)228 1652 y(\(see)g([HMM]\).)p 1692 1652 2 29 v 1694 1626 25 2 v 1694 1652 V 1719 1652 2 29 v 228 1733 a(The)f(follo)o(wing)d (statemen)o(t)j(is)g(a)g(direct)g(consequence)j(of)c(the)h(conditions)g (\(2.8\).)228 1811 y Ft(Remark)h(2.1.)21 b Fz(The)15 b(v)n(alues)e(of)g Fq(r)779 1817 y Fs(min)836 1811 y Fz(\(\000\))h(and)g Fq(r)1008 1817 y Fs(max)1071 1811 y Fz(\(\000\))g(satisfy)g(the)g(estimates)518 1902 y Fq(r)537 1908 y Fs(min)594 1902 y Fz(\(\000\))25 b Fq(<)740 1874 y Fz(1)p 740 1893 21 2 v 740 1931 a(2)772 1902 y Fq(d)794 1908 y Fs(0)813 1902 y Fz(\(\000\))7 b([1)h Fy(\000)i Fz(V)m(ar)1033 1909 y Fh(e)1025 1916 y Fo(A)1050 1920 y Fk(1)1069 1902 y Fz(\()p Fq(B)r(;)d Fz(\000\)])25 b Fq(<)h(r)1293 1908 y Fs(max)1356 1902 y Fz(\(\000\))7 b Fq(:)228 1991 y Ft(Theorem)15 b(2.2.)21 b Fi(L)n(et)e(the)g(c)n (onditions)h(of)f Fz(Theorem)f(2.1)h Fi(b)n(e)g(ful\014l)r(le)n(d)g (for)g(a)g Fq(K)1535 1997 y Fo(B)1564 1991 y Fi(-b)n(ounde)n(d)228 2041 y(c)n(ontour)c Fz(\000)d Fy(\032)h Fq(D)497 2026 y Fo(l)525 2041 y Fi(and)j(let)f Fq(X)j Fi(b)n(e)e(the)f(solution)g(of) g(the)h(e)n(quation)f Fz(\(2.4\))p Fi(.)20 b(Then)15 b Fq(X)k Fi(c)n(oincides)228 2096 y(with)12 b(the)h(analo)n(gous)h (solution)737 2086 y Fr(e)727 2096 y Fq(X)i Fi(for)d(any)h(other)f Fq(K)1061 2102 y Fo(B)1089 2096 y Fi(-b)n(ounde)n(d)h(c)n(ontour)1411 2086 y Fr(e)1410 2096 y Fz(\000)d Fy(\032)h Fq(D)1526 2081 y Fo(l)1552 2096 y Fi(satisfying)228 2146 y(the)j(estimates)364 2227 y Fz(V)m(ar)436 2234 y Fh(e)429 2241 y Fo(A)454 2245 y Fk(1)472 2227 y Fz(\()p Fq(B)r(;)543 2217 y Fz(~)540 2227 y(\000\))d Fq(<)g Fz(1)42 b Fq(and)g Fz(V)m(ar)884 2234 y Fh(e)876 2241 y Fo(A)901 2245 y Fk(1)920 2227 y Fz(\()p Fq(B)r(;)990 2217 y Fz(~)988 2227 y(\000\))p Fy(k)1061 2217 y Fr(e)1051 2227 y Fq(A)1082 2233 y Fs(1)1100 2227 y Fy(k)12 b Fq(<)1182 2199 y Fz(1)p 1182 2218 V 1182 2256 a(4)1220 2216 y Fr(e)1214 2227 y Fq(d)1236 2233 y Fs(0)1254 2227 y Fz([1)d Fy(\000)g Fz(V)m(ar)1410 2234 y Fh(e)1402 2241 y Fo(A)1427 2245 y Fk(1)1445 2227 y Fz(\()p Fq(B)r(;)1516 2217 y Fz(~)1513 2227 y(\000)q(\)])1568 2210 y Fs(2)228 2323 y Fi(wher)n(e)15 b Fz(0)e Fq(<)432 2312 y Fr(e)426 2323 y Fq(d)448 2329 y Fs(0)480 2323 y Fz(=)h(dist)p Fy(f)p Fq(\033)q Fz(\()665 2312 y Fr(e)655 2323 y Fq(A)686 2329 y Fs(1)705 2323 y Fz(\))p Fq(;)7 b(\033)765 2308 y Fx(0)776 2323 y Fz(\()p Fq(A)823 2329 y Fs(0)842 2323 y Fz(\))j Fy([)907 2312 y Fr(e)906 2323 y Fz(\000)p Fy(g)j(\024)h Fq(d)1034 2329 y Fs(0)1052 2323 y Fz(\(\000\))p Fi(.)23 b(Mor)n(e)n(over,)15 b(this)h(solution)g (satis\014es)228 2373 y(the)f(ine)n(quality)g Fy(k)p Fq(X)s Fy(k)c(\024)h Fq(r)638 2379 y Fs(0)656 2373 y Fz(\()p Fq(B)r Fz(\))k Fi(wher)n(e)460 2445 y Fq(r)479 2451 y Fs(0)497 2445 y Fz(\()p Fq(B)r Fz(\))d(:=)e(inf)681 2411 y Fr(\010)705 2445 y Fq(r)724 2451 y Fs(min)781 2445 y Fz(\(\000)823 2451 y Fo(l)835 2445 y Fz(\))19 b(:)f(V)m(ar)972 2452 y Fh(e)965 2459 y Fo(A)990 2463 y Fk(1)1008 2445 y Fz(\()p Fq(B)r(;)7 b Fz(\000)1102 2451 y Fo(l)1115 2445 y Fz(\))k Fq(<)h Fz(1)7 b Fq(;)13 b(!)q Fz(\()p Fq(B)r(;)7 b Fz(\000)1360 2451 y Fo(l)1374 2445 y Fz(\))k Fq(>)h Fz(0)1466 2411 y Fr(\011)p eop %%Page: 7 7 7 6 bop 565 147 a Fl(F)m(actorization)15 b(Theorem)e(for)g(the)g(T)m (ransfer)g(F)m(unction)320 b(7)228 246 y Fi(with)14 b Fq(r)337 252 y Fs(min)394 246 y Fz(\(\000)436 252 y Fo(l)449 246 y Fz(\))h Fi(given)g(by)g Fz(\(2.9\))f Fi(and)418 331 y Fq(!)q Fz(\()p Fq(B)r(;)7 b Fz(\000)539 337 y Fo(l)552 331 y Fz(\))27 b(:=)f Fq(d)687 337 y Fs(0)705 331 y Fz(\(\000)747 337 y Fo(l)760 331 y Fz(\))7 b([1)h Fy(\000)i Fz(V)m(ar)938 338 y Fh(e)931 345 y Fo(A)956 349 y Fk(1)974 331 y Fz(\()p Fq(B)r(;)d Fz(\000)1068 337 y Fo(l)1081 331 y Fz(\)])1109 314 y Fs(2)1136 331 y Fy(\000)j Fz(4)p Fy(k)1229 320 y Fr(e)1220 331 y Fq(A)1251 337 y Fs(1)1269 331 y Fy(k)p Fz(V)m(ar)1362 338 y Fh(e)1354 345 y Fo(A)1379 349 y Fk(1)1398 331 y Fz(\()p Fq(B)r(;)d Fz(\000)1492 337 y Fo(l)1504 331 y Fz(\))p Fq(:)228 414 y Fi(The)15 b(value)g(of)g Fq(r)488 420 y Fs(0)506 414 y Fz(\()p Fq(B)r Fz(\))h Fi(do)n(es)f(not)g(dep)n(end)h(on)f Fq(l)q Fi(.)278 496 y Fz(So,)23 b(for)e(a)g(giv)o(en)g(holomorph)o(y)f(domain)f Fq(D)1013 481 y Fo(l)1048 496 y Fz(\()p Fq(l)26 b Fz(=)e Fy(\006)p Fz(1\))e(the)h(solutions)e Fq(X)k Fz(and)d Fq(H)1693 502 y Fs(1)1711 496 y Fz(,)228 551 y Fq(H)263 557 y Fs(1)300 551 y Fz(=)361 540 y Fr(e)352 551 y Fq(A)383 557 y Fs(1)414 551 y Fz(+)12 b Fq(X)q(;)18 b Fz(do)h(not)f(dep)q(end)i (on)e(the)h Fq(K)986 557 y Fo(B)1015 551 y Fz(-b)q(ounded)g(con)o (tours)g(\000)1401 557 y Fo(l)1433 551 y Fy(\032)g Fq(D)1519 536 y Fo(l)1551 551 y Fz(satisfying)228 600 y(the)12 b(conditions)f(\(2.8\).)17 b(But)12 b(when)g(the)h(index)e Fq(l)i Fz(c)o(hanges,)f Fq(X)k Fz(and)11 b Fq(H)1324 606 y Fs(1)1354 600 y Fz(can)h(also)f(c)o(hange.)18 b(F)m(or)228 650 y(this)13 b(reason)g(w)o(e)g(shall)f(supply)g(them)g(in)g(the)h (follo)o(wing,)d(when)j(it)g(is)f(necessary)m(,)i(with)f(the)g(in-)228 706 y(dex)f Fq(l)i Fz(writing)e Fq(X)508 691 y Fs(\()p Fo(l)p Fs(\))559 706 y Fz(and)g Fq(H)676 684 y Fs(\()p Fo(l)p Fs(\))673 717 y(1)715 706 y Fz(,)g Fq(H)777 684 y Fs(\()p Fo(l)p Fs(\))774 717 y(1)827 706 y Fz(=)881 695 y Fr(e)871 706 y Fq(A)902 712 y Fs(1)927 706 y Fz(+)6 b Fq(X)1002 691 y Fs(\()p Fo(l)p Fs(\))1041 706 y Fz(.)18 b(Surely)m(,)12 b(the)h(equations)f(\(2.3\))g(and)g(\(2.4\))228 756 y(are)j(nonlinear)f(equations)h(and,)f(outside)h(the)g(balls)f Fy(k)p Fq(X)s Fy(k)f Fq(<)h(r)1232 762 y Fs(max)1295 756 y Fz(\(\000)1337 762 y Fo(l)1349 756 y Fz(\),)h(they)g(ma)o(y)m(,)d (in)i(prin-)228 814 y(ciple,)f(ha)o(v)o(e)g(other)g(solutions,)g (di\013eren)o(t)h(from)d(the)j Fq(X)1091 799 y Fs(\()p Fo(l)p Fs(\))1143 814 y Fz(or)f Fq(H)1231 792 y Fs(\()p Fo(l)p Fs(\))1228 825 y(1)1283 814 y Fz(the)h(existence)h(of)d(whic)o (h)h(is)228 866 y(guaran)o(teed)i(b)o(y)f(Theorem)g(2.1.)20 b(In)14 b(the)h(follo)o(wing)d(w)o(e)j(only)f(deal)g(with)h(the)g (solutions)f Fq(X)1683 851 y Fs(\()p Fo(l)p Fs(\))228 921 y Fz(or)g Fq(H)317 900 y Fs(\()p Fo(l)p Fs(\))314 932 y(1)369 921 y Fz(for)g Fq(l)e Fz(=)g Fy(\006)p Fz(1.)703 1041 y(3.)22 b FA(F)-5 b(a)o(ctoriza)m(tion)15 b(theorem)278 1116 y Fz(No)o(w)f(w)o(e)g(pro)o(v)o(e)h(a)f Fi(factorization)h(the)n (or)n(em)f Fz(for)g(the)h(transfer)g(function)g Fq(M)1475 1122 y Fs(1)1494 1116 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1576 1122 y Fo(l)1588 1116 y Fz(\).)20 b(Note)228 1165 y(that)14 b(this)g(theorem)f(recalls)h(the)h(corresp)q(onding)g(statemen)o(ts)f (from)e([MrMt,)h(VM)q(].)228 1247 y Ft(Theorem)i(3.1.)21 b Fi(L)n(et)e Fz(\000)628 1253 y Fo(l)661 1247 y Fi(b)n(e)h(a)g Fq(K)793 1253 y Fo(B)822 1247 y Fi(-b)n(ounde)n(d)h(c)n(ontour)g (satisfying)e(the)i(c)n(onditions)f Fz(\(2.8\))p Fi(.)228 1298 y(Supp)n(ose)j Fq(X)430 1283 y Fs(\()p Fo(l)p Fs(\))491 1298 y Fi(is)f(the)g(solution)g(of)g(the)g(b)n(asic)g(e)n(quation)g Fz(\(2.4\))p Fi(,)h Fy(k)p Fq(X)1376 1283 y Fs(\()p Fo(l)p Fs(\))1415 1298 y Fy(k)h(\024)h Fq(r)1536 1304 y Fs(0)1554 1298 y Fz(\()p Fq(B)r Fz(\))p Fi(,)f(and)228 1356 y Fq(H)266 1335 y Fs(\()p Fo(l)p Fs(\))263 1367 y(1)322 1356 y Fz(=)381 1346 y Fr(e)372 1356 y Fq(A)403 1362 y Fs(1)433 1356 y Fz(+)12 b Fq(X)514 1341 y Fs(\()p Fo(l)p Fs(\))553 1356 y Fi(.)29 b(Then,)19 b(for)f Fq(z)h Fy(2)e Fp(C)k Fy(n)11 b Fz(\000)973 1362 y Fo(l)986 1356 y Fi(,)19 b(the)f(tr)n(ansfer)f(function)i Fq(M)1455 1362 y Fs(1)1474 1356 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1556 1362 y Fo(l)1568 1356 y Fz(\))18 b Fi(admits)228 1406 y(the)d(factorization)666 1491 y Fq(M)706 1497 y Fs(1)725 1491 y Fz(\()p Fq(z)r(;)7 b Fz(\000)807 1497 y Fo(l)820 1491 y Fz(\))26 b(=)h Fq(W)960 1497 y Fs(1)979 1491 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1061 1497 y Fo(l)1073 1491 y Fz(\))g(\()p Fq(H)1150 1470 y Fs(\()p Fo(l)p Fs(\))1147 1503 y(1)1198 1491 y Fy(\000)j Fq(z)r Fz(\))360 b(\(3.1\))228 1573 y Fi(wher)n(e)14 b Fq(W)384 1579 y Fs(1)403 1573 y Fz(\()p Fq(z)r(;)7 b Fz(\000)485 1579 y Fo(l)497 1573 y Fz(\))15 b Fi(is)g(a)g(b)n(ounde)n (d)h(op)n(er)n(ator)e(in)h Fy(H)1016 1579 y Fs(1)1035 1573 y Fi(,)477 1674 y Fq(W)516 1680 y Fs(1)535 1674 y Fz(\()p Fq(z)r(;)7 b Fz(\000)617 1680 y Fo(l)630 1674 y Fz(\))41 b(=)h Fq(I)779 1680 y Fs(1)807 1674 y Fy(\000)849 1618 y Fr(Z)843 1737 y Fs(\000)864 1741 y Fj(l)897 1674 y Fq(d\026)7 b(K)989 1657 y Fx(0)986 1684 y Fo(B)1014 1674 y Fz(\()p Fq(\026)p Fz(\))g(\()p Fq(H)1132 1653 y Fs(\()p Fo(l)p Fs(\))1129 1685 y(1)1181 1674 y Fy(\000)i Fq(\026)p Fz(\))1263 1657 y Fx(\000)p Fs(1)761 1828 y Fz(+)p Fq(z)828 1772 y Fr(Z)823 1891 y Fs(\000)844 1895 y Fj(l)877 1828 y Fq(d\026)e(K)969 1811 y Fx(0)966 1838 y Fo(B)994 1828 y Fz(\()p Fq(\026)p Fz(\)\()p Fq(z)12 b Fy(\000)d Fq(\026)p Fz(\))1180 1811 y Fx(\000)p Fs(1)1225 1828 y Fz(\()p Fq(H)1279 1807 y Fs(\()p Fo(l)p Fs(\))1276 1839 y(1)1327 1828 y Fy(\000)h Fq(\026)p Fz(\))1410 1811 y Fx(\000)p Fs(1)1461 1828 y Fq(:)1637 1769 y Fz(\(3.2\))228 1974 y Fi(If)24 b Fz(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()460 1963 y Fr(e)450 1974 y Fq(A)481 1980 y Fs(1)500 1974 y Fz(\))p Fy(g)29 b(\024)g Fq(d)649 1980 y Fs(0)668 1974 y Fz(\(\000)710 1980 y Fo(l)723 1974 y Fz(\)[1)8 b Fy(\000)i Fz(V)m(ar)894 1981 y Fh(e)886 1988 y Fo(A)911 1992 y Fk(1)930 1974 y Fz(\()p Fq(B)r(;)d Fz(\000)1024 1980 y Fo(l)1036 1974 y Fz(\)])p Fq(=)p Fz(2)p Fi(,)26 b(then)g(the)e(op)n(er)n(ator)g Fq(W)1538 1980 y Fs(1)1557 1974 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1639 1980 y Fo(l)1652 1974 y Fz(\))24 b Fi(is)228 2025 y(b)n(ounde)n(d)r(ly)16 b(invertible)e(and)307 2115 y Fr(\015)307 2140 y(\015)330 2150 y Fz([)p Fq(W)381 2156 y Fs(1)400 2150 y Fz(\()p Fq(z)r(;)7 b Fz(\000)482 2156 y Fo(l)494 2150 y Fz(\)])522 2133 y Fx(\000)p Fs(1)566 2115 y Fr(\015)566 2140 y(\015)616 2150 y Fy(\024)674 2079 y Fr( )707 2150 y Fz(1)i Fy(\000)784 2116 y Fz(4)e(V)m(ar)884 2123 y Fh(e)876 2130 y Fo(A)901 2134 y Fk(1)919 2116 y Fz(\()p Fq(B)r(;)g Fz(\000)1013 2122 y Fo(l)1026 2116 y Fz(\))1049 2082 y Fr(\002)1066 2116 y Fq(d)1088 2122 y Fs(0)1107 2116 y Fz(\(\000)1149 2122 y Fo(l)1162 2116 y Fz(\))i(+)g Fy(k)1259 2105 y Fr(e)1249 2116 y Fq(A)1280 2122 y Fs(1)1299 2116 y Fy(k)1320 2082 y Fr(\003)p 784 2140 554 2 v 827 2189 a Fq(d)849 2195 y Fs(0)867 2189 y Fz(\(\000)909 2195 y Fo(l)922 2189 y Fz(\))938 2155 y Fr(\002)955 2189 y Fz(1)g(+)h(V)m(ar)1099 2196 y Fh(e)1092 2203 y Fo(A)1117 2207 y Fk(1)1135 2189 y Fz(\()p Fq(B)r(;)d Fz(\000)1229 2195 y Fo(l)1242 2189 y Fz(\))1258 2155 y Fr(\003)1275 2164 y Fs(2)1342 2079 y Fr(!)1375 2085 y Fx(\000)p Fs(1)1446 2150 y Fq(<)26 b Fy(1)p Fq(:)79 b Fz(\(3.3\))228 2274 y Fi(Pr)n(o)n(of.)20 b Fz(First)14 b(w)o(e)g(pro)o(v)o(e)g(the)h(form)o(ula)c(\(3.1\).)17 b(Note)e(that,)e(according)h(to)g(\(2.1\))f(and)h(\(2.4\),)279 2365 y Fr(e)269 2375 y Fq(A)300 2381 y Fs(1)344 2375 y Fz(=)26 b Fq(H)440 2354 y Fs(\()p Fo(l)p Fs(\))437 2386 y(1)488 2375 y Fy(\000)9 b Fq(V)553 2381 y Fs(1)572 2375 y Fz(\()598 2365 y Fr(e)588 2375 y Fq(A)619 2381 y Fs(1)647 2375 y Fz(+)h Fq(X)726 2358 y Fs(\()p Fo(l)p Fs(\))765 2375 y Fq(;)d Fz(\000)810 2381 y Fo(l)822 2375 y Fz(\))25 b(=)h Fq(H)959 2354 y Fs(\()p Fo(l)p Fs(\))956 2386 y(1)1007 2375 y Fy(\000)1048 2319 y Fr(Z)1043 2438 y Fs(\000)1064 2442 y Fj(l)1097 2375 y Fq(d\026)7 b(K)1189 2358 y Fx(0)1186 2385 y Fo(B)1214 2375 y Fz(\()p Fq(\026)p Fz(\))p Fq(H)1309 2354 y Fs(\()p Fo(l)p Fs(\))1306 2386 y(1)1348 2375 y Fz(\()p Fq(H)1402 2354 y Fs(\()p Fo(l)p Fs(\))1399 2386 y(1)1450 2375 y Fy(\000)j Fq(\026)p Fz(\))1533 2358 y Fx(\000)p Fs(1)1584 2375 y Fq(:)41 b Fz(\(3.4\))p eop %%Page: 8 8 8 7 bop 228 147 a Fl(8)377 b(V.)12 b(Hardt,)h(R.)g(Mennic)o(k)o(en,)h (A.)e(K.)h(Moto)o(vilo)o(v)228 246 y Fz(Th)o(us,)h(in)f(view)h(of)f (the)i(represen)o(tations)g(\(1.3\))f(and)f(\(1.4\),)g(the)h(function)g Fq(M)1460 252 y Fs(1)1479 246 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1561 252 y Fo(l)1573 246 y Fz(\))14 b(can)h(b)q(e)228 296 y(written)f(as)390 393 y Fq(M)430 399 y Fs(1)449 393 y Fz(\()p Fq(z)r(;)7 b Fz(\000)531 399 y Fo(l)544 393 y Fz(\))41 b(=)685 382 y Fr(e)675 393 y Fq(A)706 399 y Fs(1)734 393 y Fy(\000)10 b Fq(z)h Fz(+)848 336 y Fr(Z)842 455 y Fs(\000)863 459 y Fj(l)896 393 y Fq(d\026)c(K)988 376 y Fx(0)985 403 y Fo(B)1013 393 y Fz(\()p Fq(\026)p Fz(\))1114 365 y Fq(z)p 1075 383 97 2 v 1075 421 a(z)12 b Fy(\000)e Fq(\026)601 537 y Fz(=)42 b Fq(H)713 516 y Fs(\()p Fo(l)p Fs(\))710 548 y(1)761 537 y Fy(\000)10 b Fq(z)h Fy(\000)874 481 y Fr(Z)869 600 y Fs(\000)890 604 y Fj(l)923 537 y Fq(d\026)c(K)1015 520 y Fx(0)1012 547 y Fo(B)1040 537 y Fz(\()p Fq(\026)p Fz(\)\()p Fq(H)1151 516 y Fs(\()p Fo(l)p Fs(\))1148 548 y(1)1199 537 y Fy(\000)j Fq(\026)p Fz(\))1282 520 y Fx(\000)p Fs(1)1327 504 y Fr(\000)1346 537 y Fq(H)1384 516 y Fs(\()p Fo(l)p Fs(\))1381 548 y(1)1431 537 y Fy(\000)g Fq(z)1494 504 y Fr(\001)675 685 y Fz(+)p Fq(z)736 629 y Fr(Z)730 748 y Fs(\000)751 752 y Fj(l)784 685 y Fq(d\026)d(K)876 668 y Fx(0)873 695 y Fo(B)901 685 y Fz(\()p Fq(\026)p Fz(\))965 626 y Fr(\024)1031 657 y Fz(1)p 992 675 V 992 713 a Fq(z)12 b Fy(\000)d Fq(\026)1104 685 y Fy(\000)g Fz(\()p Fq(H)1199 663 y Fs(\()p Fo(l)p Fs(\))1196 696 y(1)1247 685 y Fy(\000)g Fq(\026)p Fz(\))1329 668 y Fx(\000)p Fs(1)1374 626 y Fr(\025)601 829 y Fz(=)675 796 y Fr(\000)694 829 y Fq(H)732 808 y Fs(\()p Fo(l)p Fs(\))729 840 y(1)780 829 y Fy(\000)h Fq(z)843 796 y Fr(\001)871 829 y Fy(\000)912 773 y Fr(Z)907 892 y Fs(\000)928 896 y Fj(l)961 829 y Fq(d\026)d(K)1053 812 y Fx(0)1050 839 y Fo(B)1078 829 y Fz(\()p Fq(\026)p Fz(\)\()p Fq(H)1189 808 y Fs(\()p Fo(l)p Fs(\))1186 840 y(1)1237 829 y Fy(\000)j Fq(\026)p Fz(\))1320 812 y Fx(\000)p Fs(1)1365 796 y Fr(\000)1384 829 y Fq(H)1422 808 y Fs(\()p Fo(l)p Fs(\))1419 840 y(1)1469 829 y Fy(\000)g Fq(z)1532 796 y Fr(\001)675 973 y Fz(+)p Fq(z)736 917 y Fr(Z)730 1036 y Fs(\000)751 1040 y Fj(l)784 973 y Fq(d\026)d(K)876 956 y Fx(0)873 984 y Fo(B)901 973 y Fz(\()p Fq(\026)p Fz(\)\()p Fq(z)12 b Fy(\000)e Fq(\026)p Fz(\))1088 956 y Fx(\000)p Fs(1)1132 973 y Fz(\()p Fq(H)1186 952 y Fs(\()p Fo(l)p Fs(\))1183 984 y(1)1234 973 y Fy(\000)g Fq(\026)p Fz(\))1317 956 y Fx(\000)p Fs(1)1362 940 y Fr(\000)1381 973 y Fq(H)1419 952 y Fs(\()p Fo(l)p Fs(\))1416 984 y(1)1466 973 y Fy(\000)g Fq(z)1529 940 y Fr(\001)1548 973 y Fq(:)228 1116 y Fz(whic)o(h)19 b(pro)o(v)o(es)g(the)h(equation)e(\(3.1\).)33 b(The)20 b(b)q(oundeness)h(of)d(the)i(op)q(erator)f Fq(W)1524 1122 y Fs(1)1543 1116 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1625 1122 y Fo(l)1637 1116 y Fz(\))20 b(for)228 1166 y Fq(z)13 b Fy(2)f Fp(C)18 b Fy(n)9 b Fz(\000)395 1172 y Fo(l)421 1166 y Fz(is)14 b(ob)o(vious.)278 1216 y(F)m(urther,)i(w)o(e)f(giv)o(e) g(a)g(sk)o(etc)o(h)h(of)f(the)h(pro)q(of)f(that)g(the)h(factor)f Fq(W)1309 1222 y Fs(1)1328 1216 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1410 1222 y Fo(l)1423 1216 y Fz(\))15 b(is)g(a)g(b)q (oundedly)228 1271 y(in)o(v)o(ertible)20 b(op)q(erator)h(if)f(the)h (condition)f(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()1083 1260 y Fr(e)1073 1271 y Fq(A)1104 1277 y Fs(1)1122 1271 y Fz(\))p Fy(g)23 b(\024)g Fq(d)1259 1277 y Fs(0)1277 1271 y Fz(\(\000)1319 1277 y Fo(l)1332 1271 y Fz(\))7 b([1)i Fy(\000)g Fz(V)m(ar)1510 1278 y Fh(e)1503 1285 y Fo(A)1528 1289 y Fk(1)1546 1271 y Fz(\()p Fq(B)r(;)e Fz(\000)1640 1277 y Fo(l)1653 1271 y Fz(\)])p Fq(=)p Fz(2)228 1322 y(holds.)18 b(The)c(form)o(ula)536 1417 y Fy(k)p Fz(\()582 1407 y Fr(e)573 1417 y Fq(A)604 1423 y Fs(1)632 1417 y Fz(+)9 b Fq(X)710 1400 y Fs(\()p Fo(l)p Fs(\))759 1417 y Fy(\000)g Fq(\026)p Fz(\))841 1400 y Fx(\000)p Fs(1)886 1417 y Fy(k)25 b(\024)1179 1389 y Fz(1)p 995 1408 390 2 v 995 1453 a(dist)p Fy(f)p Fq(\026;)7 b(\033)q Fz(\()1177 1443 y Fr(e)1168 1453 y Fq(A)1199 1459 y Fs(1)1217 1453 y Fz(\))p Fy(g)i(\000)h(k)p Fq(X)s Fy(k)1396 1417 y Fq(;)229 b Fz(\(3.5\))228 1535 y(the)14 b(de\014nitions)g(of)f Fq(d)569 1541 y Fs(0)588 1535 y Fz(\(\000)630 1541 y Fo(l)642 1535 y Fz(\))h(and)g Fq(r)772 1541 y Fs(min)829 1535 y Fz(\(\000)871 1541 y Fo(l)884 1535 y Fz(\))g(and)g(Remark)e(2.1)h(imply)e(that)509 1586 y Fr(\015)509 1611 y(\015)509 1636 y(\015)509 1661 y(\015)509 1685 y(\015)509 1710 y(\015)545 1614 y(Z)541 1733 y Fs(\000)562 1737 y Fj(l)594 1671 y Fq(d\026)c(K)686 1654 y Fx(0)683 1681 y Fo(B)711 1671 y Fz(\()p Fq(\026)p Fz(\))g(\()p Fq(H)829 1649 y Fs(\()p Fo(l)p Fs(\))826 1682 y(1)877 1671 y Fy(\000)j Fq(\026)p Fz(\))960 1654 y Fx(\000)p Fs(1)1005 1586 y Fr(\015)1005 1611 y(\015)1005 1636 y(\015)1005 1661 y(\015)1005 1685 y(\015)1005 1710 y(\015)1053 1671 y Fy(\024)1138 1637 y Fz(2)d(V)m(ar)1237 1644 y Fh(e)1230 1651 y Fo(A)1255 1655 y Fk(1)1273 1637 y Fz(\()p Fq(B)r(;)g Fz(\000)1367 1643 y Fo(l)1380 1637 y Fz(\))p 1116 1661 303 2 v 1116 1699 a(1)i(+)g(V)m(ar)1259 1706 y Fh(e)1252 1713 y Fo(A)1277 1717 y Fk(1)1295 1699 y Fz(\()p Fq(B)r(;)e Fz(\000)1389 1705 y Fo(l)1402 1699 y Fz(\))1430 1671 y Fq(:)195 b Fz(\(3.6\))228 1814 y(Using)13 b(again)g(the)i(inequalit)o(y)d(\(3.5\))h(and)h(Remark)e(2.1)h(w)o(e)h (\014nd)259 1864 y Fr(\015)259 1889 y(\015)259 1914 y(\015)259 1939 y(\015)259 1964 y(\015)259 1989 y(\015)282 1950 y Fq(z)310 1893 y Fr(Z)305 2012 y Fs(\000)326 2016 y Fj(l)358 1950 y Fq(d\026)7 b(K)450 1932 y Fx(0)447 1960 y Fo(B)476 1950 y Fz(\()p Fq(\026)p Fz(\))g(\()p Fq(H)594 1928 y Fs(\()p Fo(l)p Fs(\))591 1961 y(1)642 1950 y Fy(\000)i Fq(\026)p Fz(\))724 1932 y Fx(\000)p Fs(1)769 1950 y Fz(\()p Fq(z)j Fy(\000)d Fq(\026)p Fz(\))898 1932 y Fx(\000)p Fs(1)943 1864 y Fr(\015)943 1889 y(\015)943 1914 y(\015)943 1939 y(\015)943 1964 y(\015)943 1989 y(\015)991 1950 y Fy(\024)26 b(j)p Fq(z)r Fy(j)1127 1915 y Fz(2)7 b(V)m(ar)1227 1922 y Fh(e)1219 1929 y Fo(A)1244 1933 y Fk(1)1262 1915 y Fz(\()p Fq(B)r(;)g Fz(\000)1356 1921 y Fo(l)1369 1915 y Fz(\))p 1105 1940 V 1105 1978 a(1)i(+)g(V)m(ar)1249 1985 y Fh(e)1241 1992 y Fo(A)1266 1996 y Fk(1)1284 1978 y Fz(\()p Fq(B)r(;)e Fz(\000)1378 1984 y Fo(l)1391 1978 y Fz(\))1433 1950 y(sup)1426 1984 y Fo(\026)p Fx(2)p Fs(\000)1489 1988 y Fj(l)1508 1950 y Fy(j)p Fq(z)k Fy(\000)f Fq(\026)p Fy(j)1629 1932 y Fx(\000)p Fs(1)1680 1950 y Fq(:)228 2102 y Fz(The)k(inequalit)o(y)f(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()684 2091 y Fr(e)675 2102 y Fq(A)706 2108 y Fs(1)724 2102 y Fz(\))p Fy(g)12 b(\024)f Fq(d)838 2108 y Fs(0)857 2102 y Fz(\(\000)899 2108 y Fo(l)912 2102 y Fz(\)[1)d Fy(\000)i Fz(V)m(ar)1083 2109 y Fh(e)1075 2116 y Fo(A)1100 2120 y Fk(1)1119 2102 y Fz(\()p Fq(B)r(;)d Fz(\000)1213 2108 y Fo(l)1225 2102 y Fz(\)])p Fq(=)p Fz(2)14 b(yields)377 2215 y Fy(j)p Fq(z)r Fy(j)24 b(\024)i(k)535 2205 y Fr(e)525 2215 y Fq(A)556 2221 y Fs(1)575 2215 y Fy(k)9 b Fz(+)g(dist)713 2181 y Fr(\010)737 2215 y Fq(z)r(;)e(\033)q Fz(\()828 2205 y Fr(e)818 2215 y Fq(A)849 2221 y Fs(1)868 2215 y Fz(\))884 2181 y Fr(\011)920 2215 y Fy(\024)12 b(k)994 2205 y Fr(e)985 2215 y Fq(A)1016 2221 y Fs(1)1034 2215 y Fy(k)d Fz(+)1111 2187 y(1)p 1111 2206 21 2 v 1111 2244 a(2)1136 2215 y Fq(d)1158 2221 y Fs(0)1177 2215 y Fz(\(\000)1219 2221 y Fo(l)1232 2215 y Fz(\)[1)f Fy(\000)i Fz(V)m(ar)1403 2222 y Fh(e)1395 2229 y Fo(A)1420 2233 y Fk(1)1439 2215 y Fz(\()p Fq(B)r(;)d Fz(\000)1533 2221 y Fo(l)1545 2215 y Fz(\)])228 2313 y(and)14 b(one)g(obtains)f(for)h Fq(\026)d Fy(2)g Fz(\000)695 2319 y Fo(l)722 2313 y Fz(that)596 2416 y(sup)589 2451 y Fo(\026)p Fx(2)p Fs(\000)652 2455 y Fj(l)671 2416 y Fy(j)p Fq(z)g Fy(\000)f Fq(\026)p Fy(j)792 2399 y Fx(\000)p Fs(1)861 2416 y Fy(\024)1124 2388 y Fz(2)p 924 2407 421 2 v 924 2445 a Fq(d)946 2451 y Fs(0)964 2445 y Fz(\(\000)1006 2451 y Fo(l)1019 2445 y Fz(\)[1)e(+)i(V)m(ar)1190 2452 y Fh(e)1183 2459 y Fo(A)1208 2463 y Fk(1)1226 2445 y Fz(\()p Fq(B)r(;)d Fz(\000)1320 2451 y Fo(l)1333 2445 y Fz(])1349 2416 y Fq(:)p eop %%Page: 9 9 9 8 bop 565 147 a Fl(F)m(actorization)15 b(Theorem)e(for)g(the)g(T)m (ransfer)g(F)m(unction)320 b(9)228 246 y Fz(Hence,)15 b(for)e(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()605 236 y Fr(e)595 246 y Fq(A)626 252 y Fs(1)645 246 y Fz(\))p Fy(g)k(\024)h Fq(d)759 252 y Fs(0)777 246 y Fz(\(\000)819 252 y Fo(l)832 246 y Fz(\)[1)d Fy(\000)g Fz(V)m(ar)1004 253 y Fh(e)996 260 y Fo(A)1021 264 y Fk(1)1039 246 y Fz(\()p Fq(B)r(;)e Fz(\000)1133 252 y Fo(l)1146 246 y Fz(\)])p Fq(=)p Fz(2)o(,)278 352 y Fy(k)p Fq(W)338 358 y Fs(1)356 352 y Fz(\()p Fq(z)r(;)g Fz(\000)438 358 y Fo(l)451 352 y Fz(\))i Fy(\000)g Fq(I)535 358 y Fs(1)554 352 y Fy(k)42 b(\024)717 317 y Fz(2)7 b(V)m(ar)817 324 y Fh(e)810 331 y Fo(A)835 335 y Fk(1)853 317 y Fz(\()p Fq(B)r(;)g Fz(\000)947 323 y Fo(l)960 317 y Fz(\))p 695 342 303 2 v 695 380 a(1)i(+)h(V)m(ar)839 387 y Fh(e)831 394 y Fo(A)856 398 y Fk(1)875 380 y Fz(\()p Fq(B)r(;)d Fz(\000)969 386 y Fo(l)982 380 y Fz(\))690 506 y(+)727 463 y(4)g(V)m(ar)828 470 y Fh(e)820 477 y Fo(A)845 481 y Fk(1)863 463 y Fz(\()p Fq(B)r(;)g Fz(\000)957 469 y Fo(l)970 463 y Fz(\))1000 416 y Fr(n)1028 463 y Fy(k)1058 452 y Fr(e)1049 463 y Fq(A)1080 469 y Fs(1)1098 463 y Fy(k)i Fz(+)1175 446 y Fs(1)p 1175 453 17 2 v 1175 477 a(2)1203 463 y Fq(d)1225 469 y Fs(0)1243 463 y Fz(\(\000)1285 469 y Fo(l)1298 463 y Fz(\)[1)g Fy(\000)g Fz(V)m(ar)1470 470 y Fh(e)1462 477 y Fo(A)1487 481 y Fk(1)1505 463 y Fz(\()p Fq(B)r(;)e Fz(\000)1599 469 y Fo(l)1612 463 y Fz(\)])1640 416 y Fr(o)p 727 496 940 2 v 966 534 a Fq(d)988 540 y Fs(0)1007 534 y Fz(\(\000)1049 540 y Fo(l)1061 534 y Fz(\))g([1)i(+)h(V)m(ar)1240 541 y Fh(e)1232 548 y Fo(A)1257 552 y Fk(1)1276 534 y Fz(\()p Fq(B)r(;)d Fz(\000)1370 540 y Fo(l)1382 534 y Fz(\)])1410 522 y Fs(2)617 644 y Fz(=)695 609 y(4)g(V)m(ar)795 616 y Fh(e)788 623 y Fo(A)813 627 y Fk(1)831 609 y Fz(\()p Fq(B)r(;)g Fz(\000)925 615 y Fo(l)938 609 y Fz(\))14 b([)p Fq(d)1002 615 y Fs(0)1020 609 y Fz(\(\000)1062 615 y Fo(l)1074 609 y Fz(\))c(+)f Fy(k)1172 599 y Fr(e)1162 609 y Fq(A)1193 615 y Fs(1)1212 609 y Fy(k)p Fz(])p 695 634 549 2 v 739 672 a Fq(d)761 678 y Fs(0)779 672 y Fz(\(\000)821 678 y Fo(l)834 672 y Fz(\))e([1)h(+)i(V)m(ar)1012 679 y Fh(e)1004 686 y Fo(A)1029 690 y Fk(1)1048 672 y Fz(\()p Fq(B)r(;)d Fz(\000)1142 678 y Fo(l)1154 672 y Fz(\)])1182 660 y Fs(2)1274 644 y Fq(<)26 b Fz(1)7 b Fq(:)228 743 y Fz(The)15 b(last)g(inequalit)o(y)e(is)i(a)g(direct)h(consequence)h(of)d(the)i (second)g(assumption)e(in)g(\(2.8\).)21 b(W)m(e)228 793 y(conclude)14 b(that)g Fq(W)527 799 y Fs(1)546 793 y Fz(\()p Fq(z)r(;)7 b Fz(\000)628 799 y Fo(l)641 793 y Fz(\))14 b(is)f(in)o(v)o(ertible)h(and)g(that)f(the)i(inequalit)o(y)e (\(3.3\))g(holds.)p 1692 793 2 29 v 1694 767 25 2 v 1694 793 V 1719 793 2 29 v 228 868 a(The)h(follo)o(wing)d(theorems)j(can)g (b)q(e)h(pro)o(v)o(ed)f(in)f(the)i(same)e(w)o(a)o(y)g(as)h(Theorem)g (4.4)e(and)i(Theo-)228 918 y(rem)f(4.7)g(in)g([MM2].)228 1001 y Ft(Theorem)i(3.2.)21 b Fi(The)15 b(sp)n(e)n(ctrum)g Fq(\033)q Fz(\()p Fq(H)865 979 y Fs(\()p Fo(l)p Fs(\))862 1012 y(1)904 1001 y Fz(\))g Fi(of)h(the)f(op)n(er)n(ator)g Fq(H)1256 979 y Fs(\()p Fo(l)p Fs(\))1253 1012 y(1)1307 1001 y Fz(=)1361 990 y Fr(e)1352 1001 y Fq(A)1383 1007 y Fs(1)1411 1001 y Fz(+)10 b Fq(X)1490 986 y Fs(\()p Fo(l)p Fs(\))1545 1001 y Fi(b)n(elongs)15 b(to)228 1051 y(the)g(close)n(d)g Fq(r)437 1057 y Fs(0)455 1051 y Fz(\()p Fq(B)r Fz(\))p Fi(-neighb)n(ourho)n(o)n(d)523 1123 y Fy(O)556 1130 y Fo(r)571 1134 y Fk(0)588 1130 y Fs(\()p Fo(B)q Fs(\))642 1123 y Fz(\()668 1113 y Fr(e)658 1123 y Fq(A)689 1129 y Fs(1)708 1123 y Fz(\))27 b(:=)821 1089 y Fr(\010)845 1123 y Fq(z)14 b Fy(2)d Fp(C)21 b Fz(:)d(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()1167 1113 y Fr(e)1158 1123 y Fq(A)1189 1129 y Fs(1)1207 1123 y Fz(\))p Fy(g)12 b(\024)f Fq(r)1318 1129 y Fs(0)1337 1123 y Fz(\()p Fq(B)r Fz(\))1402 1089 y Fr(\011)228 1200 y Fi(of)j(the)g(sp)n(e)n(ctrum)f(of) 575 1189 y Fr(e)565 1200 y Fq(A)596 1206 y Fs(1)615 1200 y Fi(.)18 b(If)c(the)g(c)n(ontour)g Fz(\000)933 1206 y Fo(l)957 1200 y Fy(\032)e Fq(D)1036 1185 y Fo(l)1063 1200 y Fi(satis\014es)i Fz(\(2.8\))p Fi(,)f(then)h(the)g(nonr)n(e)n(al) g(sp)n(e)n(c-)228 1258 y(trum)f(of)g Fq(H)414 1236 y Fs(\()p Fo(l)p Fs(\))411 1269 y(1)466 1258 y Fi(b)n(elongs)h(to)f Fq(D)690 1243 y Fo(l)710 1258 y Fy(\\)6 b(O)777 1265 y Fo(r)792 1269 y Fk(0)809 1265 y Fs(\()p Fo(B)q Fs(\))863 1258 y Fz(\()889 1247 y Fr(e)879 1258 y Fq(A)910 1264 y Fs(1)929 1258 y Fz(\))p Fi(.)18 b(Mor)n(e)n(over,)c(the)f(sp)n(e)n (ctrum)g Fq(\033)q Fz(\()p Fq(H)1490 1236 y Fs(\()p Fo(l)p Fs(\))1487 1269 y(1)1529 1258 y Fz(\))h Fi(c)n(oincides)228 1308 y(with)f(a)g(subset)h(of)f(the)h(sp)n(e)n(ctrum)f(of)h(the)f(tr)n (ansfer)g(function)h Fq(M)1232 1314 y Fs(1)1251 1308 y Fz(\()p Fy(\001)p Fq(;)7 b Fz(\000)1324 1314 y Fo(l)1335 1308 y Fz(\))p Fi(.)19 b(Mor)n(e)13 b(pr)n(e)n(cisely,)g(the)228 1358 y(sp)n(e)n(ctrum)h(of)h Fq(M)491 1364 y Fs(1)510 1358 y Fz(\()p Fy(\001)p Fq(;)7 b Fz(\000)583 1364 y Fo(l)595 1358 y Fz(\))15 b Fi(in)g(the)g(set)339 1438 y Fy(O)q Fz(\()399 1427 y Fr(e)389 1438 y Fq(A)420 1444 y Fs(1)439 1438 y Fq(;)7 b Fz(\000)484 1444 y Fo(l)496 1438 y Fz(\))27 b(:=)609 1391 y Fr(n)637 1438 y Fq(z)14 b Fy(2)d Fp(C)27 b Fz(:)18 b(dist)p Fy(f)p Fq(z)r(;)7 b(\033)q Fz(\()966 1427 y Fr(e)956 1438 y Fq(A)987 1444 y Fs(1)1006 1438 y Fz(\))p Fy(g)k(\024)h Fq(d)1120 1444 y Fs(0)1138 1438 y Fz(\(\000)1180 1444 y Fo(l)1193 1438 y Fz(\))7 b([1)i Fy(\000)g Fz(V)m(ar)1371 1445 y Fh(e)1364 1452 y Fo(A)1389 1456 y Fk(1)1407 1438 y Fz(\()p Fq(B)r(;)e Fz(\000)1501 1444 y Fo(l)1514 1438 y Fz(\)])p Fq(=)p Fz(2)1584 1391 y Fr(o)228 1530 y Fi(e)n(quals)15 b(the)g(sp)n(e)n (ctrum)f(of)h Fq(H)680 1508 y Fs(\()p Fo(l)p Fs(\))677 1541 y(1)719 1530 y Fi(,)f(i.)7 b(e.,)646 1606 y Fq(\033)671 1572 y Fr(\000)691 1606 y Fq(M)731 1612 y Fs(1)749 1606 y Fz(\()p Fy(\001)p Fq(;)g Fz(\000)822 1612 y Fo(l)834 1606 y Fz(\))850 1572 y Fr(\001)879 1606 y Fy(\\)i(O)q Fz(\()976 1595 y Fr(e)966 1606 y Fq(A)997 1612 y Fs(1)1016 1606 y Fq(;)e Fz(\000)1061 1612 y Fo(l)1073 1606 y Fz(\))18 b(=)h Fq(\033)q Fz(\()p Fq(H)1237 1584 y Fs(\()p Fo(l)p Fs(\))1234 1617 y(1)1276 1606 y Fz(\))p Fq(:)333 b Fz(\(3.7\))228 1673 y Fi(In)15 b(fact)g(such)g(a)g(statement)g(sep)n(ar)n(ately)g (holds)g(for)f(the)h(p)n(oint)g(and)h(c)n(ontinuous)f(sp)n(e)n(ctr)n (a.)278 1754 y Fz(In)g(the)h(follo)o(wing)d(lemma)f(w)o(e)k(state)g(a)f (simple)f(but)i(useful)g(relation)e(b)q(et)o(w)o(een)j Fq(H)1601 1732 y Fs(\()p Fo(l)p Fs(\))1598 1765 y(1)1655 1754 y Fz(and)228 1812 y(the)g(adjoin)o(t)f(op)q(erator)h(of)f Fq(H)704 1791 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))701 1823 y(1)769 1812 y Fz(.)26 b(According)17 b(to)g(our)f(con)o(v)o(en)o(tion) h(\000)1374 1819 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1455 1812 y Fy(\032)f Fq(D)1538 1797 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1620 1812 y Fz(is)h(the)228 1862 y(con)o(tour)d(whic)o(h)g(is)f (conjugate)h(to)g(the)h(con)o(tour)f(\000)1027 1868 y Fo(l)1040 1862 y Fz(.)228 1937 y Ft(Lemma)h(3.1.)21 b Fi(L)n(et)d Fz(\000)593 1943 y Fo(l)623 1937 y Fy(\032)f Fq(D)707 1922 y Fo(l)738 1937 y Fi(b)n(e)h(a)g Fq(K)866 1943 y Fo(B)895 1937 y Fi(-b)n(ounde)n(d)h(c)n(ontour)g(for)e(which)h (the)g(c)n(onditions)g(of)228 1987 y Fz(Theorem)12 b(2.1)h Fi(ar)n(e)g(ful\014l)r(le)n(d.)18 b(Then)c(for)f(any)h Fq(z)g Fy(2)d Fp(C)i Fy(n)7 b Fz(\000)1110 1993 y Fo(l)1139 1987 y Fi(the)13 b(fol)r(lowing)g(e)n(quality)h(holds)g(true:)465 2067 y Fq(W)504 2073 y Fs(1)523 2067 y Fz(\()p Fq(z)r(;)7 b Fz(\000)605 2073 y Fo(l)618 2067 y Fz(\))648 2020 y Fr(\020)672 2067 y Fq(H)710 2045 y Fs(\()p Fo(l)p Fs(\))707 2078 y(1)758 2067 y Fy(\000)j Fq(z)821 2020 y Fr(\021)872 2067 y Fz(=)931 2020 y Fr(\020)955 2067 y Fq(H)993 2045 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))p Fx(\003)990 2078 y Fs(1)1084 2067 y Fy(\000)g Fq(z)1147 2020 y Fr(\021)1185 2033 y(\002)1203 2067 y Fq(W)1242 2073 y Fs(1)1261 2067 y Fz(\()p 1277 2044 22 2 v Fq(z)r(;)d Fz(\000)1343 2074 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1407 2067 y Fz(\))1423 2033 y Fr(\003)1440 2042 y Fx(\003)1473 2067 y Fq(:)152 b Fz(\(3.8\))228 2159 y Fi(F)m(urther)14 b(the)h(sp)n(e)n(ctrum)f(of)h Fq(H)708 2137 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))p Fx(\003)705 2170 y Fs(1)805 2159 y Fi(c)n(oincides)g(with)f(the)h(sp)n(e)n(ctrum)f (of)h Fq(H)1403 2137 y Fs(\()p Fo(l)p Fs(\))1400 2170 y(1)1442 2159 y Fi(.)228 2233 y(Pr)n(o)n(of.)20 b Fz(Let)14 b Fq(z)g Fy(2)d Fp(C)18 b Fy(n)9 b Fz(\000)599 2239 y Fo(l)612 2233 y Fz(.)18 b(By)c(de\014nition)p 892 2211 V 14 w Fq(z)f Fy(2)e Fp(C)19 b Fy(n)9 b Fz(\000)1059 2240 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1137 2233 y Fz(and)723 2301 y Fq(M)763 2307 y Fs(1)782 2301 y Fz(\()p Fq(z)r(;)e Fz(\000)864 2307 y Fo(l)876 2301 y Fz(\))892 2284 y Fx(\003)937 2301 y Fz(=)25 b Fq(M)1034 2307 y Fs(1)1053 2301 y Fz(\()p 1069 2278 V Fq(z)s(;)7 b Fz(\000)1136 2308 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1200 2301 y Fz(\))p Fq(:)409 b Fz(\(3.9\))228 2371 y(Therefore,)15 b(the)f(relation)f(\(3.8\))g(follo) o(ws)g(from)f(the)i(factorizations)678 2446 y Fq(M)718 2452 y Fs(1)737 2446 y Fz(\()p Fq(z)r(;)7 b Fz(\000)819 2452 y Fo(l)831 2446 y Fz(\))19 b(=)f Fq(W)955 2452 y Fs(1)974 2446 y Fz(\()p Fq(z)r(;)7 b Fz(\000)1056 2452 y Fo(l)1069 2446 y Fz(\))g(\()p Fq(H)1146 2425 y Fs(\()p Fo(l)p Fs(\))1143 2457 y(1)1194 2446 y Fy(\000)i Fq(z)r Fz(\))p eop %%Page: 10 10 10 9 bop 228 147 a Fl(10)358 b(V.)12 b(Hardt,)h(R.)g(Mennic)o(k)o(en,)h (A.)e(K.)h(Moto)o(vilo)o(v)228 246 y Fz(and)607 307 y Fq(M)647 313 y Fs(1)666 307 y Fz(\()p 682 284 22 2 v Fq(z)r(;)7 b Fz(\000)748 314 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))812 307 y Fz(\))19 b(=)g Fq(W)937 313 y Fs(1)955 307 y Fz(\()p 971 284 V Fq(z)s(;)7 b Fz(\000)1038 314 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1102 307 y Fz(\))g(\()p Fq(H)1179 285 y Fs(\()p Fx(\000)p Fo(l)p Fs(\))1176 318 y(1)1253 307 y Fy(\000)p 1294 284 V 9 w Fq(z)r 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1409 y Fz(+)d(\012)1475 1392 y Fs(\()p Fo(l)p Fs(\))1514 1409 y Fz(\))1530 1392 y Fx(\000)p Fs(1)1582 1409 y Fq(;)22 b Fz(\(3.13\))228 1541 y Fi(wher)n(e)12 b Fq(\015)j Fi(stands)f(for)e (an)h(arbitr)n(ary)e(r)n(e)n(cti\014able)h(close)n(d)h(c)n(ontour)g (going)g(ar)n(ound)h(the)f(sp)n(e)n(ctrum)228 1596 y(of)j Fq(H)316 1575 y Fs(\()p Fo(l)p Fs(\))313 1607 y(1)372 1596 y Fi(inside)g(the)h(set)f Fy(O)q Fz(\()691 1586 y Fr(e)681 1596 y Fq(A)712 1602 y Fs(1)731 1596 y Fq(;)7 b Fz(\000)776 1602 y Fo(l)789 1596 y Fz(\))16 b Fi(in)h(the)g(p)n (ositive)f(dir)n(e)n(ction.)24 b(The)16 b(inte)n(gr)n(ation)g(along)h Fq(\015)228 1646 y Fi(is)d(understo)n(o)n(d)i(in)f(the)g(str)n(ong)f (sense.)228 1727 y(Pr)n(o)n(of.)20 b Fz(The)c(estimate)g(in)f(\(3.11\)) h(can)g(b)q(e)g(pro)o(v)o(ed)h(b)o(y)e(using)h(the)h(relation)e (\(3.8\))h(follo)o(wing)228 1778 y(the)h(pro)q(of)g(of)f(the)i (estimate)f(\(3.3\).)26 b(This)17 b(estimate)g(yields)f(that)h(the)h (sum)e Fq(I)1497 1784 y Fs(1)1527 1778 y Fz(+)c(\012)1601 1763 y Fs(\()p Fo(l)p Fs(\))1657 1778 y Fz(is)k(a)228 1828 y(b)q(oundedly)e(in)o(v)o(ertible)f(op)q(erator)i(in)e Fy(H)865 1834 y Fs(1)884 1828 y Fz(.)278 1878 y(T)m(o)g(pro)o(v)o(e)i (the)g(form)o(ula)d(\(3.12\))i(w)o(e)h(recall)g(that)f(due)h(to)g(the)g (factorization)f(theorem)g(3.1)228 1936 y(and)9 b(the)h(form)o(ula)d (\(3.8\))i(the)h(follo)o(wing)d(factorization)h(holds)h(for)h(for)f Fq(z)k Fy(2)e(O)q Fz(\()1443 1925 y Fr(e)1432 1936 y Fq(A)1463 1942 y Fs(1)1483 1936 y Fq(;)c Fz(\000)1528 1942 y Fo(l)1540 1936 y Fz(\))p Fy(n)p Fq(\033)q Fz(\()p Fq(H)1656 1914 y Fs(\()p Fo(l)p Fs(\))1653 1947 y(1)1695 1936 y Fz(\):)497 2035 y([)p Fq(M)549 2041 y Fs(1)567 2035 y Fz(\()p Fq(z)r(;)g Fz(\000)649 2041 y Fo(l)662 2035 y Fz(\)])690 2020 y Fx(\000)p Fs(1)775 2035 y Fz(=)849 1989 y Fr(\020)874 2035 y Fq(H)912 2014 y Fs(\()p Fo(l)p Fs(\))909 2046 y(1)960 2035 y Fy(\000)i Fq(z)1022 1989 y Fr(\021)1047 1998 y Fx(\000)p Fs(1)1105 2035 y Fz([)p Fq(W)1156 2041 y Fs(1)1175 2035 y 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Fz(w)o(e)f(ha)o(v)o(e)h Fy(k)824 1567 y Fr(e)827 1578 y Fq(b)845 1584 y Fx(\006)872 1578 y Fz(\()p Fq(\026;)7 b Fy(\001)p Fz(\))p Fy(k)20 b Fz(=)h Fy(k)p Fq(b)p Fy(k)p Fz(.)34 b(Since)1284 1567 y Fr(e)1274 1578 y Fq(A)1305 1584 y Fs(1)1343 1578 y Fz(is)19 b(b)q(ounded,)i(one)f (can)228 1628 y(alw)o(a)o(ys)13 b(c)o(ho)q(ose)i(a)f Fq(K)563 1634 y Fo(B)592 1628 y Fz(-b)q(ounded)g(con)o(tour)h(\000)f (lying)f(in)g(the)i(domain)d(\(4.3\).)18 b(Indeed,)d(for)f(the)228 1677 y Fq(K)263 1683 y Fo(B)292 1677 y Fz(-b)q(oundedness)h(of)e(the)i (con)o(tour)f(\000)f(it)h(is)f(su\016cien)o(t)i(to)e(ha)o(v)o(e)h(its)g (in\014nite)f(part)h(presen)o(ted)228 1727 y(b)o(y)h(an)f(appropriate)i (semi-in\014nite)d(real)i(in)o(terv)n(al.)21 b(Th)o(us,)15 b(if)f(the)i(function)f Fq(b)f Fz(is)h(su\016cien)o(tly)228 1777 y(small)d(in)i(the)h(sense)g(that)g(the)g(conditions)f(\(2.8\))f (hold,)g(one)i(can)f(apply)g(all)f(the)i(statemen)o(ts)228 1827 y(of)e(the)i(Section)f(2)f(and)h(3)g(to)f(the)i(corresp)q(onding)g (transfer)g(function)e Fq(M)1387 1833 y Fs(1)1406 1827 y Fz(\()p Fq(z)r(;)7 b Fz(\000\).)854 1919 y FA(References)228 1990 y Fl([AL])100 b Ff(Ad)o(amy)n(an,)13 b(V.)8 b(M.,)k Fl(and)h Ff(Langer,)g(H.)p Fl(:)k(Sp)q(ectral)d(prop)q(erties)h(of)d(a) g(class)i(of)e(op)q(erator-)403 2035 y(v)n(alued)i(functions,)g(J.)f (Op)q(erator)g(Theory)h Fe(33)f Fl(\(1995\),)g(259{277.)228 2081 y([ALMSa])25 b Ff(Ad)o(amy)n(an,)d(V.)8 b(M.,)22 b(Langer,)h(H.,)g(Mennicken,)f(R.,)g Fl(and)e Ff(Sa)o(urer,)k(J.:)31 b Fl(Sp)q(ec-)403 2127 y(tral)12 b(comp)q(onen)o(ts)i(of)d(selfadjoin)o (t)j(blo)q(c)o(k)f(op)q(erator)g(matrices)g(with)f(un)o(b)q(ounded)i (en)o(tries,)403 2172 y(Math.)f(Nac)o(hr.)g Fe(178)g Fl(\(1996\),)g(43{80.)228 2218 y([A)o(G])95 b Ff(Akhiezer,)17 b(N.)7 b(I.)p Fl(,)15 b(and)h Ff(Glazman,)h(I.)7 b(M.)p Fl(:)21 b(Theory)15 b(of)g(linear)i(op)q(erators)g(in)f(Hilb)q(ert)403 2264 y(space,)d(V)m(ol.)g Fe(2)p Fl(,)f(Ungar,)h(New)g(Y)m(ork,)f (1963.)228 2309 y([G])123 b Ff(Goedbloed,)12 b(J.)7 b(P.)p Fl(:)16 b(Lecture)c(notes)g(on)g(ideal)h(magnetoh)o(ydro)q(dynami)q (cs,)h(Rijnhiuzen)403 2355 y(Rep)q(ort,)f(F)m(orm)g(Instutuut)h(v)o(o)q (or)f(Plasmafysica,)i(Nieww)o(egein,)f(1983,)f(83{145.)228 2401 y([GK])93 b Ff(Gohber)o(g,)14 b(I.)7 b(C.,)14 b Fl(and)g Ff(Krein,)i(M.)7 b(G.)p Fl(:)18 b(In)o(tro)q(duction)d(to)f (the)g(theory)g(of)f(linear)j(non-)403 2446 y(selfadjoin)o(t)e(op)q (erators,)g(American)g(Mathematical)h(So)q(ciet)o(y)m(,)f(Pro)o (vidence,)h(1988.)p eop %%Page: 14 14 14 13 bop 228 147 a Fl(14)358 b(V.)12 b(Hardt,)h(R.)g(Mennic)o(k)o(en,) h(A.)e(K.)h(Moto)o(vilo)o(v)228 246 y([GKMT])30 b Ff(Gesztesy,)22 b(F.,)h(Kal)n(ton,)g(N.)7 b(J.,)23 b(Makar)o(o)o(v,)g(K.)7 b(A.,)22 b Fl(and)f Ff(Tsekano)o(vski)q(i,)g(E.)p Fl(:)403 292 y(Some)13 b(applicati)q(ons)j(of)d(op)q(erator-v)n(alued)j (Herglotz)e(functions,)h(LANL)e(E-prin)o(t)h(math.)403 337 y(F)l(A/9802103.)228 383 y([HMM])54 b Ff(Hardt,)20 b(V.,)f(Mennicken,)g(R.)d Fl(and)i Ff(Moto)o(vilo)o(v,)g(A.)8 b(K.)p Fl(:)25 b(A)16 b(factorization)j(theo-)403 429 y(rem)c(for)h(the)g(transfer)g(function)h(asso)q(ciated)g(with)g(a)e(2) c Fd(\002)f Fl(2)16 b(op)q(erator)h(matrix)f(ha)o(ving)403 474 y(un)o(b)q(ounded)f(couplings,)g(in)f(preparation.)228 520 y([K])123 b Ff(Ka)n(to,)12 b(T.)p Fl(:)k(P)o(erturbation)c(theory)f (for)g(linear)h(op)q(erators,)g(Springer-V)m(erlag)h(New)d(Y)m(ork,)403 566 y(1966.)228 611 y([KL])99 b Ff(Krein,)18 b(M.)7 b(G.)16 b Fl(and)g Ff(Langer,)i(H.)p Fl(:)983 602 y(\177)978 611 y(Ub)q(er)e(die)h Fc(Q)p Fl({F)m(unktion)h(eines)f(\005{hermitesc)o (hen)403 657 y(Op)q(erators)c(im)h(Raume)f(\005)798 661 y Fb({)822 657 y Fl(,)g(Acta)g(Sci.)g(Math.)g(Szeged)h Fe(34)p Fl(,)f(\(1973\),)g(191{230.)228 703 y([L])129 b Ff(Lifschitz,)16 b(A.)7 b(E.)p Fl(:)20 b(Magnetoh)o(ydro)q(dynami)q (cs)e(and)d(sp)q(ectral)h(theory)m(,)f(Klu)o(w)o(er)g(Acad-)403 748 y(emic)e(Publishers,)j(Dordrec)o(h)o(t,)d(1989.)228 794 y([MrMt])53 b Ff(Markus,)14 b(A.)7 b(S.,)12 b Fl(and)i Ff(Ma)n(tsaev,)f(V.)7 b(I.:)17 b Fl(On)c(the)g(basis)h(prop)q(ert)o(y)g (for)f(a)g(certain)h(part)403 840 y(of)9 b(the)i(eigen)o(v)o(ectors)g (and)g(asso)q(ciated)h(v)o(ectors)e(of)g(a)g(selfadjoin)o(t)i(op)q (erator)f(p)q(encil,)h(Math.)403 885 y(USSR)h(Sb.)g Fe(61)g Fl(\(1988\),)g(289{307.)228 931 y([MM1])64 b Ff(Mennicken,)21 b(R.,)g Fl(and)f Ff(Moto)o(vilo)o(v,)i(A.)8 b(K.)p Fl(:)30 b(Op)q(erator)20 b(in)o(terpretation)i(of)d(reso-)403 977 y(nances)11 b(generated)g(b)o(y)f(2)r Fd(\002)r Fl(2)h(matrix)g (Hamiltonians,)i(Theor.)d(Math.)g(Ph)o(ys.)g Fe(116)g Fl(\(1998\),)403 1022 y(867{880.)228 1068 y([MM2])64 b Ff(Mennicken,)21 b(R.,)g Fl(and)f Ff(Moto)o(vilo)o(v,)i(A.)8 b(K.)p Fl(:)30 b(Op)q(erator)20 b(in)o(terpretation)i(of)d(reso-)403 1114 y(nances)c(arising)h(in)g(sp)q(ectral)f(problems)h(for)f(2)9 b Fd(\002)h Fl(2)k(op)q(erator)h(matrices,)h(Math.)e(Nac)o(hr.)403 1159 y Fe(201)f Fl(\(1999\),)g(117{181)h(\(LANL)e(E-prin)o(t)i (funct-an/9708001)s(\).)228 1205 y([MS])97 b Ff(Mennicken,)12 b(R.,)f Fl(and)h Ff(Shkalik)o(o)o(v,)g(A.)7 b(A.:)17 b Fl(Sp)q(ectral)c(decomp)q(osition)h(of)d(symmetric)403 1251 y(op)q(erator)j(matrices,)f(Math.)g(Nac)o(hr.)g Fe(179)g Fl(\(1996\),)g(259{273.)228 1296 y([M1])99 b Ff(Moto)o(vilo)o(v,)32 b(A.)8 b(K.:)46 b Fl(P)o(oten)o(tials)30 b(app)q(earing)g(after)d(remo)o(v)n(al)i(of)e(the)h(energy{)403 1342 y(dep)q(endence)12 b(and)g(scattering)g(b)o(y)g(them,)f(In:)16 b(Pro)q(c.)10 b(of)h(the)g(In)o(tern.)g(W)m(orkshop)h(\\Mathe-)403 1388 y(matical)i(asp)q(ects)f(of)f(the)h(scattering)h(theory)f(and)h (applications")q(,)h(St.)d(P)o(etersburg)i(Uni-)403 1433 y(v)o(ersit)o(y)m(,)f(St.)g(P)o(etersburg)h(\(1991\),)f(101{108.)228 1479 y([M2])99 b Ff(Moto)o(vilo)o(v,)14 b(A.)7 b(K.:)19 b Fl(Remo)o(v)n(al)c(of)e(the)g(resolv)o(en)o(t-lik)o(e)k(energy)d(dep) q(endence)h(from)e(in-)403 1525 y(teractions)j(and)g(in)o(v)n(arian)o (t)i(subspaces)e(of)f(a)h(total)f(Hamiltonian,)k(J.)14 b(Math.)h(Ph)o(ys.)h Fe(36)403 1570 y Fl(\(1995\),)d(6647{6664)h (\(LANL)f(E-prin)o(t)h(funct-an/9606002\).)228 1616 y([N])124 b Ff(Nabok)o(o,)11 b(S.)c(N.)p Fl(:)16 b(Non)o(tangen)o(tial)d(b)q (oundary)g(v)n(alues)f(of)f(op)q(erator-v)n(alued)i Fc(R)p Fl(-functions)403 1662 y(in)g(a)g(half-plane,)i(Leningrad)g(Math.)e(J.) g Fe(1)f Fl(\(1990\),)i(1255{1278.)228 1707 y([RS])104 b Ff(Reed,)21 b(M.,)e Fl(and)g Ff(Simon,)i(B.)p Fl(:)28 b(Metho)q(ds)19 b(of)g(mo)q(dern)g(mathematical)i(ph)o(ysics,)g(I)q(I)q (I:)403 1753 y(Scattering)14 b(theory)m(,)f(Academic)h(Press,)f(N.Y.,)f (1979.)228 1799 y([VM])89 b Ff(Vir)o(ozub,)13 b(A.)7 b(I.,)12 b Fl(and)h Ff(Ma)n(tsaev,)f(V.)c(I.)p Fl(:)16 b(The)c(sp)q(ectral)h(prop)q(erties)g(of)f(a)g(certain)h(class)403 1844 y(of)f(selfadjoin)o(t)j(op)q(erator)f(functions,)g(F)m(unct.)f (Anal.)h(Appl.)f Fe(8)g Fl(\(1974\),)g(1{9.)278 1922 y Fv(V.)g(Hardt,)g(Dep)n(ar)n(tment)e(of)i(Ma)n(thema)n(tics,)f (University)i(of)g(Regensbur)o(g,)g(D-93040)g(Regens-)228 1964 y(bur)o(g,)f(Germany)278 2005 y Fn(E-mail)g(addr)n(ess)s Fw(:)i Fa(volker.hard)o(t@m)o(ath)o(em)o(ati)o(k.u)o(ni-)o(reg)o(en)o (sbu)o(rg.)o(de)278 2076 y Fv(R.)d(Mennicken,)i(Dep)n(ar)n(tment)d(of)i (Ma)n(thema)n(tics,)g(University)h(of)g(Regensbur)o(g,)g(D-93040)h(Re-) 228 2117 y(gensbur)o(g,)f(Germany)278 2159 y Fn(E-mail)f(addr)n(ess)s Fw(:)i Fa(reinhard.me)o(nni)o(cke)o(n@)o(mat)o(hem)o(ati)o(k.u)o(ni)o (-re)o(gen)o(sbu)o(rg.)o(de)278 2229 y Fv(A.K.)f(Moto)o(vilo)o(v,)j (Labora)n(tor)n(y)f(of)g(Theoretical)f(Physics,)h(Joint)f(Institute)h (f)o(or)f(Nuclear)228 2271 y(Resear)o(ch,)c(Dubna,)i(141980,)i(R)o (ussia)278 2312 y Fn(E-mail)e(addr)n(ess)s Fw(:)i Fa(motovilv@th)o(sun) o(1.j)o(in)o(r.r)o(u)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9912300445719--