% $Header: /cvsroot/latex-beamer/latex-beamer/examples/beamerexample3.tex,v 1.8 2004/10/07 20:53:07 tantau Exp $ \documentclass{beamer} %\usepackage[pdftex]{graphicx} %\usepackage{background} %\usepackage{pdfslide} %\usepackage[pdftex]{hyperref} %\usepackage{amsthm,amsfonts,amssymb,latexsym} %\usepackage{color} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\nor#1{||{#1}||} % \def\md#1{|{#1}|} % \def\sp#1#2{\(#1,#2\)} % \def\ove#1{\overline{#1}} % \def\ovs#1#2{\overset{#1}{#2}} % \def\ovs#1#2{\overset{#1}\to{#2}} \def\({\left(} \def\al{\alpha} \def\lee{\leqslant} \def\){\right)} \def\e{\varepsilon} \def\gee{\geqslant} \def\[{\left[} \def\la{\lambda} \def\]{\right]} \def\ffi{\varphi} %\def\be{\beta} %\def\{{\left\{} \def\ot{\otimes} \def\<{\langle} \def\wh{\widehat} \def\>{\rangle} \def\wt{\widetilde} \def\sbs{\subset} \def\tr{\operatorname{trace}\,} %%%%%%%%%%%%% after 30.01.00 02:44:40 Sat: \def\Gr{\operatorname{Gr}} \def\AP{\operatorname{AP}} \def\BAP{\operatorname{BAP}} \def\N{\operatorname{N}} \def\I{\operatorname{I}} \def\id{\operatorname{id}} \def\L{\operatorname{L}} \def\QN{\operatorname{QN}} \def\J{\operatorname{J}} \def\R{\operatorname{R}} \def\reg{\operatorname{reg}} \def\dual{\operatorname{dual}} \def\sbs{\subset} %%%%%%%%%%%%%%%%%%%%%%%%% % \def\{\quad\blacksquare} \def\med{\medpagebreak} \def\QQ{$\quad\blacksquare$} \def\small{\smallpagebreak} \def\Q{\quad\blacksquare} \def\bigp{\bigpagebreak} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Copyright 2003 by Till Tantau . % % This program can be redistributed and/or modified under the terms % of the LaTeX Project Public License Distributed from CTAN % archives in directory macros/latex/base/lppl.txt. % % The purpose of this example is to show how \part can be used to % organize a lecture. % \usetheme{Warsaw} \usepackage[english]{babel} \usepackage[latin1]{inputenc} \usepackage{graphicx} \setbeamercovered{transparent} \renewcommand{\phi}{\varphi} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\ban}{\begin{eqnarray*}} \newcommand{\ean}{\end{eqnarray*}} \newcommand{\too}{\mathop{\longrightarrow}} \newcommand{\nul}{{\bf0}} \newcommand{\odin}{{\bf1}} \newcommand{\cupm}{\mathop{\cup}} \newcommand{\capm}{\mathop{\cap}} \renewcommand{\t}{{\Bbb T}} \newcommand{\td}{{\Bbb T}^d} \newcommand{\rd}{{\Bbb R}^d} \newcommand{\zdd}{{\Bbb Z}^{d-1}} \newcommand{\zddp}{{\Bbb Z}^{d-1}_+} \newcommand{\rdd}{{\Bbb R}^{d-1}} \newcommand{\zd}{{\Bbb Z}^{d}} \newcommand{\tdd}{{\Bbb T}^{d-1}} \newcommand{\ld}{L(\td)} \renewcommand{\r}{{\Bbb R}} \newcommand{\z} {{\Bbb Z}} \newcommand{\n} {{\Bbb N}} \newcommand{\calt}{{\cal T}} \newcommand{\call}{{\cal L}} \newcommand{\cons}{{C\,}} \newcommand{\ddd}{,\dots,} \renewcommand{\lll}{\left(} \newcommand{\rrr}{\right)} \newcommand{\ex}[1]{e^{2\pi i{#1}}} \newcommand{\exm}[1]{e^{-2\pi i{#1}}} \newcommand{\sml}[3]{\sum\limits_{{#1}={#2}}^{#3}} \newcommand{\Ldvad}{L_2(\rd)} \newcommand{\h}{\widehat} \newcommand{\smzd}[1]{\sum\limits_{#1\in\zd}} \renewcommand{\span}{\mbox{span}\,} \newcommand{\supp}{\mbox{supp}\,} \newcommand{\smi}[1]{\sum\limits_{#1=1}^\infty} \newcommand{\cA}{{\mathcal A}} \newcommand{\cB}{{\mathcal B}} \newcommand{\cD}{{\mathcal D}} \newcommand{\cE}{{\mathcal E}} %\newcommand{\cL}{{\mathcal L}} \newcommand{\cS}{{\mathcal S}} \newcommand{\bN}{{\mathbb N}} \newcommand{\bZ}{{\mathbb Z}} \newcommand{\bQ}{{\mathbb Q}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bC}{{\mathbb C}} \newtheorem{Proposition}[theorem]{Proposition} \newtheorem{coro}[theorem]{Corollary} % % The following info should normally be given in you main file: % \newtheorem{defi}[theorem]{Definition} \title{On nuclearity of operators with $s$-nuclear adjoints} \author{Oleg Reinov} %\institute{ %Department of Applied Mathematics and Control Processes, %Saint Petersburg State University} \date{} \begin{document} \frame{\titlepage %\pause %\vspace{-1.3cm}%\hspace{3cm} %\includegraphics[width=4cm]{3-adic_disk.png} } %\section{p-Adic Multiresolution analysis} %\section[Introduction]{Introduction} %\subsection[p-adic numbers]{p-adic numbers} % \frame{ {\color{red}\bf Nuclear operators} \vspace{0.5cm} %%%%%%%%%%%%%%%%%%%%%%% An operator $T:X\to Y$ is nuclear if it is of the form $$ Tx=\sum_{k=1}^\infty \langle x'_k,x\rangle y_k $$ for all $x\in X,$ where $(x'_k)\subset X^*, (y_k)\subset Y,\, \sum_k ||x'_k||\,||y_k||<\infty.$ We use the notation $N(X,Y)$ \smallskip If $T$ is nuclear, then $T^*$ is nuclear. \smallskip } \frame{ \begin{thebibliography}{99} \bibitem{5} A. Grothendieck, Produits tensoriels topologiques et espases nucl\'eaires, Mem. Amer. Math. Soc., Volume 16, 1955, 196 + 140. \end{thebibliography} Suppose $T$ is a bounded linear operator acting between Banach spaces $X$ and $Y,$ Is it true that if $T^*$ is nuclear then $T$ is nuclear too? \smallskip As is well known, a negative answer was obtained already by T. Figiel and W.B. Johnson in: \begin{thebibliography}{99} \bibitem{4} T. Figiel, W.B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc., Volume 41 (1973), 197--200. \end{thebibliography} } \frame{ {\color{red}\bf Grothendieck's Approximation} \vspace{0.3cm} % \begin{thebibliography}{99} %\bibitem{5} A. Grothendieck, Produits tensoriels topologiques et espases nucl\'eaires, %Mem. Amer. Math. Soc., Volume 16, %1955, 196 + 140. % \end{thebibliography} \begin{itemize} \item \begin{defi} $X\in AP$ iff $$ \forall\ Y, \forall\ \text{ compact }\ K\sbs X,\ \forall\ \e>0,\ \forall\ T: X\to Y,$$ $$\exists\ R\in X^*\ot Y:\ \sup_{x\in K} ||Rx-Tx||\le \e. $$ \end{defi} \pause \item Or, the same: \begin{defi} $X\in AP$ iff for every $(x_n)\in c_0(X)$ and for every $\varepsilon>0$ there exists a finite rank operator $R$ in $X$ such that $\sup_n ||Rx_n-x_n||\le \varepsilon.$ \smallskip \end{defi} \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Grothendieck's Approximation} \begin{thebibliography}{99} \bibitem{5} A. Grothendieck, Produits tensoriels topologiques et espases nucl\'eaires, Mem. Amer. Math. Soc., Volume 16, 1955, 196 + 140. \end{thebibliography} First part of "la proposition 15,2; chap. I, p. 86": \smallskip \noindent {\bf Case $\mathbf X^*$.}\ Let $ T\in L(X,Y)$ and assume that $\, X^*$ has the \,$AP. $ If $ T^*\in N(Y^*, X^{*}),$ then $T\in N(X,Y).$ %\endproclaim \medskip A proof can be found in \begin{thebibliography}{99} \bibitem{Diestel} J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, RI, 1977. \end{thebibliography} } \frame{ {\color{red}\bf Grothendieck's Approximation} Second part of "la proposition 15,2; chap. I, p. 86": \noindent {\bf Case $\mathbf Y^{**}.$}\ Let $ T\in L(X,Y)$ and assume that $\, Y^{**}\in \,AP.$ If $ T^*\in N(Y^*, X^{*}),$ then $T\in N(X,Y).$ %\endproclaim \medskip } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Eve Oja} \begin{thebibliography}{99} \bibitem{8} E. Oja, O.I. Reinov, Un contre-exemple \`a une affirmation de A.Grothendieck, C. R. Acad. Sc. Paris. --- Serie I, Volume 305 (1987), 121--122. \end{thebibliography} \begin{itemize} \item \noindent {\bf I.}\ Let $ T\in L(X,Y)$ and assume that $\, Y^{***}\in \,AP.$ If $ T^*\in N(Y^*, X^{*}),$ then $T\in N(X,Y).$ %\endproclaim \smallskip \pause \item {\bf II.}\ There exist Banach spaces $X,Y$ and a non-nuclear operator $T: X\to Y$ so that $X$ and $Y^{**}$ have the metric approximation property and $T^*$ is nuclear. \smallskip \end{itemize} } \frame{ {\color{red}\bf $s$-nuclear operators -- Applications de puissance p.\'eme sommable} \begin{itemize} \item An operator $T:X\to Y$ is $s$-nuclear $(00$ there exists a finite rank operator $R$ in $X$ such that $\sup_n ||Rx_n-x_n||\le \varepsilon.$ \pause \item {\bf Theorem 1.}\ Let $\, s\in (0,1],$ $ T\in L(X,Y)$ and assume that either $\, X^*\in \,AP_s\ $ or $\, Y^{***}\in \,AP_s.$ If $ T\in N_s(X, Y^{**}),$ then $T\in N_1(X,Y).$ In other words, under these conditions, from the $ s$-nuclearity of the conjugate operator $ T^*,$ it follows that the operator $ T$ is nuclear. \end{itemize} } %---------------------------------------------------------------------------------------------- \frame{ {\color{red}\bf Theorem} The examples in the following result show that the condition "$X^*$ or $Y^{***}$ has the approximation property of order $s$" is essential. \smallskip \begin{itemize} \item {\bf Theorem.} For each $s\in (2/3, 1]$ there exist a Banach space $Z_s$ and a non-nuclear operator $T_s: Z_s^{**}\to Z_s$ so that $ Z_s^{**}$ has the metric approximation property, $Z_s^{***}$ has the $AP_r$ for every $r\in (0,s)$ and $T_s^*$ is $s$-nuclear. \smallskip \vskip 0.1cm \pause \item {\it Remark}:\, The space $Z_1^{***}$ is isomorphic to a space of type $Z_1^*\oplus E,$ where $E$ is an asymptotically Hilbertian space. This gives us one more example of an asymptotically Hilbertian space which fails the approximation property. \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Example we use} \begin{itemize} \item Let $r\in(2/3,1], q\in[2,\infty), 1/r=3/2-1/q.$ There exist a separable reflexive Banach space $Y_0$ and a tensor element $w\in Y_0^*\widehat\otimes_r Y_0$ so that $w\neq0, \tilde w=0,$ the space $Y_0$ (as well as $Y_0^*)$ has the $AP_s$ for every $sq.$ \vskip0.3cm \pause \item %\vskip 0.1cm For $q=2$ (that is, $r=1)$, the space $Y_0$ is a subspace of a space of the type $\left(\sum_j l_{p_j}^{k_j}\right)_{l_2}$ with $p_j\searrow 2$ and $k_j\nearrow \infty.$ Every such space is an asymptotically Hilbertian space (for definitions and some discussion, see \begin{thebibliography}{99} \bibitem{1} P.~G. Casazza, C.~L. Garc\'{\i}a, W.~B. Johnson, An example of an asymptotically Hilbertian space which fails the approximation property, Proc. Amer. Math. Soc,, Volume 129, No. 10 (2001), 3017-3024. \end{thebibliography} ). \end{itemize} } %%%%%%%%%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ \frame{ {\color{red}\bf Reference} %\bigskip %\bigskip %\medskip \begin{thebibliography}{09} %\bigskip \medskip \bibitem{21} O.~I. Reinov, On linear operators with $s$-nuclear adjoints, $0