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

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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}}

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                 \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}}
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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}}
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i{#1}}} \newcommand{\sml}[3]{\sum\limits_{{#1}={#2}}^{#3}}
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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{$H^\infty$ 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 $L^p, 1<p<\infty.$

             \end{thebibliography}

\begin{thebibliography}{99}

  \bibitem{5} P. Billard, Bases dans H et bases de sous espaces de dimension finie dans A, Proc. 
Conf.,Oberwolfach (August 14-22, 1971), ISNM Vol. 20, Birkhauser, Basel and Stuttgart, 
1972.

             \end{thebibliography}

\begin{thebibliography}{99}

  \bibitem{5} S. V. Bockarev, Existence of a basis in the space of functions in the disk, and 
some properties of the Franklin system, Mat. Sb. (N.S.) 95 (137) (1974), 3-18 == Math. 
USSR Sbornik 24 (1974), 1-16. 

             \end{thebibliography}
             
             }

\frame{
 {\color{red}\bf Preliminaries by A. Pe\l czy{\'n}ski, $B(H) \dots$ }
 
See, generally,

\begin{thebibliography}{99}

  \bibitem{5} Pe\l czy{\'n}ski~A.
{\it Banach spaces of analytic functions
and absolutely summing operators} \rm --
AMS Regional Conference Series in Mathematics 30,
Providence, 1977.

             \end{thebibliography}

\pause

\begin{itemize}
\item
 $B(H)\notin AP.$

\item
$L^\infty/H^\infty \in AP.$

\item
$H^\infty $ --- ???

\end{itemize}

%\pause
  % \item

}

\frame{
 {\color{red}\bf Absolutely summing operators}
 

\begin{defi}
An operator $T$ from a Banach space $X$ into a Banach space $Y$
is said to be {\it $p$-absolutely summing}, notation $T\in \Pi_p(X,Y),$ where
$0<p\le \infty,$
if there is a constant $C\in (0,\infty)$ such that for all finite families $(x_i)_{i=1}^n\subset X$
$$
 \sum_{i=1}^n ||Tx_i||^p \le C^p\, \sup \{\sum_{i=1}^n |\langle x',x_i\rangle|^p:\ x'\in X,\, ||x'||\le1\}.
$$
The $p$-summing norm $\pi_p(T)$ is defined as $\inf C.$
\end{defi}



%\end{itemize}

}



\frame{
 {\color{red}\bf Properties of order p}

  %   \begin{itemize}
  %\item
 \begin{defi}
  For $p>1,$ 
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 $0<q\le \infty$ and $1/s=1/q+1.$
 We say that $X$ has the approximation property of order $s,$\ $X\in AP_s,$
 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
 $\sup_n ||Rx_n-x_n||\le \varepsilon.$
\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 $1<p<\infty.$

\end{itemize}

}

\frame{
 {\color{red}\bf A generalization}

\begin{thebibliography}{99}

  \bibitem{5} J.M. Delgado, E. Oja, C. Pineiro, E. Serrano
The p-approximation property in terms of density of finite rank operators,
J. Math. Anal. Appl. 354 (2009) 159-164

             \end{thebibliography}

}

\frame{
 {\color{red}\bf General situation}

% On AP_s s<1, below defs....    Now on log


Main theorem on AP for $H^\infty:$ the space has the AP "up to logarithm".

\begin{itemize}
  \item
{\bf Theorem.}
 Let $(x_n)_n$ be a sequence in $H^\infty$ such that
$$
 ||x_n|| \le \frac1{\log (1+n)}.
$$
Then  for every $\e>0$ 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, $0<s\le1,$
  J. Math. Anal. Appl., Volume 415 (2014) 816-824.



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

\end{thebibliography}

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

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


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