% $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{Operator frames in Banach spaces} \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 Definition} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% A system $\mathcal F:=((x'_k)_{k=1}^\infty, (y_k)_{k=1}^\infty)$ is an O-frame ({\it operator frame}) for $T\in L(X,Y),$ if for every $x\in X$ the series $\sum_{k=1}^\infty \langle x'_k, x\rangle y_k$ converges in $Y$ and $$ Tx= \sum_{k=1}^\infty \langle x'_k, x\rangle y_k, \ \ x\in X. $$ %We use %the notation $N(X,Y)$ \pause {\bf Examples.}\ 1.\, $\Delta: l_\infty\to l_1,$ a diagonal, $(\delta_k)\in l_1.$ Then $\Delta x=\sum \delta_k\,\langle e_k, x\rangle e_k.$ 2.\, $X$ has a basis $(f_k)_{k=1}^\infty$ %то для любого $x\in X$ имеем: $x=\sum_{k=1}^\infty \langle f'_k, x\rangle f_k,$ где %$(f'_k)$ --- биортогональная к $(f_k)$ система. %Следовательно, для любого банахова пространства $W$ и каждого оператора If $T:X\to W,$ then $$ Tx= \sum_{k=1}^\infty \langle f'_k, x\rangle Tf_k,\ \ x\in X. $$ 3.\, $W$ has a basis $(w_k).$ %$(w'_k),$ то всякий элемент $w\in W$ разлагается в ряд $w=\sum_{k=1}^\infty \langle w'_k, w\rangle w_k$ %и для $x\in X$ и If $T: X\to W,$ then $\langle Tx, w'_k\rangle= \langle x, T^*w'_k\rangle,$ hence $Tx=\sum_{k=1}^\infty \langle T^*w'_k, x\rangle w_k.$ %Отметим, что в этом примере пространство $W$ %сепарабельно, а пространство $X$ не обязано быть сепарабельным. 4.\, If $X$ (or $Y)$ is separable and has BAP. Every $T\in L(X,W)$ has O-frame. %\smallskip } \frame{ {\color{red}\bf Ideal property} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% %\begin{itemize} % \item \begin{Proposition} %{\bf Lemma.}\, {\it $T\in L(X,W),$ $A\in L(W,V), B\in L(Z,X).$ If $T$ has an O-frame, then $ATB: Z\to V$ has an O-frame. } \end{Proposition} \pause %\item \begin{coro} %{\bf Corollary.}\, {\it If $T\in L(X,W)$ factors through a Banach space with a basis, then $T$ has an O-frame. } %\smallskip \end{coro} \pause %\item \begin{coro} %{\bf Corollary.}\, {\it If $T\in L(X,W)$ factors through a Banach space with the BAP, then $T$ has an O-frame. } \end{coro} %\end{itemize} %If %$T$ is nuclear, then $T^*$ is nuclear. %\smallskip } %%% \frame{ {\color{red}\bf Dual situation} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% One more property: %\smallskip %\begin{itemize} %\item \begin{Proposition} %{\bf Proposition 1.1.}\, Let $\mathcal F:=((x'_k), (w_k))$ be an O-frame for $T\in L(X.W).$ Then the dual system $\mathcal F^d:=((w_k), (x'_k))$ is a weak${}^*$ O-frame for $T^*,$ i.e. $$ T^*w'= w^*\text{-}\lim_N \sum_{k=1}^N \langle w', w_k\rangle x'_k,\ \ w'\in W^*. $$ \end{Proposition} %\end{itemize} \pause {\it Proof}.\, For $w'\in W^*$ and $x\in X$ we have: $$ \langle Tx,w'\rangle = \langle \sum_{k=1}^\infty \langle x'_k, x\rangle w_k, w'\rangle =\langle \sum_{k=1}^\infty \langle w', w_k\rangle x'_k, x\rangle, $$ hence $T^*w' = w^*\text{-}\lim_N \sum_{k=1}^N x'_k \langle w', w_k\rangle $ (the limit is in the topology $\sigma(X^*,X)$). %\smallskip } %% \frame{ {\color{red}\bf Dual O-frame} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% % \begin{itemize} %\item \begin{defi} % {\bf Определение 1.3.}\, O-frame $((x'_k), (w_k))$ for $T$ is shrinking if for every $w'\in W^{*}$ the norm $||\sum_{k=n+1}^\infty x'_k \langle w', w_k\rangle||\to 0$ as $n\to \infty.$ \end{defi} \pause %\item \begin{Proposition} %{\bf Предложение 1.2.}\, Let $\mathcal F:=((x'_k), (w_k))$ be an O-frame for $T\in L(X.W).$ The dual system $\mathcal F^d:=((w_k), (x'_k))$ is an O-frame for $T^*$ iff the O-frame $\mathcal F$ is shrinking. \end{Proposition} %\end{itemize} } %% \frame{ {\color{red}\bf Boundedly complete O-frame} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% % \begin{itemize} %\item % {\bf Определение 1.2.}\, \begin{defi} O-frame $((x'_k), (w_k))$ for $T$ is boundedly complete if for every $x''\in X^{**}$ the series $\sum_{k=1}^\infty \langle x'', x'_k\rangle w_k$ converges in the space $W.$ \end{defi} \pause %\item \begin{Proposition} %{\bf Предложение 1.3.}\, Let $\mathcal F:=((x'_k), (w_k))$ be an O-frame for $T\in L(X.W).$ TFAE: $1)$\, O-frame $\mathcal F$ is boundedly complete; $2)$\, for every $x''\in X^{**},$ it follows from the boudedness of the partial sums $(\sum_{k=1}^N \langle x'', x'_k\rangle w_k)_{N=1}^\infty$ the convergence of the series $\sum_{k=1}^\infty \langle x'', x'_k\rangle w_k$ in the space $W.$ \end{Proposition} %\end{itemize} } %% \frame{ {\color{red}\bf O-frames and weak compactness} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% % \begin{itemize} % \item \begin{Theorem} % {\bf Теорема 1.1.}\, Let $\mathcal F:=((x'_k), (w_k))$ be an O-frame for $T\in L(X.W).$ If this O-frame $\mathcal F$ is boundedly complete and shrinking, then the operator $T$ is weakly compact. \end{Theorem} %\end{itemize} } %% \frame{ {\color{red}\bf O-frames and basis-factorization} \vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% % \begin{itemize} \begin{Theorem} %{\bf Теорема 1.2.}\, {\it Let $T\in L(X,W).$ TFAE: $1)$\, $T$ has an O-frame; $2)$\, the operator $T$ factors through a Banach space with a basis; $3)$\, $T$ factors through a Banach sequence space with a basis. } \end{Theorem} } %% \frame{ {\color{red}\bf O-frames and basis-factorization: Proof} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{Proof} % {\it Доказательство}.\, % 1) implies 3). $T,$ O-frame $\mathcal F:=((x'_k), (w_k)),\, w_k\neq0.$ % $w_k\neq0.$% for each $k.$ \ %The series $\sum_{k=1}^\infty \langle x'_k, x\rangle w_k$ converges (to $Tx)$ for every $x\in X,$ %so $\exists \, K>0:\ \forall N\ ||\sum_{k=1}^N x'_k\otimes w_k||\le K.$ %(равномерно) ограничены (например, %константой $K>0).$ %Put $ t:=\{a=(a_k)_{k=1}^\infty:\ \text{series}\ \ \sum_{k=1}^\infty a_kw_k\ \ \text{converges\ in}\ W\}, %!!! русский не идет! $ %and let $e_k$ --- $k$-й единичный вектор в $t$ %(т.е., $(e_k)_s=0$ при $k\neq s$ и $(e_k)_k=1).$ %Put $ |||a|||_t:= \sup_N ||\sum_{k=1}^N a_kw_k||\ \, (\ge \lim_N ||\sum_{k=1}^N a_kw_k||). $ For $a=(a_1, a_2,\dots, a_{N+s}, 0, 0,\dots),$ $ |||\sum_{k=1}^N a_ke_k||| \le |||\sum_{k=1}^{N+s} a_ke_k||| $ and the linear span of $(e_k)_{k=1}^\infty$ is dense in $t.$ Thus, $(e_k)$ is a monotonr basis %(см. [2]) в %!!! Ref in the Banach space $t.$ %(ср. [Sing, Prop. 3.1]). If $j: t\to W$ is a natural map $a\mapsto \sum_{k=1}^\infty a_kw_k,$ then $||j||\le1.$ %(по определению нормы $|||\cdot|||$ в $t).$ Set $Ax:= (\langle x'_k,x\rangle)_{k=1}^\infty;$ %так как ряд %$\sum_{k=1}^\infty \langle x'_k, x\rangle w_k$ сходится, то then $Ax\in t.$ Furthermore, $$ |||Ax|||_t=\sup_N ||\sum_{k=1}^N \langle x'_k, x\rangle w_k||\le K\, ||x||,\ \ \forall\, x\in X. $$ Thus, $A\in L(X,t)$ and $T=jA: X\to t\to W.$ \end{Proof} } %% \frame{ {\color{red}\bf Unconditional O-frames} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{defi} % {\bf Определение 1.4.}\, Let $T\in L(X,W),$ $(x'_k)_{k=1}^\infty\subset X^*, (w_k)_{k=1}^\infty\subset W.$ We say that $\mathcal F:=((x'_k)_{k=1}^\infty, (w_k)_{k=1}^\infty)$ is an UO-frame (unconditional operator frame) for $T,$ if for every $x\in X$ the series $\sum_{k=1}^\infty \langle x'_k, x\rangle w_k$ converges unconditionally in $W$ and $$ Tx= \sum_{k=1}^\infty \langle x'_k, x\rangle w_k, \ \ x\in X. $$ %If there exists a UO-frame for $T,$ then we say that $T$ has a UO-frame. \end{defi} \pause \begin{Theorem} %{\bf Теорема 1.3.}\, {\it Let $T\in L(X,W).$ TFAE: $1)$\, $T$ has a UO-frame; $2)$\, $T$ the operator $T$ factors through a Banach space with an unconditional basis; $3)$\, $T$ the operator $T$ factors through a Banach sequence space with an unconditional basis basis. } \end{Theorem} } %% \frame{ {\color{red}\bf O-frames and bounded approximation property} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{defi} % {\bf Определение 2.1.}\, Let $T\in L(X,W),$ $C\ge1.$ We say that $T$ has the C-BAP if for every compact subset $K$ of $X,$ for every $\varepsilon>0$ there is a finite rank operator $R: X\to W$ such that $||R||\le C\, ||T||$ and $\sup_{x\in K} ||Rx-Tx||\le \varepsilon.$ $T$ has the BAP, if it has the C-BAP for some $C\in[1,\infty).$ \end{defi} \pause \begin{Theorem} Let $X$ be a separable Banach space, $W$ be any Banach space and $T\in L(X,W).$ TFAE: $(1)$\, $T$ has an O-frame; $(2)$\, $T$ has the BAP; $(3)$\, $T$ factors through a Banach space with a basis. \end{Theorem} } %% \frame{ {\color{red}\bf Comparing Banach frames and O-frames} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{itemize} \item%{Concluding remarks} {\bf Comparing the usual Banach frames with O-frames.} \item Known: If $X$ has an unconditional Banach frame, then: 1. The frame is shrinking iff $X$ does not contain $l_1$ iff $X$ is almost reflexive. 2. $X$ is reflexive iff it does not contain both $l_1$ and $c_0.$ \smallskip \pause For O-frames, the situation is different. We have \begin{Example} There exists an operator $T: l_1\to C[0,1]$ such that 1. $T$ is conditionally weakly compact and, thus, does not contain $l_1.$ $T$ has no shrinking O-frame. 2. $T$ does not contain also $c_0,$ but is not weakly compact. \end{Example} \end{itemize} } %% \frame{ {\color{red}\bf Comparing Banach frames and O-frames} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{itemize} \item Trivially, If $X$ does not have the approximation property, then it can not have a Banach frame. \pause For O-frames, the situation is different. We have \begin{Example} There exist two separable reflexive Banach spaces $X,Y$ and and an operator $T: X\to Y$ so that: Both $X$ and $Y$ do not have the approximation property, but $T$ has an unconditional O-frame. \end{Example} \end{itemize} } %% \frame{ {\color{red}\bf Comparing Banach frames and O-frames} %\vspace{0.1cm} %%%%%%%%%%%%%%%%%%%%%%% \begin{itemize} \item Known: If a reflexive space has the approximation property, then it has a Banach frame. \smallskip \pause For O-frames, the situation is different. We have \begin{Example} There exists a weakly compact operator, which has the approximation property, but has no operator frame. \end{Example} \end{itemize} } %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ \LARGE Thank you for your attention! } \end{document} %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%555555555555555555555555555555555555555555555555555555555555555555 \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