% $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{$H^\infty$ and the Grothendieck approximation property } \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 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} \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} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %Or, the same: \begin{defi} $X\in BAP$ iff $\exists\ C\ge1:$ $$ \forall\ \text{ compact }\ K\sbs X,\ \forall\ \e>0,$$ $$\exists\ R\in X^*\ot X:\ \sup_{x\in K} ||Rx-x||\le \e,\ ||R||\le C. $$ \smallskip We say also $C$-$MAP.$ If $C=1,$ \ $X\in MAP.$ \end{defi} \pause Easy reformulation: \pause %\item X has the property $C$-$MAP,$ if given $\e>0,$ a Banach space $Y,$ an operator $T\in\L(X,Y)$ and any finite sequence $(x_k)\sbs X,$ there exists a finite rank operator $R$ from $X$ to $Y$ such that 1)\ $||Tx_k-Rx_k||<\e$ for all $k,$ 2)\ $||R||\le C ||T||.$ %\end{itemize} } %%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Classical Spaces} \vspace{0.3cm} Separable classical Banach spaces and $L^\infty$ have the MAP. E.g.: $C(K), L_p(\mu), 1\le p <\infty \dots$ [Groth.] Each of spaces $H^p, L^p/H^p_0\, (1\le p<\infty), A, C/A_0$ have the MAP. Moreover, all these spaces have bases. %\begin{itemize} %\item \begin{thebibliography}{99} \bibitem{5} R. P. Boas, Jr., Isomorphism between $H^p$ and $L^p,$ Amer. J. Math. 77 (1955), 655-656. + a result of Marcinkiewicz and Paley on $L^p, 11,$ X has the property $AP_p$ (repectively, the property $K-MAP_p),$ if given $\e>0,$ a Banach space $Y,$ an operator $T\in\Pi_{p'}(X,Y)$ and a weakty $p'$- summable sequence $(x_k)\sbs X,$ there exists a finite rank operator $R$ from $X$ to $Y$ such that $$\sum ||Tx_k-Rx_k||^{p'}<\e$$ (respectively,. and $\pi_{p'}(R)\le K\pi_{p'}(T)$). \end{defi} Easy to see: X has the property $K-MAP_p,$ if given $\e>0,$ a Banach space $Y,$ an operator $T\in\Pi_{p'}(X,Y)$ and any finite sequence $(x_k)\sbs X,$ there exists a finite rank operator $R$ from $X$ to $Y$ such that 1)\ $||Tx_k-Rx_k||<\e$ for all $k,$ 2)\ $\pi_{p'}(R)\le K\pi_{p'}(T).$ %\end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} Recall: %\begin{itemize} %\item \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 $\sup_n ||Rx_n-x_n||\le \varepsilon.$ \end{defi} \pause % \item A generalization: \begin{defi} Let $00$ 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 $H^\infty$ -- the simplest} \begin{itemize} \item The space $H^\infty$ has the property $AP_p$ for any $p>0, p\neq1.$ Moreover, if $p>1,$ then $H^\infty$ and all its even duals have the property $1-MAP_p;$ if $p<1,$ then all the duals of $H^\infty$ have the property $AP_p.$ \smallskip \pause \item \begin{thebibliography}{99} \bibitem{5} Bourgain~J., Reinov~O.I. {\it On the approximation properties for the space $ H ^ {\infty} $} // Math. Nachr. - 122 (1985). - P.~19-27. \end{thebibliography} \medskip \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf A generalization} \begin{thebibliography}{99} \bibitem{5} Manuel D. Contreras and Santiago Diaz-Madrigal, Uniform Approximation Properties for Spaces of Analytic Functions, Math. Nachr. 210 (2000), 85 -91 \end{thebibliography} \pause X has the uniform $K-MAP_p,$ if given $\e>0,$ there is a function $N\in \mathbf N\to m(N)\in \mathbf N$ such that for every Banach space $Y,$ any operator $T\in\Pi_{p'}(X,Y)$ and any finite sequence $(x_k)_1^N \sbs X,$ there exists a finite rank operator $R$ from $X$ to $Y$ such that 1)\ $||Tx_k-Rx_k||<\e$ for all $k\le N,$ 2)\ $\pi_{p'}(R)\le K\pi_{p'}(T),$ 3)\ $\dim R(X) \le m(N).$ %% \pause \begin{itemize} \item {\it Theorem}.\, Let $\delta$ be a positive number. Then the space $H^\infty$ has the uniform $(1+\delta)$-bounded approximation property of order $p$ for every $10$ there is a finite rank operator $R$ in $H^\infty$ with $$ \sup_n ||Rx_n-x_n||\le \e. $$ \end{itemize} \pause Moreover, we can control both the rank and the norm of $R:$ %\pause % } \frame{ {\color{red}\bf General situation} \begin{itemize} \item {\bf Theorem.}\, There is a function $B(\e), \e>0,$ such that if $(x_n)_n$ is a sequence in $H^\infty,$ satisfying $$ ||x_n|| \le \frac1{\log (1+n)}, $$ then there exists an operator $R: H^\infty\to H^\infty$ such that 1)\ $\sup_n ||Rx_n-x_n||\le \e,$ 2)\ $rank T< B(\e),$ 3)\ $||T||< B(\e.)$ \end{itemize} \begin{thebibliography}{99} \bibitem{5} Bourgain~J., Reinov~O.I. {\it On the approximation properties for the space $ H ^ {\infty} $} // Math. Nachr. - 122 (1985). - P.~19-27. \end{thebibliography} %%%%%%%%%%%%%%%%%%%%%%%%%% } %---------------------------------------------------------------------------------------------- \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 $$T: X\to c_0\to l_1 \to Y.$$ \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} % \smallskip } %%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Nuclear operators: a question} \vspace{0.5cm} %%%%%%%%%%%%%%%%%%%%%%% {\bf Question}:\, Let $T$ map $X^{**}$ into $X$ %being weak${ }^*$-to-weak continuous, and $\pi_X$ be the natural isometric injection from $X$ to $X^{**}.$ Suppose that $$ \pi_X T: X^{**}\to X \to X^{**} $$ is nuclear. Is it true that $$ T: X^{**}\to X $$ is nuclear too? \vspace{0.2cm} In general, the answer is NO. } %%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Application} \begin{itemize} \item %%%%%%%%%%%%%%%% %\vskip 0.3 cm {\bf Theorem.} {\it Let a linear operator $ T: H ^ {\infty} \to A $ be such that there are two sequences of functions $ \{g _ n \}\subset L ^ 1 $ and $ \{f _ n \}\subset H ^ {\infty}, $ for which $ \sum _ k \int | g _ k | \, dm < \infty, $ $ \| f _ n \| < 1/\log {(n + 1)} $ for each $ n $ and $$ T (f) = \sum _ {k = 1} ^ {\infty} \int g _ k (t) \, f (t) \, dm (t) \, f _ k. $$ Then the operator $ T $ is nuclear as an operator, acting from $ H ^ {\infty} $ into the disk-algebra $ A. $ } \vskip 0.5 cm %\pause \end{itemize} } \frame{ {\color{red}\bf $n$-dimensional case} \vskip 0.5 cm For the case of $H^\infty(M)$ on complex manifolds $M,$ see \vskip 0.5 cm \begin{thebibliography}{99} \bibitem{5} Alexander Brudnyi, On the approximation property for Banach spaces predual to $H^\infty$-spaces, Journal of Functional Analysis 263 (2012) 2863-2875 \end{thebibliography} } %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ \LARGE {\color{red} Thank you for your attention! } } \end{document} %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \frame{ {\color{red}\bf Example we use} \begin{itemize} \item %%%%%%%%%%%%%%%% \vskip 0.3 cm {\bf Theorem.} {\it Let a linear operator $ T: H ^ {\infty} \to A $ be such that there are two sequences of functions $ \{g _ n \}\subset L ^ 1 $ and $ \{f _ n \}\subset H ^ {\infty}, $ for which $ \sum _ k \int | g _ k | \, dm < \infty, $ $ \| f _ n \| < 1/\log {(n + 1)} $ for each $ n $ and $$ T (f) = \sum _ {k = 1} ^ {\infty} \int g _ k (t) \, f (t) \, dm (t) \, f _ k. $$ Then the operator $ T $ is nuclear as an operator, acting from $ H ^ {\infty} $ into the disk-algebra $ A. $ } \vskip 0.5 cm \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