Content-Type: multipart/mixed; boundary="-------------0705051049158" This is a multi-part message in MIME format. ---------------0705051049158 Content-Type: text/plain; name="07-114.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-114.keywords" adjoints, Banach space embeddings, Hilbert spaces ---------------0705051049158 Content-Type: application/x-tex; name="adjoint-II" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="adjoint-II" \documentclass{amsart} \usepackage{amssymb, latexsym} \renewcommand{\baselinestretch}{2} \parskip .5ex \newcommand{\bm}[1]{\mb{\boldmath ${#1}$}} \newcommand{\beqa}{\begin{eqnarray*}} \newcommand{\eeqa}{\end{eqnarray*}} \newcommand{\beqn}{\begin{eqnarray}} \newcommand{\eeqn}{\end{eqnarray}} \newcommand{\bu}{\bigcup} \newcommand{\bi}{\bigcap} \newcommand{\iy}{\infty} \newcommand{\lt}{\left} \newcommand{\rt}{\right} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\Lra}{\Leftrightarrow} \newcommand{\lgra}{\longrightarrow} \newcommand{\Lgra}{\Longrightarrow} \newcommand{\lglra}{\longleftrightarrow} \newcommand{\Lglra}{\Longleftrightarrow} \newcommand{\R}{\mathbb R} \newcommand{\K}{\mathbb K} \newcommand{\N}{\mathbb N} \newcommand{\ov}{\overline} \newcommand{\mb}{\makebox} \newcommand{\es}{\emptyset} \newcommand{\ci}{\subseteq} \newcommand{\rec}[1]{\frac{1}{#1}} \newcommand{\ld}{\ldots} \newcommand{\ds}{\displaystyle} \newcommand{\f}{\frac} \newcommand{\et}[2]{#1_{1}, #1_{2}, \ldots, #1_{#2}} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\G}{\Gamma} \newcommand{\e}{\varepsilon} \newcommand{\ph}{\phi} \newcommand{\de}{\delta} \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\m}{\mu} \newcommand{\Om}{\Omega} \newcommand{\Si}{\Sigma} \newcommand{\s}{\sigma} \newcounter{cnt1} \newcounter{cnt2} \newcounter{cnt3} \newcommand{\blr}{\begin{list}{$($\roman{cnt1}$)$} {\usecounter{cnt1} \setlength{\topsep}{0pt} \setlength{\itemsep}{0pt}}} \newcommand{\bla}{\begin{list}{$($\alph{cnt2}$)$} {\usecounter{cnt2} \setlength{\topsep}{0pt} \setlength{\itemsep}{0pt}}} \newcommand{\bln}{\begin{list}{$($\arabic{cnt3}$)$} {\usecounter{cnt3} \setlength{\topsep}{0pt} \setlength{\itemsep}{0pt}}} \newcommand{\el}{\end{list}} \newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{ex}[thm]{Example} \newtheorem{Q}[thm]{Question} \newtheorem{Def}[thm]{Definition} \newtheorem{prop}[thm]{Proposition} \newtheorem{rem}[thm]{Remark} \newcommand{\Rem}{\begin{rem} \rm} \newcommand{\bdfn}{\begin{Def} \rm} \newcommand{\edfn}{\end{Def}} \newcommand{\TFAE}{the following are equivalent~: } \newcommand{\etc}[3]{#1_{#3 1}, #1_{#3 2}, \ld, #1_{#3 #2}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\alto}[3]{\lt\{ \ba {ll}#1 & \mb{ if \quad}#2 \\ #3 & \mb{ otherwise} \ea \rt.} \newtheorem*{acknowledgements}{Acknowledgements} \sloppy \date{} \begin{document} \title{\bf Adjoint for Operators in Banach Spaces II} \author[Gill]{T. L. Gill} \address[Tepper L. Gill]{ Department of Electrical Engineering Howard University\\ % [-1ex] \normalsize \sc Howard University\\ %\and Washington DC 20059 \\ USA, {\it E-mail~:} {\tt tgill@howard.edu}} \author[Zachary]{W. W. Zachary} \address[Woodford W. Zachary]{ Department of Electrical Engineering \\ Howard University\\ Washington DC 20059 \\ USA, {\it E-mail~:} {\tt wwzachary@earthlink.net}} %\abstract{ write abstract here} \date{} %thispagestyle{empty} \subjclass{Primary (45) Secondary(46) } \keywords{Adjoints, Banach space embeddings, Hilbert spaces} \maketitle \begin{abstract} In a previous paper [GBZS] it was shown that each bounded linear operator $A$, defined on a separable Banach space $\mathcal{B}$, has a natural adjoint $A^*$. In this paper we prove that, for each closed linear operator $C$ defined on $ \mathcal{B}$, there exists a pair of contractions $A,\;B$ such that $C=AB^{-1}$. We also prove that, if $C$ is densely defined, then $B= (I-A^*A)^{-1/2}$. This result allows us to amend an oversight of [GBZS] by showing that every closed densely defined linear operator on $\mathcal{B}$ has a natural adjoint. \end{abstract} \section{Introduction} In a previous paper [GBZS], we used the fact that every separable Banach space ${\mathcal{B}}$ may be continuously embedded in a separable Hilbert space ${\mathcal{H}}$ to show that the bounded linear operators $L[{\mathcal{B}}]$ are continuously embedded in $L[{\mathcal{H}}]$, and that each operator $A \in L[{\mathcal{B}}]$ has a natural adjoint operator $A^* \in L[{\mathcal{B}}]$. This result has a number of interesting implications about $L[{\mathcal{B}}]$. For example, every ideal is a star ideal, and such notions as unitary, self-adjoint, normal, etc, may be defined in the same manner as for a Hilbert space. Furthermore, if ${\mathcal{B}}$ has the approximation property, then the embedding is dense and natural definitions of the Schatten classes can be given as restrictions of the intersections of the same classes on ${\mathcal{H}}$ to ${\mathcal{B}}$ (see [GZ]). In another direction, this theorem has also allowed us to provide a simple extension proof of the Fourier transform to $L^p$ spaces without any advanced methods of real analysis (see [GZ]). \section*{purpose} In [GBZS], we stated, but did not prove, that every closed densely defined linear operator on ${\mathcal{B}}$ has a natural adjoint. This result was used later in the paper. In this paper we correct that oversight and, in the process, extend a few results on operator ranges that, to our knowledge, have only been known for Hilbert spaces (see Fillmore and Williams [FW]). \section{Preliminaries} As above, $L[{\mathcal{B}}],L[{\mathcal{H}}]$ denote the bounded linear operators on ${\mathcal{B}},{\mathcal{H}}$ respectively, and ${\mathcal{B}}$ is a continuous dense embedding in ${\mathcal{H}}$. We state the next two theorems for reference. A proof can be found in Gill et al [GBZS]. The first theorem generalizes the well-known result of von Neumann [VN] for bounded operators on Hilbert spaces, while the second one extends a result of Lax [LX]. \begin{thm}\label{V: von} Let ${\mathcal{B}}$ be a separable Banach space and let $A$ be a bounded linear operator on ${\mathcal{B}}$. Then $A$ has a well-defined adjoint $A^ * $ defined on ${\mathcal{B}}$ such that: \begin{enumerate} \item the operator $ A^ * A \ge 0$ (maximal accretive), \item $(A^ * A)^ * = A^ * A$, and \item $I + A^ * A$ has a bounded inverse. \end{enumerate} \end{thm} \begin{thm}\label{L*: lax*} Let $A$ be a bounded linear operator on ${\mathcal{B}}$. Then $A$ has a bounded extension to $L[{\mathcal{H}} ]$, with $ \left\| A \right\|_{\mathcal{H}} \le k\left\| A \right\|_\mathcal{B}$ with $k$ constant; \end{thm} \begin{Def} If $B$ is a bounded linear operator on ${\mathcal{B}}$, we define $B^{-1}$ to be the inverse of the restriction of $B$ to the closure of $B^*(\mathcal{B})$. \end{Def} \section{\bf{Closed operators on ${\mathcal{B}}$} } We want to show that Theorem \ref{V: von} extends to closed operators, while avoiding individual domain considerations, and so, we take an indirect approach. We begin with two theorems on the range of operators, the proofs of which follow closely the Hilbert space case (see [FW]). \begin{thm}\label{EI: ex1} Suppose that $ {\mathcal{S}}$ is a subset of $({\mathcal{B}},\; \left\| {\, \cdot \,} \right\|)$, and $({\mathcal{S}}, \; \left\| {\, \cdot \,} \right\|^\prime )$ is a Banach space with $\left\| \psi \right\|^\prime \ge \left\| \psi \right\|$, for each $\psi \in \mathcal{S}$. Then ${\mathcal{S}}$ is the range of a nonnegative bounded linear operator in ${\mathcal{B}}.$ \end{thm} \begin{proof} Since ${\mathcal{S}}$ is a subset of ${\mathcal{B}}$, the inclusion map $T$ from $({\mathcal{S}}, \; \left\| {\, \cdot \,} \right\|^\prime )$ into $({\mathcal{B}}\; \left\| {\, \cdot \,} \right\|)$ is bounded. It follows that the adjoint of $T$, $T^* $, is bounded from $({\mathcal{B}}\; \left\| {\, \cdot \,} \right\|)$ to $({\mathcal{S}}, \; \left\| {\, \cdot \,} \right\|^\prime )$. If $T^* = U[TT^* ]^{1/2} $ is the polar decomposition of $T^* $, then $U$ is a partial isometry mapping ${\mathcal{B}}$ onto ${\mathcal{S}}$. Since $T$ is nonnegative, so is $U$. \end{proof} \begin{thm} If $A,B \in L({\mathcal{B}})$, then \[ R(A^* ) + R(B^* ) = R([A^* A + B^* B]^{1/2} ). \] \end{thm} \begin{proof} Let $T^*$ act on ${\mathcal{B}} \oplus {\mathcal{B}}$ in the normal way and represent it as $T^* = \left( {\begin{array}{*{20}c} {A^*} & {B^*} \\ 0 & 0 \\ \end{array} } \right)$, so that $T = \left( {\begin{array}{*{20}c} {A } & 0 \\ {B } & 0 \\ \end{array} } \right)$, and $T^*T = \left( {\begin{array}{*{20}c} {A^*A + B^*B } & 0 \\ 0 & 0 \\ \end{array} } \right)$. It follows that: \[ \begin{gathered} \left[ {R(A^* ) + R(B^* )} \right] \oplus \left\{ 0 \right\} = R(T^*) = R([T^*T ]^{1/2} ) = R\left( {\begin{array}{*{20}c} {[A^* A + B^* B]^{1/2} } & 0 \\ 0 & 0 \\ \end{array} } \right) \hfill \\ {\text{ }} = R([A^* A + B^* B]^{1/2} ) \oplus \left\{ 0 \right\}. \hfill \\ \end{gathered} \] \end{proof} \begin{thm}\label{EII: ex2} Let $C$ be a closed linear operator on $\mathcal{B}$. Then there exists a pair of bounded linear contraction operators $A,B \in L[\mathcal{B}]$ such that $ C = AB^{ - 1}$, with $B$ nonnegative. Furthermore, $D(C) = R(B)$, $R(C) = R(A)$ and $P = A^* A + B^* B$ is the orthogonal projection $B^{ - 1} B$ onto $\bar R(B^* ) = R(A^* ) + R(B^* )$. \end{thm} \begin{proof} Let ${\mathcal{S}} = D(C)$ be the domain of $C$ and endow it with the graph norm, so that $\left\| \varphi \right\|^\prime = \left\| \psi \right\| + \left\| {C \psi } \right\|$. Since $C$ is linear and closed, $({\mathcal{S}}, \; \left\| {\, \cdot \,} \right\|^\prime )$ is a Banach space and $\left\| \psi \right\|^\prime \ge \left\| \psi \right\|$. We will have use of the fact that, since ${\mathcal{S}}$ is a Banach subspace of ${\mathcal{B}}$, it is embedded in a Hilbert subspace $({\mathcal{S'}},\left\langle { \cdot , \cdot } \right\rangle ^\prime)$ of ${\mathcal{H}}$, where $\left\langle {\varphi ,\psi } \right\rangle ^\prime = \left\langle {\varphi ,\psi } \right\rangle + \left\langle {C\varphi ,C\psi } \right\rangle $. By Theorem \ref{EI: ex1}, there is a bounded nonnegative contraction $B$ with $B({\mathcal{B}}) = {\mathcal{S}}$ and, for $ \psi \in {\mathcal{S}}$, $\left\| \psi \right\|^\prime = \left\| {B^{ - 1} \psi } \right\| $. Now let $A = CB$ so that, for $\psi \in {\mathcal{B}}$, we have: \[ \begin{gathered} \left\| {A\psi } \right\| = \left\| {CB\psi } \right\| \leqslant \left\| {B\psi } \right\| + \left\| {CB\psi } \right\| \hfill \\ {\text{ = }}\left\| {B\psi } \right\|^\prime = \left\| {B^{ - 1} B\psi } \right\| = \left\| {P\psi } \right\| \leqslant \left\| \psi \right\|. \hfill \\ \end{gathered} \] Hence, $\left\| {A\varphi } \right\| \le \left\| \varphi \right\| $ so that $A$ is a contraction and $A = CB = (AB^{ - 1} )B = A(B^{ - 1} B) = AP$. Also, since $A$ and $B$ are bounded on $\mathcal{B}$, they have extensions to $\mathcal{H}$. With the same notation, we also have on $\mathcal{H}$: \[ \begin{gathered} \left\langle {\varphi ,[A^* A + B^* B]\psi } \right\rangle = \left\langle {B\varphi ,B\psi } \right\rangle + \left\langle {CB\varphi ,CB\psi } \right\rangle \hfill \\ = \left\langle {B\varphi ,B\psi } \right\rangle ^\prime = \left\langle {B^{ - 1} B\varphi ,B^{ - 1} B\psi } \right\rangle = \left\langle {P\varphi ,P\psi } \right\rangle = \left\langle {\varphi ,P\psi } \right\rangle . \hfill \\ \end{gathered} \] Hence, $A^* A + B^* B = P$ and, since $R(A^* ) + R(B^* ) = R([A^* A + B^* B]^{1/2} )$, $R(A^* ) + R(B^* ) $ is closed and equal to the closure of $R(B)$ on $\mathcal{H}$, the same is true for the restriction to $\mathcal{B}$ (note that $B$ is selfadjoint). \end{proof} Let ${\mathbf{V}}(\mathcal{B})$ be the set of contractions and ${\mathbf{C}}(\mathcal{B})$ the set of closed densely defined linear operators on ${\mathcal{B}}$. The following improvement of Theorem 5 is possible when the operator $C$ is also densely defined. This extends a theorem of Kaufman [KA] to Banach spaces. \begin{thm} \label{K: Kauf}The equation ${\rm K}(A) = A(I - A^* A)^{ - 1/2}$ defines a bijection from ${\mathbf{V}}(\mathcal{B})$ onto ${\mathbf{C}}(\mathcal{B})$, with inverse ${\rm K}^{ - 1} (C) = C(I + C^* C)^{ - 1/2} $. \end{thm} \begin{proof} Let $A \in {\mathbf{V}}(\mathcal{B})$ and set $B = (I - A^* A)^{1/2} $, which is easily seen to be positive and in ${\mathbf{V}}(\mathcal{B})$. It follows that ${\rm K}(A) = AB^{ - 1} $ and $A^* A + B^2 = I$ so that, by the proof of Theorem \ref{EII: ex2}, we see that ${\rm K}(A)$ is a closed linear operator on $\mathcal{B}$. Since the domain of ${\rm K}(A)$ is $B({\mathcal{B}})$, which is dense in ${\mathcal{B}}$, ${\rm K}(A)$ is in ${\mathbf{C}}(\mathcal{B})$. For the opposite direction, if $C \in {\mathbf{C}} ({\mathcal{B}})$, using the same notation, let $C$ be the extension to $\mathcal{H}$. Then, by Theorem \ref{EII: ex2} there exists a pair of bounded linear contraction operators $A,B \in L[\mathcal{H}]$ such that $C = AB^{ - 1} $ with $B$ positive with range $D(C)$ and $A^* A + B^2 = I$. Furthermore, for each nonzero $\varphi \in \mathcal{H}$, $\left\| \varphi \right\|_{\mathcal{H}}^2 - \left\| {A\varphi } \right\|_{\mathcal{H}}^2 = \left\| {B\varphi } \right\|_{\mathcal{H}}^2 > 0$. Thus, $A \in {\mathbf{V}}(\mathcal{H})$ with ${\rm K}(A) = C$, so that the restriction of $A \in {\mathbf{V}}(\mathcal{B})$ and ${\rm K}(A) = C$ on $\mathcal{B}$. Now, the graph of $C$ in $\mathcal{H}$ is the set of all $\{ (B\varphi ,A\varphi ),\varphi \in {\mathcal{H}}{\text{\} }}$, so that ${\text{C}}^{\text{*}} = \{ (\phi ,\psi ) \in {\mathcal{H}} \times {\mathcal{H}}\} $ such that $(\phi ,A\varphi )_{\mathcal{H}} = (\psi ,B\varphi )_{\mathcal{H}} $, or $(A^* \phi ,\varphi )_{\mathcal{H}} = (B\psi ,\varphi )_{\mathcal{H}} $ for all $\varphi \in {\mathcal{H}}$, so that $C^* = B^{ - 1} A^* $. Thus, the same is true for the restriction of $C^*$ to $\mathcal{B}$. It is clear that $I + C^* C$ is an invertible linear operator with bounded inverse and, for each $\varphi \in {\mathcal{B}}$, we have that $$ \begin{gathered} \varphi = B^2 \varphi + B^{ - 1} (I - B^2 )B^{ - 1} B^2 \varphi \hfill \\ {\text{ }} = (I + B^{ - 1} A^* AB^{ - 1} )B^2 \varphi = (I + C^* C)B^2 \varphi . \hfill \\ \end{gathered} $$ It follows that $(I + C^* C)^{ - 1} = B^2 $ and therefore, $A = CB = C(I + C^* C)^{ - 1/2} = K^{ - 1} (C)$. \end{proof} \begin{cor} Let ${\mathcal{B}}$ be a separable Banach space and let $A$ be a closed densely defined linear operator on ${\mathcal{B}}$. Then $A$ has a well-defined adjoint $A^ * $ defined on ${\mathcal{B}}$ such that: \begin{enumerate} \item the operator $ A^ * A \ge 0$ (maximal accretive), \item $(A^ * A)^ * = A^ * A$, and \item $I + A^ * A$ has a bounded inverse. \end{enumerate} \end{cor} From the proof of Theorem 6, we also have: \begin{cor} Let ${\mathcal{B}}$ be a separable Banach space. Then the mapping ${\rm K}(A) = A(I - A^* A)^{-1/2}$ is a bijection from the set of contractions, ${\mathbf{V}}(\mathcal{B})$, onto ${\mathbf{C}}(\mathcal{B})$, with inverse given by ${\rm K}(C)^{-1} = C(I + C^* C)^{-1/2}$. \end{cor} \section*{\textbf{Acknowledgement}} We would sincerely like to thank Jerry Goldstein for kindly pointing out our missing proof. \newpage \begin{thebibliography}{99} \small %\addcontentsline{toc}{chapter}{\protect\numberline{} \bibitem[FW]{FW} P. A. Fillmore and J. P. Williams, { \it On operator ranges,} Adv. Math. {\bf 7} (1971), 254-281. \bibitem[GBZS]{GBZS} T. Gill, S. Basu, W. W. Zachary and V. Steadman, { \it On natural adjoint operators in Banach spaces}, Proceedings of the American Mathematical Society, { \bf 132} (2004), 1429--1434. \bibitem[GZ]{GZ} T. L. Gill and W. W. Zachary, {\it Constructive representation theory for the Feynman operator calculus on Banach spaces}, to appear, Integration: Mathematical Theory and Applications. \bibitem[KA]{KA} W. \ E. \ Kaufman, { \it A Stronger Metric for Closed Operators in Hilbert Spaces,} Proc. Amer. Math. Soc. {\bf 90} (1984), 83--87. \bibitem[LX]{LX} P. D. Lax, { \it Symmetrizable linear tranformations,} Comm. Pure Appl. Math. {\bf 7} (1954), 633-647. \bibitem [VN] {VN} J.\ von Neumann, {\it {\"{U}}ber adjungierte Funktionaloperatoren,} Annals of Mathematics {\bf 33} (1932), 294--310. \end{thebibliography} \end{document} ---------------0705051049158--