% $Header: /cvsroot/latex-beamer/latex-beamer/examples/beamerexample3.tex,v 1.8 2004/10/07 20:53:07 tantau Exp $

\documentclass{beamer}

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\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}

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 %       \def\{\quad\blacksquare}
\def\med{\medpagebreak}
    \def\QQ{$\quad\blacksquare$}      \def\small{\smallpagebreak}
    \def\Q{\quad\blacksquare}         \def\bigp{\bigpagebreak}
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% Copyright 2003 by Till Tantau <tantau@cs.tu-berlin.de>.
%
% 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}}
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%\newcommand{\cL}{{\mathcal L}}
\newcommand{\cS}{{\mathcal S}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bZ}{{\mathbb Z}}
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\newcommand{\bR}{{\mathbb R}}
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\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 $(0<s\le1)$ if it is of the form
$$
 Tx=\sum_{k=1}^\infty \langle x'_k,x\rangle y_k
$$
for all $x\in X,$ where $(x'_k)\subset X^*, (y_k)\subset Y,\, \sum_k ||x'_k||^s\,||y_k||^s<\infty.$ We use
the notation $N_s(X,Y).$
 
\smallskip


\pause
%\vspace{1cm}

  \item
\begin{thebibliography}{99} 

  \bibitem{6} A. Hinrichs, A. Pietsch, $p$-nuclear operators in the sense of Grothendieck,
 Math. Nachr., Volume 283, No. 2 (2010), 232--261.
 
 
             \end{thebibliography}
             
 We are interested in the following question  [Problem 10.1]:        
{\it Suppose $T$ is a (bounded linear) operator acting between Banach spaces $X$ and $Y,$
and let $s\in(0,1).$ Is it true that if $T^*$ is $s$-nuclear then $T$ is $s$-nuclear too}?
%}

\end{itemize}

}

\frame{
 {\color{red}\bf $s$-nuclear operators -- Applications  de puissance p.\'eme sommable}


 It is not difficult to see that if 
$T^*$ is $s$-nuclear, then $T$ is $p$-nuclear with 
$1/s=1/p+1/2.$

 This is the best possible general result 
one can obtain without imposing any conditions on the Banach spaces involved.
The sharpness of the assertion $1/s=1/p+1/2,$ 
for $s\in (2/3, 1],$
can be seen, for instance, in 
 
   \begin{thebibliography}{99} 

  \bibitem{14} O.I. Reinov,
Approximation properties $ \mathrm{AP_s}$ and $p$-nuclear
operators {\rm(}the case  $ 0<s\le1)$,
Journal of Mathematical Sciences, Volume 115, No. 2 (2003), 2243-2250. 
 [Zapiski Nauchnykh Seminarov POMI, Vol. 270, 2000, pp. 277-291.]
             \end{thebibliography}
 
  So, 
 we consider a slightly
different question:
{\it Under which conditions  on the Banach spaces involved %$X$ and $Y$
is it valid that

$(*)$\  an operator $T\in L(X,Y)$ is nuclear if its adjoint $T^*$ is $s$-nuclear}?
 
 
 }



\frame{
 {\color{red}\bf $AP_s,\, 0<s\le1.$ }

To formulate the theorem, we need a definition: 

\begin{itemize}
  \item
Let $0<q\le \infty$ and $1/s=1/q+1.$
 We say that $X$ has the approximation property of order $s,$
 if for every $(x_n)\in l_q(X)$ 
 (where $l_q(X)$ means $c_0(X)$ for $q=\infty)$
 and for every $\varepsilon>0$ 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 $s<r$
(but, evidently, does not have the $AP_r).$
Moreover, $Y_0$ is of type 2 and of cotype $q_0$ for any $q_0>q.$
\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, $0<s\le1,$
  J. Math. Anal. Appl., Volume 415 (2014) 816-824.
 


%%%%%%%%%%%%%%

\end{thebibliography}

}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame{
\LARGE
Thank you for your attention!
}
\end{document}


%%%%%%%%%%%%%%%%%%%%%
