% $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 <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}}
\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 a question of Boris Mitjagin
}
\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 $$T: X\to c_0\to l_1 \to Y.$$

\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{


Let A be a compact operator in $H.$ Then A has the norm convergent
expansion
$$A=\sum_{n=1}^{N} \mu_n(A)\, (f_n, \cdot) h_n,$$
where $(f_n),$ $(h_n)$ are ONS's, $\mu_1(A)\ge \mu_2(A)\ge \dots >0)$
 \smallskip

The $\mu_n(A)$ are called the singular values of  $A.$ Notation $s_n(A)$ or just $s_n.$
 \smallskip

 \begin{thebibliography}{99}

  \bibitem{5} Simon B., Trace ideals and their applications, London Math. Soc.
Lecture Notes 35, Cambridge University Press, 1979.

             \end{thebibliography}

}

\frame{
 {\color{red}\bf R. Schatten and J. von Neumann}

            \vspace{0.3cm}


\begin{itemize}
  \item
$$
 A\in S_p(H):\   \sum s_n^p(A)<\infty,\ p>0.
$$

\pause
   \item

$$
S_p\circ S_q \subset S_r,\  1/r=1/p+1/q;
$$

$p, q\in (0,\infty)$

$$
N(H)=S_1(H).
$$
\smallskip





\end{itemize}

}



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

$$N_p(H)=S_p(H), 0<p\le1.$$

\smallskip

  \item
\begin{thebibliography}{99}

  \bibitem{6} R. Oloff, p-normierte Operatorenideale, Beitr\"age Anal. 4, 105-108 (1972).


             \end{thebibliography}



\end{itemize}


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

\frame{
 {\color{red}\bf On  products of nuclear operators}

 A natural question (due to Boris Mitjagin):

      \begin{itemize}
  \item
Is it true that a product of two nuclear operators in Banach spaces
can be factored through a trace class (i.e., $S_1$-) operator in a Hilbert space?


\smallskip


\pause
%\vspace{1cm}

 %\item


\smallskip

  \item
  By using an example from

\begin{thebibliography}{99}

  \bibitem{6} Carleman T.,
 \"Uber die Fourierkoeffizienten einer stetigen Funktion, A. M., 41
(1918), 377-384.


             \end{thebibliography}

 it can be shown that

 \item
 The answer is negative.


\end{itemize}

}



\frame{
 {\color{red}\bf Explanation}


$f$ is Carleman's continuous function:

$\hat{f}\in l_2\setminus \cup_{p<2} l_p.$
\pause

$$T: C\overset{*f}\to C.$$
\pause

$T$ is nuclear.
\smallskip

Consider the product $TT.$ Note that eigenvalues $(\la_k(TT))\in l_1$ and not better.
\smallskip
\pause

Suppose, there is an $S_1$-operator $U\in S_1(H)$ so that
$$TT: C\overset A\to H\overset U \to H\overset B\to C.$$


\medskip



}


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



\frame{
 {\color{red}\bf Explanation (continued)}

Consider
$$H\overset B\to C\overset A\to H\overset U \to H\overset B\to C.$$

Eigenvalues of $UAB$ $=$ eigenvalues of $TT=BUA$ (and, so, in $l_1).$
\pause

BUT:
$$A\in \Pi_2;\  \text{ so, }\ AB\in S_2;\  U\in S_1.$$
\pause

Hence,
$$UAB\in S_{2/3}.$$
\smallskip

Contradiction.


\pause

\begin{itemize}
  \item
{\it Remark}.\,
Sharp fact is that if $V\in NN,$ then it factors through an operator $U\in S_2.$
\end{itemize}

}





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

\begin{itemize}
  \item
 Let $\alpha, \beta\in (0,1].$ If $T\in N_\alpha\circ N_\beta,$ then it
 factors through an $S_r$-operator, where
 $$
 \frac1r=\frac1\alpha+\frac1\beta-\frac32.
 $$

\smallskip


\pause
%\vspace{1cm}

  \item
Particular cases:
$$
\alpha=1, \beta=\frac23 \implies r=1;
$$
$$
\alpha=1, \beta=1 \implies r=2.
$$
\end{itemize}

}




\frame{
 {\color{red}\bf M. I. Zelikin}

To formulate the theorem, we need a definition:

\begin{itemize}
  \item
The spectrum of A is central-symmetric, if together with any eigenvalue $\la\neq0$ it has the
eigenvalue $-\la$ of the same multiplicity.
\pause

It was proved in a paper by M. I. Zelikin
\smallskip

  \begin{thebibliography}{99}

  \bibitem{14} M. I. Zelikin,
A criterion for the symmetry of a spectrum",
Dokl. Akad. Nauk 418 (2008), no. 6, 737-740
             \end{thebibliography}

\item
{\bf Theorem.}\
The spectrum of a nuclear operator $A$
acting on a separable Hilbert space is central-symmetric iff
$trace\,  A^{2n - 1} = 0, \, n \in \mathbf N.$



\end{itemize}

}


%----------------------------------------------------------------------------------------------
\frame{
 {\color{red}\bf Generalization}

We can proof:
\smallskip

\begin{itemize}
  \item
{\bf Theorem.} Let $Y$ be a subspace of a quotient (or a quotient of a subspace)
of an $L_p$-space, $1\le p\le\infty$
and $T\in N_s(Y,Y)$ $(s$-nuclear), where $1/s=1+|1/2-1/p|,$
The spectrum of $T$
 is central-symmetric iff\
$trace\,  T^{2n - 1} = 0, n =1,2,\dots.$
\smallskip
\vskip 0.1cm

\pause

\item
{\it Remark}:\,
In the theorem "trace" is well defined. The result is sharp.

\pause

See also

\item
\begin{thebibliography}{99}

  \bibitem{5} Boris S. Mityagin, Criterion for $Z_d$-symmetry of a Spectrum of a Compact Operator,
 arXiv: 1504.05242 [math.FA].

             \end{thebibliography}

\end{itemize}


}


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

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


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




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

\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}


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