

%%%%%  To jest referat na jubileuszowy Zjazd na 100 lat PTM wrzesień 2019 Kraków 



\documentclass{beamer}

\usepackage{amssymb}
\usepackage{beamerthemesplit}
\usepackage[T1]{fontenc}                %%  Ta i poniższa linia: by działało specjalne kodowanie w 
\usepackage[cp1250]{inputenc}           %%          Windows-ach przey naciskaniu  "Alt" - np. do polskich liter.
\usepackage[polish, english]{babel}     %%  To by określić jakie języki z pakietu  'babel' będą użyte, ten ostatni (tu ang.)- PODSTAWOWY!
%%\usepackage[english, polish]{babel}   %%  Tu tylko zmana ról języków w stos do pow. -- tak polski jest podstawowy!
%%\usepackage{polski}                   %%  By dodać jeszcze conieco z polskiej specjalności - głównie dzielenie wyrazów.


%%%%%%%%%%%DODAWANIE RYSUNKOW:
\usepackage{graphics}                   %%  To by można było potem dołączać rysunki w formatach PDF, PNG, JPEG, MPS (do pdflatexa) lub EPS (do 
\usepackage{graphicx}                   %%  samego  latexa - by były widoczne już już w .dvi); dołączanie przez kom. \includegraphics :
                                        %%  \includegraphics[parametry]{nazwa_pliku}, parametry (oddzielane przez ,) to: 
                                        %%  height=długość  -- wysokość obszaru zajmowanego przez grafikę - rysunek zostanie 
                                        %%                      przeskalowany  do  zadanego rozmiaru;          
                                        %%  width=długość  --	szerokość obszaru zajmowanego przez grafikę - rysunek zostanie przeskalowany 
                                        %%                do zadanego rozmiaru (jeżeli podamy tylko jeden wymiar pozostały zostanie przeskalowany tak, 
                                        %%                aby zachować proporcje); 
                                        %%  viewport=wartość wartość wartość wartość --- (4 liczby) parametr pozwala na wycięcie z dołączanej 
                                        %%                grafiki zadanego prostokąta, określonego przez lewy dolny i prawy górny róg. 


%%\usepackage[dvips]{graphics}          %% to dawało błąd "option clash..." przy kompilacji


%%%%%%%%%%%%%%%%%%%%%% KOLORY:
\usepackage{color}          %%   w pliku używać np. .... {\color{red}  lola}... inne parametry do \color - standardowo:
                                      %%  black, white, red, green, blue, cyan, magenta, yellow

%%\usepackage[usenames]{color}       %% to dawało błąd "option clash..." przy kompilacji



%%%%%%%%%%%%%%%%%%%%%% OPCJE  BEAMERA


%%\setbeamertemplate{headline}[default] %% To do wywalania górnego kolorowego paska.
\setbeamertemplate{footline}[default]   %% A to do dolnego.








%%%%%%\documentclass[11pt]{article}

%%%%%\usepackage{amssymb}

%%%%%\usepackage{amsmath}

\usepackage{relsize}


%%%%%\textwidth=14.9cm

%%%%%%%\hoffset=-2.3cm

%%%%%\textheight=24.7cm

%%%%%\voffset=-2.75cm








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%%%%%%%%%%%"PRYWATNE"    KOMENDY
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%%%%%%FONTY%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%LITERY POGRUBIONE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Ti}{\mathbb{T}} %%  zbiór czasów (\R lub \N z zerem)
\newcommand{\UC}{\mathbb{U}}  %% unit circle

\newcommand{\B}{\mathbb{B}}  %% unit circle



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%Różne dalsze SYMBOLE OGOLNE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}  %%% definicja \cal -- brak jej np. w stylu birkjour


\newcommand{\eps}{\epsilon}
\newcommand{\Eps}{{\cal E}}
\newcommand{\Epsi}{{\cal E}}
\newcommand{\si}{\sigma}
\newcommand{\al}{\alpha}
\newcommand{\la}{\lambda}
\newcommand{\ga}{\gamma}
\newcommand{\La}{\Lambda}
\newcommand{\de}{\delta}
\newcommand{\be}{\beta}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\ups}{\upsilon}
\newcommand{\phiv}{\varphi}
\renewcommand{\notin}{\not\in}
\newcommand{\llll}{\ell}

\newcommand{\impl}{\Longrightarrow}
\newcommand{\ldn}{\ell^2(\N)}
\newcommand{\ldz}{\ell^2(\Z)}
\newcommand{\lfin}{\ell_{\indrm{fin}}(\N)}
\newcommand{\eln}{\ell(\N)}
\newcommand{\linf}{\ell^{\infty}}
\newcommand{\ogroddz}{\ell^{\infty}_*}
\newcommand{\indrm}[1]{\mbox{\scriptsize\rm #1}}
\newcommand{\ac}{_{\indrm{ac}}}
\newcommand{\sic}{_{\indrm{sc}}}
\newcommand{\pp}{_{\indrm{pp}}}
\newcommand{\ace}{\mbox{\rm ac}}
\newcommand{\sce}{\mbox{\rm sc}}
\newcommand{\ppe}{\mbox{\rm pp}}
\newcommand{\sing}{_{\indrm{sing}}}
\newcommand{\lin}{\mbox{\rm lin}}
\newcommand{\tr}{\mbox{\rm tr}\,}
\newcommand{\supp}{\mbox{\rm supp}\,}
\newcommand{\discr}{\mbox{\rm discr}\,}
\newcommand{\dist}{\mbox{\rm dist}\,}
\newcommand{\sgn}{\mbox{\rm sgn}}
\newcommand{\Ent}{\mbox{\rm Ent}}
\newcommand{\Int}{\mbox{\rm Int}}
\newcommand{\im}{\mbox{\rm Im}}
\newcommand{\re}{\mbox{\rm Re}}
\DeclareMathOperator{\Ree}{Re\,}

\newcommand{\Emptyset}{\varnothing}  %%ładniejszy zbiór pusty niż \emptyset


\newcommand{\const}{\mbox{\rm const}}
%\newcommand{\mod}{\mbox{\rm mod}}
\newcommand{\Zer}{0\hspace{-0.435em}0}                     %%pogrubione zero

\newcommand{\TT}{{\cal T}} %% do ozanaczania np.  C_0 polgrup
\newcommand{\tiTT}{\tilde{\TT}}

\newcommand{\tiX}{\tilde{X}}


\newcommand{\ginv}[1]{#1^{-1\bullet }} %%% np. odwracanie operatorów nie koniecznie "na"


\newcommand{\goradol}[2]{\substack{#1\\#2}}%% To wygodne do dwupiętrowych indeksów



\newcommand{\diag}{\mbox{\rm diag}}
\DeclareMathOperator*{\dia}{diag}
\newcommand{\di}[2]{\dia\limits_{#1}#2}   %%DIRECT SUM of oper-s = Diagonal oper....
\newcommand{\diA}{\di{j\in\N}{A_j}}
\newcommand{\diAk}{\di{k\in\N}{A_k}}

\newcommand{\Diag}[2]{\matr{#1}{0}{0}{#2}} %%diagonalna 2x2




\newcommand{\fracd}{\displaystyle\frac}
\newcommand{\half}{\frac{1}{2}}
%\newcommand{\Forall}{\mathlarger{\mathlarger{\mathlarger{\forall}}}}%{\mbox{\Large\mbox{$\forall$}}}%%%Uwaga! \mathlarger działa gdy jest  \usepackage{relsize}
%\newcommand{\Exists}{\mathlarger{\mathlarger{\mathlarger{\exists}}}}%{\mbox{\Large\mbox{$\exists$}}}%%%Uwaga! \mathlarger działa gdy jest  \usepackage{relsize}
\newcommand{\Forall}{\mathlarger{\mathlarger{\forall}}}%{\mbox{\Large\mbox{$\forall$}}%%%Uwaga! \mathlarger działa gdy jest  \usepackage{relsize}
\newcommand{\Exists}{\mathlarger{\mathlarger{\exists}}}%{\mbox{\Large\mbox{$\exists$}}%%%Uwaga! \mathlarger działa gdy jest  \usepackage{relsize}

%\newcommand{\wt}{\widetilde}
\newcommand{\till}{\widetilde}
\newcommand{\adj}{^{*}}
\newcommand{\sprz}[1]{\overline{#1}}
\newcommand{\clos}[1]{\overline{#1}}
\newcommand{\obc}[2]{#1 \!\mid_{#2}}%%obciecie dziedziny funkcji
\newcommand{\res}[2]{\left(#1\right)_{#2}}
\newcommand{\resac}[1]{\res{#1}{\ace}}
\newcommand{\respp}[1]{\res{#1}{\ppe}}
\newcommand{\ressc}[1]{\res{#1}{\sce}}
\newcommand{\plusminus}{\pm}
\newcommand{\opcje}[4]{\left\{
    \begin{array}{cc}#1&\mbox{for $#2$}\\#3&\mbox{for $#4$}\end{array}\right.}
    
\newcommand{\opcjepl}[4]{\left\{
    \begin{array}{cc}#1&\mbox{dla $#2$}\\#3&\mbox{dla $#4$}\end{array}\right.}    
\newcommand{\leftopcjepl}[4]{\left\{
    \begin{array}{lc}#1&\mbox{dla $#2$}\\#3&\mbox{dla $#4$}\end{array}\right.}       
    
\newcommand{\opcjee}[4]{\left\{
    \begin{array}{rc}#1&\mbox{for $#2$}\\#3&\mbox{for $#4$}\end{array}\right.}    
\newcommand{\opcjetr}[6]{\left\{
    \begin{array}{cl}#1&\mbox{for $#2$}\\#3&\mbox{for $#4$}\\
#5&\mbox{for $#6$}\end{array}\right.}
\newcommand{\matr}[4]{\left(
    \begin{array}{cc}#1&#2\\#3&#4\end{array}\right)}
\newcommand{\col}[2]{\left(
    \begin{array}{c}#1\\#2\end{array}\right)}

\newcommand{\len}{\left}
\newcommand{\pn}{\right}


%%%strzałki, granice, zbieznosc
\newcommand{\strz}{\longrightarrow}
\newcommand{\nadstrz}[1]{\stackrel{#1}{\longrightarrow}} %% costam (tzn. parametr #1) nad  strz.
%\newcommand{\nadpodstrz}[2]{-\!\!\!\frac{\scriptstyle #1}{\scriptstyle #2}\!\!\!\!\!\!\!\!\rightarrow} %% coďż˝ nad i coďż˝ pod
\newcommand{\nadpodstrz}[2]{\frac{{\scriptstyle \hspace{0.8em}#1}\hspace{0.5em}}{\scriptstyle #2}\!\!\!\!\!\!\!\rightarrow} %% coďż˝ nad i coďż˝ pod
\newcommand{\strzn}{\nadpodstrz{}{n}}
\newcommand{\strzk}{\nadpodstrz{}{k}}
\newcommand{\strzj}{\nadpodstrz{}{j}}   
\newcommand{\slabo}{\nadstrz{\rm w}} %%% slaba zbieznosc: "w" nad strz.


\newcommand{\limn}{\lim_{n\to+\infty}}
\newcommand{\limsupn}{\limsup_{n\to+\infty}}
\newcommand{\liminfn}{\liminf_{n\to+\infty}}

\newcommand{\limk}{\lim_{k\to+\infty}}
\newcommand{\limsupk}{\limsup_{k\to+\infty}}
\newcommand{\liminfk}{\liminf_{k\to+\infty}}

\newcommand{\limj}{\lim_{j\to+\infty}}
\newcommand{\limsupj}{\limsup_{j\to+\infty}}
\newcommand{\liminfj}{\liminf_{j\to+\infty}}

\newcommand{\LP}{\rm LIM} %zb. wartoďż˝ci  granicznych ciďż˝gu (teďż˝ niesk.)
\newcommand{\LPF}{{\rm LIM}_{\C}} % jw. ale tylko skoďż˝cz.

\newcommand{\limkk}[1]{\lim_{#1k\to+\infty}}
\newcommand{\limsupkk}[1]{\limsup_{#1k\to+\infty}}




%%%%%%%%%%%%%%%%Sumy
\newcommand{\sumn}{\sum_{n=1}^{+\infty}}
\newcommand{\sumnz}{\sum_{n=0}^{+\infty}}
\newcommand{\sumk}{\sum_{k=1}^{+\infty}}
\newcommand{\suml}{\sum_{l=1}^{+\infty}}
\newcommand{\sumnnz}{\sum_{n=n_0}^{+\infty}}




%%%%%%%%% duże Sumy proste  

%%%%%%%%%%%%%%%%%%%O pisaniu operatorów matem. - fragment z Internetu...
%%%%%%%%%%%%%%So what is the "right" way? As usually, there is more than one; but the simplest and cleanest one is to use the amsmath package:
%%%%%%%%%%%%%%\usepackage{amsmath}
%%%%%%%%%%%%%%%%(this is not necessary if you use one of the amsart, amsbook or amsrep classes) and then say
%%%%%%%%%%%%\operatorname{sgn}
%%%%%%%%%But there is one caveat. Some time ago I saw in my friend's file the following code:
%%%%%%%%%%%%%%%%\newcommand{sgn}{{\operatorname{sgn}}}
%%%%%%%%%%%%The result is disastrous: the space vanishes again! The explanation lies in the fact that TeX's math mode treats everything in curly %%%%%%%%%%%%braces as an ordinary symbol, and we need an operator and not an ordinary symbol. The right way would be to define
%%%%%%%%%%%%\newcommand{sgn}{\operatorname{sgn}}
%%%%%%%%%%%—i.e., this would be the right way if there were no special command for that in the amsmath package. If you use some operator more often, %%%%%%%%%it's best to define it using the following construct:
%%%%%%%%%\DeclareMathOperator{\sgn}{sgn}
%%%%%%%%%%where the first argument is the command and the second one the name to be typeset.
%%%%%%%%%%%Again, there is a bit more to say. Assume that you want to have an operator like lim, which takes "limits" (in TeX's terminology, this %%%%%%%%%%%%means that the sub- or superscript is to be placed below or above when in "displayed" mode). Then, you should use the starred version of
%%%%%%%%%%%%%% \DeclareMathOperator:
%%%%%%%%%%%%%%%%%%%%\DeclareMathOperator*{\esssup}{ess\,sup}
%%%%%%%%%%%%%(notice the thin space—\,—which gives the resultess sup shown). Of course, there is an analogous operatorname* variant.
%%%%%%%%%%%%%%(This article is a rough translation of a Polish version, published on the author's web site in 2008.)
%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator*{\dsF}{\goradol{\scriptscriptstyle F}{\vspace{0.7ex}}\!\!\bigoplus}
\newcommand{\ds}[1]{\bigoplus_{#1}}  %%DIRECT SUM
\newcommand{\dsX}{\ds{j\in\N}X_j}
\newcommand{\dsFX}{\dsF_{j\in\N}X_j}

%%%\newcommand{\dsXFF}{\ds{j\in\N}{\scriptstyle F\,}X_j}
%%\newcommand{\dsXF}[1]{{\bigoplus_{j\in\N}\!\!\!\!\!\!\goradol{\vspace{0.7ex}}{\scriptscriptstyle #1\,}}X_j}



%%%CIAGI
\newcommand{\se}[2]{\left\{#1\right\}_{#2}}
\newcommand{\sse}[1]{\left\{#1\right\}}
               %%ciďż˝gi dla indeksu n
\newcommand{\sen}[2]{\se{#1}{n\geq #2}}
\newcommand{\senj}[1]{\sen{#1}{1}}
\newcommand{\senz}[1]{\sen{#1}{0}}
\newcommand{\sennz}[1]{\sen{#1}{n_0}}
                %%ciďż˝gi dla indeksu k
\newcommand{\sek}[2]{\se{#1}{k\geq #2}}
\newcommand{\sekj}[1]{\sek{#1}{1}}
\newcommand{\sekz}[1]{\sek{#1}{0}}
\newcommand{\sekkz}[1]{\sek{#1}{k_0}}
               %%ciďż˝gi dla indeksu r
\newcommand{\ser}[2]{\se{#1}{r\geq #2}}
\newcommand{\serj}[1]{\ser{#1}{1}}
\newcommand{\serz}[1]{\ser{#1}{0}}
\newcommand{\serrz}[1]{\ser{#1}{r_0}}

%%ciďż˝gi dla indeksu j
\newcommand{\sej}[2]{\se{#1}{j\geq #2}}
\newcommand{\sejj}[1]{\sej{#1}{1}}
\newcommand{\sejz}[1]{\sej{#1}{0}}
\newcommand{\sejnz}[1]{\sen{#1}{n_0}}
%%CIAGI dla indeksu j w N
\newcommand{\sejn}[1]{\se{#1}{j\in\N}}










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  Dla  SPECJALNYCH  DZIAŁÓW MATEM.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%       Przestrzenie unormowane, Banacha



\newcommand{\zerosp}{\{0\}}
\newcommand{\Zerosp}{\{\Zer\}}

\newcommand{\LL}{{\cal L}} %% do ozanaczania  rodziny liniowych op.
\newcommand{\CC}{{\cal C}} %% do ozanaczania rodziny  domk. op.
\newcommand{\GG}{{\cal G}} %% do ozanaczania rodziny  generujacych op. 9zasadniczo dla C_0 półgrup op.
\newcommand{\BB}{{\cal B}} %% do ozanaczania rodziny ogr. op.



\newcommand{\Ker}{\mbox{\rm Ker\,}}
\newcommand{\Ran}{\mbox{\rm Ran\,}}


\newcommand{\norm}[2]{\|#1\|_{#2}}%% norma z indeksem
\newcommand{\nor}[1]{\norm{#1}{}}%% norma
\newcommand{\opnor}[1]{\norm{#1}{\indrm{Op}}}%% norma operatprowa "Op"
\newcommand{\spnor}[1]{\norm{#1}{\indrm{sp}}}%% norma operatprowa "sp"





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% SYMBOLE  STANDARDOWE DLA MACIERZY JACOBIEGO %%%%%%



\newcommand{\macjac}{
%\hspace{4 em}
\left ( \begin{array}{ccccc}
q_1     & w_1 &       &        &        \\
w_1   & q_2   & w_2 &        &        \\
        & w_2 & q_3   & w_3  &        \\
        &       & w_3 & q_4    & \ddots \\
        &       &       & \ddots & \ddots
\end{array}\right )} %% ??to uzywac bez $$ juz...


\newcommand{\J}{J}
\newcommand{\Jz}{\J_0}
\newcommand{\forJ}{\mathcal{J}}
\newcommand{\forJz}{{\cal J_0}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%  Dla C_0 półgrup, chaosu itp





















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% "STARE" SYMBOLE, TU RACZEJ NIEUďż˝YTE %%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\acw}{convergence weight\ }
\newcommand{\muga}{\mu_{\ga}}
\newcommand{\calk}{^{\bullet}}
\newcommand{\subI}{\prec^I}%%%%%%%%%%%%%%%%%%%%%{\mathrm{moder}^I}
\newcommand{\subII}{\prec^{II}}%%%%%%%%%%%%%%%%%%%%%%{\mbox{\rm moder}^{II}}
\newcommand{\fle}[1]{\mbox{ for $#1$ s.l.}}
\newcommand{\fse}[1]{\mbox{ for $#1$ s.s.}}
\newcommand{\st}[1]{{\rm st}(#1)}
\newcommand{\maxgw}{\max^*}
\newcommand{\ave}[2]{\underline{#1\wr  #2}}
\newcommand{\avga}[1]{\ave{\ga}{#1}}
\newcommand{\strzga}{\nadstrz{\ga}} %%% zbieďż˝noďż˝ďż˝ w sensie uog. Cesaro z wagami \ga
\newcommand{\strzgan}{\nadpodstrz{\ga}{n}} %%% zbieďż˝noďż˝ďż˝ w sensie uog. Cesaro z wagami \ga z ind. n
\newcommand{\strzgap}{\nadstrz{\ga,  p}} %%%zbieďż˝noďż˝ďż˝ w sensie p-bezwgl. uog. Cesaro z wagami \ga
\newcommand{\strzgaj}{\nadstrz{\ga,  1}}
\newcommand{\strzgapr}{\nadstrz{\ga'}}
\newcommand{\strzgappr}{\nadstrz{\ga,  p'}}
\newcommand{\strzgaprp}{\nadstrz{\ga',  p}}
\newcommand{\strzgapn}{\nadpodstrz{\ga, p}{n}} %%%zbieďż˝noďż˝ďż˝ w sensie p-bezwgl. uog. Cesaro z wagami \ga z ind. n
\newcommand{\Cga}[1]{{\rm C}_{\ga}(#1)}
\newcommand{\equivgapp}{\stackrel{\ga,  p}{\thickapprox}}
\newcommand{\tik}{\tilde{k}}
\newcommand{\Sol}{\mathrm{Sol}}
\newcommand{\Solg}{\mathrm{Sol}_{\star}}
\newcommand{\Sub}{\mathrm{Sub}}
%%%\newcommand{\nor}[3]{\|#1\|_{#2:#3}}
\newcommand{\nornn}[1]{\nor{#1}Nn}
\newcommand{\inor}[4]{|\!|\!\!|#1|#2{|\!|\!\!|}_{#3:#4}}
\newcommand{\inornn}[2]{\inor{#1}{#2}Nn}
\newcommand{\kapppa}{\xi}
\newcommand{\rrr}{\kappa}
\newcommand{\fff}{\rho}
\newcommand{\Klaseta}{\Upsilon}


\newcommand{\UU}{{\cal U}}
%%%%%%\newcommand{\XX}{{\cal X}}
\newcommand{\Lj}{\La^{(j)}}
\newcommand{\Sj}{S^{(j)}}
\newcommand{\Rj}{R^{(j)}}
\newcommand{\Vj}{V^{(j)}}
\newcommand{\Lz}{\La^{(0)}}
\newcommand{\Sz}{S^{(0)}}
\newcommand{\Rz}{R^{(0)}}
\newcommand{\Vz}{V^{(0)}}
\newcommand{\Lkm}{\La^{(k-1)}}
\newcommand{\Skm}{S^{(k-1)}}
\newcommand{\Rkm}{R^{(k-1)}}
\newcommand{\Vkm}{V^{(k-1)}}
\newcommand{\xkm}{x^{(k-1)}}
\newcommand{\Ljp}{\La^{(j+1)}}
\newcommand{\Sjp}{S^{(j+1)}}
\newcommand{\Rjp}{R^{(j+1)}}
\newcommand{\Vjp}{V^{(j+1)}}


\newcommand{\Seq}[1]{Seq_{n_0}(#1)}
\newcommand{\sRnz}{\se{R(n)}{n_0}}
\newcommand{\sVnz}{\se{V(n)}{n_0}}
\newcommand{\sVj}{\se{\Vj(n)}{N_j}}
\newcommand{\sLj}{\se{\Lj(n)}{N_j}}
\newcommand{\sRj}{\se{\Rj(n)}{N_j}}
\newcommand{\sSj}{\se{\Sj(n)}{N_j}}
\newcommand{\sVkm}{\se{\Vkm(n)}{N_{k-1}}}
\newcommand{\sLkm}{\se{\Lkm(n)}{N_{k-1}}}
\newcommand{\sRkm}{\se{\Rkm(n)}{N_{k-1}}}
\newcommand{\sSkm}{\se{\Skm(n)}{N_{k-1}}}
\newcommand{\sVjp}{\se{\Vjp(n)}{N_{j+1}}}
\newcommand{\sLjp}{\se{\Ljp(n)}{N_{j+1}}}
\newcommand{\sRjp}{\se{\Rjp(n)}{N_{j+1}}}
\newcommand{\sSjp}{\se{\Sjp(n)}{N_{j+1}}}

\newcommand{\grla}{\GRR{\la}}
\newcommand{\grw}{\GRR{w}}
\newcommand{\grz}{\GRRR{z}}
\newcommand{\grS}{\GRR{S}}
\newcommand{\grrho}{\GRR{\rho}}
\newcommand{\grnu}{\GRR{\nu}}
\newcommand{\grnui}{\GRI{\nu}}
\newcommand{\grOm}{\GRR{\Omega}}
\newcommand{\grT}{\GRR{T}}

\newcommand{\jede}{1,\ldots,d}
\newcommand{\Jede}{\{1,\ldots,d\}}
\newcommand{\Te}{{\cal T}}
\newcommand{\DDe}{{\cal D}}
\newcommand{\tLa}{\widetilde{\La}}
\newcommand{\TIB}{\widetilde{B}}
\newcommand{\tix}{\widetilde{x}}


\newcommand{\mdwr}{M_2(\R)}
\newcommand{\mdwc}{M_2(\C)}
\newcommand{\mdk}{M_d(K)}
\newcommand{\mdc}{M_d(\C)}
\newcommand{\mdr}{M_d(\R)}

%\newcommand{\Diag}{\mbox{\rm Diag}}
\newcommand{\vi}{v_i}
\newcommand{\mui}{\la_i}
\newcommand{\laa}{w}
\newcommand{\laambda}{\laa}

\newcommand{\Beta}{{\cal B}}
\newcommand{\Bper}{B^{0}}
\newcommand{\relR}{{\bowtie}}
   %%\newcommand{\Jper}{\J\indrm{per}}
\newcommand{\ha}{{\cal H}}
\newcommand{\JZ}{\J_{0\Z}}
\newcommand{\pik}{\spadesuit}
\newcommand{\sipp}{\si_{\indrm{pp}}}
\newcommand{\silsp}{\si_{\indrm{LSP}}}
\newcommand{\sips}{\si_{\indrm{ps}}}
\newcommand{\sipgw}{\si_{\indrm{p*}}}
\newcommand{\rgw}{r_{*}}
\newcommand{\sip}{\si_{\indrm{p}}}
\newcommand{\siac}{\si_{\indrm{ac}}}
\newcommand{\sisc}{\si_{\indrm{sc}}}
\newcommand{\siess}{\si_{\indrm{ess}}}
\newcommand{\sid}{\si_{\indrm{d}}}
%\newcommand{\pik}{\spadesuit}
\newcommand{\hapik}{\ha_{\pik}}
\newcommand{\haac}{\ha_{\indrm{ac}}}
\newcommand{\hasc}{\ha_{\indrm{sc}}}
\newcommand{\happ}{\ha_{\indrm{pp}}}
\newcommand{\COT}[2]{#1^{\{#2\}}}
\newcommand{\Sigm}{S}%%% przedtem zamiast S  bylo \Sigma - moze to lepsze?
\newcommand{\Sim}{\Sigm_{<}}
\newcommand{\Simdo}{\sprz{\Sim}}
\newcommand{\Siw}{\Sigm_{>}}
\newcommand{\Simr}{\Sigm_{\leq}}
\newcommand{\Sir}{\Sigm_{=}}
\newcommand{\SiR}{\Sigm_{\relR}}

\newcommand{\asympto}{\asymp}

\newcommand{\Inv}{\mbox{\rm Inv}}
\newcommand{\ZW}{0_W}
\newcommand{\Zal}{0_{\al}}



\newcommand{\Iks}{X\!\!\!\!\!X}
\newcommand{\Iksna}[1]{\Iks_{#1}}%%{\Iks{[#1]}}%%{\Iks_{[#1]}}



%%%pogrubianie
\newlength{\szer}%pomocnicza nazwa jednostki dlugosci
%pogr. "1-go stopnia":
\newcommand{\GR}[1]{\settowidth{\szer}{$#1$}#1\hspace{-0.95\szer}#1}
%pogr. "1-go stopnia" dla indexow:
\newcommand{\GRI}[1]{\settowidth{\szer}{$\scriptstyle#1$}#1\hspace{-0.95\szer}#1}

%pogr "2-go stopnia":
\newcommand{\GRR}[1]{\settowidth{\szer}{$#1$}
#1\hspace{-0.95\szer}#1\hspace{-0.95\szer}#1}
%pogr "3-go stopnia":
\newcommand{\GRRR}[1]{\settowidth{\szer}{$#1$}
#1\hspace{-0.95\szer}#1\hspace{-0.95\szer}#1\hspace{-0.95\szer}#1}


%%%%%%abstrakcyjne własności - oznaczenia pogrubione
\newcommand{\coor}{{\bf(coor)}}
\newcommand{\inj}{{\bf(inj)}}
\newcommand{\iso}{{\bf(iso)}}
\newcommand{\simi}{{\bf(sim)}}
\newcommand{\pointbound}{{\bf(pointbound)}}
\newcommand{\projm}{{\bf(proj$-$)}}
\newcommand{\proj}{{\bf(proj)}}
\newcommand{\projmm}{{\bf(proj$--$)}}
\newcommand{\projpm}{{\bf(proj$+-$)}}
\newcommand{\projp}{{\bf(proj$+$)}}
\newcommand{\ban}{{\bf(ban)}}
\newcommand{\bel}{{\bf(bel)}}
\newcommand{\belm}{{\bf(bel$-$)}}
\newcommand{\belmp}{{\bf(bel$-+$)}}
\newcommand{\den}{{\bf(den)}}
\newcommand{\appr}{{\bf(appr)}}
\newcommand{\incr}{{\bf(incr)}}
\newcommand{\mono}{{\bf(mono)}}
\newcommand{\major}{{\bf(major)}}




\newcommand{\diT}{\di{j\in\N}{\TT_j}}
\newcommand{\diginvA}{\di{j\in\N}{\left(\ginv{(A_j)}\right)}}

\newcommand{\rezinf}[1]{\rho_{\infty}\left(\sejn{#1}\right)}
\newcommand{\rezinfA}{\rezinf{A_j}}




\newcommand{\X}[1]{X^{\leq #1}}
\newcommand{\Xfin}{X_{\indrm{fin}}}
\newcommand{\Xappr}{X_{\indrm{appr}}}

\newcommand{\DfinA}{D_{\indrm{fin}}(\sejn{A_j})}
\newcommand{\na}[2]{\widetilde{#2}^{#1}}%%%{\stackrel{\vee_{#1}}{#2}}


\newcommand{\XX}[1]{\widetilde{X_{#1}}}
\newcommand{\XXX}{{\cal X}}
\newcommand{\II}{{\cal I}} %% do ozanaczania identyfikacji X_j w sumie prostej
\newcommand{\PP}{{\cal P}} %% do ozanaczania rzutu P_r  w sumie prostej
















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\begin{document}



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\frame{%%

\begin{center}
{\LARGE 100 lat PTM\ \
Krak\'ow, 2 -- 7 IX 2019}\\
\vspace{3ex} 
{\LARGE\color{green} 
Chaos Liniowy i Analiza Spektralna Operatorów
}\\
\vspace{5ex}
{\large Marcin Moszyński}\\
\vspace{1.2ex}
{\small\em Uniwersytet Warszawski\\
Wydział Matematyki Informatyki i Mechaniki}\\
%%\vspace{4ex}
%%%% Supported by ...} 

\end{center}

}%%do\frame

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

\frameRozpod{Liniowe Układy Dynamiczne i Chaos Liniowy}{1. Liniowa dynamika i jej generator}   

\vspace{-1.5ex} {\bf Liniowa} zależność dynamiki  od warunków początkowych (NIE od czasu \dots), zatem:
   \bite
     \item { Przestrzeń stanów układu} ---  przestrzeń Banacha $X\not=\Zerosp$
     \item {\bf Liniowy Układ Dynamiczny} w $X$: rodzina  $\Te=\{T_t\}_{t\in \Ti}$  operatorów ograniczonych $T_t\in\BB(X)$, gdzie $\Ti$ --- {\em zbiór czasów}  -- tu:\vspace{-1ex}
        \bde
     \itemb dla czasu {\bf dyskretnego} \ $\Ti:=\N:=\{0,1,2,\ldots\}$
     \itemb dla czasu {\bf ciągłego} \ $\Ti:=[0;+\infty)$
        \ede    
oraz  $\Te$ jest:\vspace{-1ex}
     \bde
     \itemb   półgrupą ($T_{t+s}=T_tT_s$)
     \itemb silnie ciągła ,,po $t$'' (dla czasu ciągłego), 
     \ede
   \eite
   


\end{frame}  

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


             
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\begin{frame} 
tzn. trajektorie dla  $x\in X$ 
$$
\R\ni t\longmapsto T_tx\in X
$$ 
są ciągłe ($C_0$ - półgrupa)\\
\vspace{1ex}
{\bf Sens fizyczny}: $T_tx$ ---  ,,stan $x$ po czasie $t$''\\
\vspace{1ex}
{\bf Generator} $A$ półgrupy $\Te$: 
 \bite 
\item gdy czas dyskretny:\\
 $A\bl{:=}T_1$ \ \ \pausee \ifbl  \  $T_n=A^n$, \ $n\in\N$
\item gdy czas ciągły:\\
 $A$ ---  infinitezymalny generator $C_0$ półgrupy $\Te$ (silna pochodna $\Te=\{T_t\}_{t\in \Ti}$ \  w $t=0$) \ \ \pausee \ifbl \   formalnie ,,$T_t=e^{tA}$\ ''\\ (zawsze ---  $A$ domknięty operator liniowy  z gęstą w $X$ dziedziną, gdy $A$ jest ograniczony, to pow. ``$=$'' jest ścisła)
\eite


\end{frame}  

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\framePod{2. Chaos liniowy (,,topologiczny'')}


             
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%\begin{frame}  

\bite
\item {\bf orbita}  punktu  $x$: $\Orb(x)\bl{:=}\{T_tx:t\in\Ti\}$
\item   $x$ jest  {\bf okresowy} \iffbl \ $\exists_{t>0} \ T_tx=x $
\eite

%%\end{frame}             
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\bl{Pewne dwa pojęcia ,,topologicznego nieuporządkowania'' $\Te$:}

\plDefi
\bite
\item  $\TT$ jest {\bf tranzytywna} \iffbl  \ dla każdego otwartego  $U\subset X$   \ \vspace{-2ex}
$$
\bigcup_{x\in U}\Orb(x) \ \ \mbox{jest gęsty w  $X$},\vspace{-2ex}
$$ 
{\cblue równoważnie:} każde dwa niepuste otwarte \ $U,V\subset X$  można ,,połączyć pewną trajektorią''
\item   $\TT$ jest   {\bf hipercykliczna}  \iffbl \  istnieje  orbita gęsta w $X$.
\eite

\plRema
Dla  $X$  ośrodkowej:  hipercykliczność \ \iffbl   tranzytywność
%%%\end{Rem}

\end{frame}             
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\begin{frame}  

\plDefi (Devaney, 1986)
\bite
\item
$\Te$ jest   {\bf chaotyczna} \iffbl  jest tranzytywna i zbiór punktów okresowych dla  $\Te$ jest gęsty w $X$
\item  
$\Te$ jest  {\bf  sub-chaotyczna}  \iffbl istnieje domknięta podprzestrzeń $\tiX\not=\Zerosp$ niezmiennicza dla   $\Te$ taka, że \ \ 
$\tiTT:=\{\obc{T_t}{\tiX}\}_{t\in\Ti}$ \ \  jest chaotyczna jako półgrupa w  $\tiX$ \\ (,,$\Te$ jest chaotyczna {\bf w}  $\tiX$\ '')
\item Każda $\tiX$ j. w.:  {\bf przestrzeń chaotyczności dla   $\Te$}
\eite


\plRema
$\Te$ jest   {\bf chaotyczna} w $X$ \ifbl
\bite
\item   $\dim X=+\infty$
\item  $\Te$ spełnia tzw. ``SDIC'' --- ,,wrażliwą zależność'' od warunków  początkowych
\eite

\end{frame}             
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\framePod{3. Przykład Rolewicza} 
%%\begin{frame}  

\plExamp  
(Rolewicz  1969)
Rozważmy operator $T_{-1}$ ,,przesunięcia w lewo''  w  $X= \ell^2(\N)$\vspace{-1ex}
$$
T_{-1}(x_0, x_1, x_2, \ldots):=(x_1, x_2, x_3, \ldots).
$$ 
Półgrupa generowana przez ,,skalowany'' operator $T_{-1}$   \vspace{-1ex}
$$
A\bl{:=}\mu T_{-1}, \vspace{-1ex}
$$
 dla skalowania   $|\mu|>1$  (,,operator Rolewicza'') jest hipercykliczna.
\bite
\item Zapewne pierwszy przykład operatora liniowego o tej własności
\item nazwa ,,hipercykliczny'' --- około 1986 (>1969)\dots
\item Dowód działa dla $X= \ell^p(\N)$ z wszystkimi  $p\in[1;+\infty)$
\item \redd{Czy jest również  chaos\dots??}
\eite

 









\end{frame}             
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\frameRozpod{Spektralne kryteria chaotyczności typu DSW}{1. Selekcje wektorów własnych}
\vspace{-2.2ex}


 $A$ --- generator $\Te$, \ \ \  ,,spektralne kryteria chaotyczności'', tzn. wyniki  postaci:
\begin{center} 
 {\em pewne własności spektralne  $A$  \ \ \ifbl  \ chaotyczność / sub-chaot. $\Te$}
 \end{center}
 \vspace{0.6ex}
 
 Niech $\Emptyset\not=\Om\subset\C$ oraz \ \ $f: \Om\strz X$

\plDefi
$f$  {\bf jest selekcją wektorów własnych  $A$ (na $\Om$)} \  \iffbl  \  
$$
\Forall_{\la\in\Om} \ \ \ f(\la)\in\Ker(A-\la I) 
$$

{\bf Uwaga:} Nie każde $f(\la)$ musi być wektorem własnym! --- Może być jeszcze $f(\la)=0$\dots --- wtedy może być $\la\not\in\sip(A)$
 \vspace{1ex}
 
 
\end{frame}             
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\begin{frame}  

 
 
 Selekcja $f$ jest:
\bite
\item {\bf nietrywialna} \iffbl \ $f\not=\Zer$ ($\Zer$ ---  funkcja stale równa zero)
\item {\bf analityczna} \iffbl \ $\Om$ jest otwarty i  $f$ jest wektorową funkcją analityczną 
\eite
%%%\end{Def}


 
 
 
 
 
\end{frame}             
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \framePod{2. Podstawowe spektralne kryterium (sub-)chaotyczności} 

 
 
 
Oznaczmy:\vspace{-2ex}
$$
\B:=\leftopcjepl{\UC\pausee\bl{:=}\{z\in\C:\ |z|=1\}}{\mbox{czasu dyskretnego}}{\mbox{\bf i}\R}{\mbox{czasu ciągłego}}
$$
 
\plCrite {(Godefroy \bl{+} Shapiro, DSW = Desch \bl{+} Schappacher \bl{+} Webb, Banasiak \bl{+} M.M.)}\\
 Jeśli istnieje nietrywialna analityczna selekcja $f$ wektorów własnych  
$A$ na otwartym spójnym $\Om\subset\C$, takim że     
$$
\Om\cap \B\not=\Emptyset,
$$ 
to  $\TT$ jest sub-chaotyczna. Ponadto, gdy  $f$ jest taka jak wyżej, to  $\clos{\lin\ f(\Om)}$ jest przestrzenią chaotyczności dla $\TT$. 
\\ \gree{$\Longrightarrow$  $\TT$ --- chaotyczna, jeśli ${\lin\ f(\Om)}$ jest gęsta w $X$.}
%%%\end{Crit}

\end{frame}             
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\begin{frame}  


\Defi   $x\in X$ is  {\bf cyclic } for $B\in\BB(X)$ \ iff 
$$
\clos{\lin\{B^{n}x:\,n\in\N\}}=X.
$$
 
The above Criterion $+$ the R.-I. point spectrum results give:


 
\Theo  \bl{(``The Right-Inversion Spectral Chaos \\Criterion'':  M.M. 2014)}\\
Suppose that  $\la_0\in\sipgw(A)$ and  $\dist(\B,\la_0)<\frac1{\rgw(A,\la_0)}$. Then:
\bite
\item  $\TT$ is sub-chaotic 
\item   $\clos{\lin\{B^{n}f_0:\,n\in\N\}}$  is a space of chaoticity for $\TT$ for any $f_0\in\Ker(A-\la_0 I)\setminus\{0\}$ and $B\in\RINV(A,\la_0)$
\item If, moreover, there exists $f_0\in\Ker(A-\la_0 I)$ which is   cyclic  for some  $B\in\RINV(A,\la_0)$, then  $\TT$ is chaotic.
\eite
%%\end{Th} 




\end{frame}             
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\begin{frame} 

\Examp {\cblue (chaos for Rolewicz type operators)}
The discrete  dynamical system $\Te$ generated by  the Rolewicz type operator $\mu T_{-1}$ with $|\mu|>1$ 
is chaotic in $\ell^p(\N)$ with $p<+\infty$, and not only hypercyclic (or  only sub-chaotic). 

\Proo Recall that $\rgw(A,0)^{-1}=|\mu|$. Hence it suffices to  take $\la_0=0$, $B=\mu^{-1} T_{+1}$ and $f_0=e_0$, 
where $(e_0)_n=0$  for $n>0$, $(e_{0})_0=1$.

\Rema {\cblue (chaos for weighted left shift - generated discrete and continuous  d. s.)}
The same argumentation proves that for $p<+\infty$ both discrete and continuous d. s. 
generated by the {weighted left shift}   $A:=T_{-1, w}$ in  $\ell^p(\N)$ are chaotic, providing 
that  $s=\inf_{n\geq1}|w_n|>1$ (and that $A$ generates a d.s.).


\end{frame}             
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\framePod{4. Super-upper-triangular chaotic operators}

We shall generalize the chaoticity result of the last weighted left shift example. Here  $X=\ell^p(\N)$ with $1\leq p<+\infty$ or $X=c_0$ (--- complex sequences  converging to 0). 

\bl{Matrix terms for  $A\in\BB(X)$:}    
 $$
A(k,l):=(Ae_l)_k,\ \ \ \ k,l\in\N,,
$$
 where $e_s\in X$ 
is  the $s$-th standard ``base'' vector: $$(0,0,0 \ldots,0,\stackrel{\mbox{\bl{s}}}{1}, 0,0, \ldots)$$

\Defi
\bite
\item
$A$ is  \bff{upper-triangular}  iff $A(k,l)=0$ for any $k> l$, $k,l\in\N$
\item
$A$ is \bff{super-upper-triangular} iff $A(k,l)=0$ for any $k\geq l$, $k,l\in\N$ (i.e. $A$ posseses only zero terms below the first super diagonal);

\eite


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\begin{frame} 


\bite
\item  $A\supdia$ is  the \redd{superdiagonal part} of $A$ (i.e. the  part with only the first super diagonal of $A$) and  
 $A\off$ is \bl{off-superdiagonal part} of $A$'':
$$
(A\supdia x)_j:=A(j, j+1)x_{j+1} \ \ \  \mbox{for} \ \ x\in X, j\in\N, \ \ A\off:=A-A\supdia.
$$
\eite




$$
\supupdia
$$



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\begin{frame}  
%%We obviously have $A\supdia, A\off\in \BB(X)$, if $A\in\BB(X)$.

\Theo
Let $A$ be a bounded super-upper-triangular operator in $X$. If for some $r>0$ 
\beql{eqAsupupp}
 |A(j, j+1)|\geq 1+r, \ \ \ \ \mbox{ for any } \ \ j\in\N 
\eeq
and
\beql{eqAoff}
\|A\off\|< r,
\eeq
then the linear dynamical system $\Te$ in $X$ (discrete or continuous one) generated by $A+\be I$ is chaotic, providing   
\beqw
|\re\,\be|\leq1 & \mbox{at the continuous case},\\
|\be|\leq2 & \mbox{at the discrete case}.
\eeqw





%%%\end{Th}

\Proo
 - is based on ``The Right-Inversion Spectral Chaos Criterion'' (used for  $A+\be I$ instead of $A$ and  for $\la_0:=\be$) and on:
 
 
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\begin{frame}  

\Lemm
If $B \in\BB(X)$ satisfies:  
$$
 j=k+1\Longrightarrow B(j, k)\not=0, \ \ \  \ \  \ \ j>k+1\Longrightarrow B(j, k)=0 \ \ \ 
$$ 
for any $j, k\in\N$, then $e_0$ is  cyclic for $B$.
%%\end{Lem}


\vspace{5ex}

{\cblue\em \small This research is supported by  Polish Ministry of Science and Higher Education grant 
 no. NN 201 605640 and  National Science Centre -- Poland grant no. 2013/09/B/ST1/04319}







\begin{center}
{\LARGE \bl{THE} \gree{END}}
\end{center}

\end{frame}  















\frameRozpod{R.-I. Point Spectrum and the Local Spectral Theory}{1. Right-inversion point spectrum}   

$X$ ---  a Banach space,   $X\not=\Zerosp$. \bl{Denote:} 
\bite
\item
$\BB(X)$ --- the space  of  bounded linear operators on  $X$ (here the domain $D(A)$ of $A$ equals $X$)
\item
$\CC(X)$ --- the set  of  closed linear operators in $X$ (the domain $D(A)$ of $A$ need not be dense)
\eite

\end{frame}  

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\begin{frame}  

 
Let $A\in\CC(X)$.  We study  a particular part of  $\sip(A)$ ---  the point spectrum (= the set of eigenvalues). 

\Defi 
The {\bf  right-inversion (r.-i.) point spectrum} of $A$: 
%\vspace{-2ex}
$$
\sipgw(A):=\{\la\in\sip(A):\exists_{B\in\BB(X)}\ (A-\la I)B=I\}.  
$$ 

\Rema 
$\la\in\sipgw(A)\ \bl{\Longrightarrow}\   \Ran(A-\la I)=X$.\\
Hence, defining also \bff{the surjective point spectrum} of $A$ by \ \ 
$$
\sips(A):=\{\la\in\sip(A):\,\Ran(A-\la I)=X\},
$$
we get $\sipgw(A)\subset\sips(A)$.
\end{frame}  

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\begin{frame}  

\bl{Denote:} \ \ 
$
\RINV(A,\la):=\{B\in\BB(X):\ (A-\la I)B=I\}
$.

\bl{Note:} 
\bite
\item
$\sipgw(A)=\{\la\in\sip(A):\ \RINV(A,\la)\not=\Emptyset\}.$ 
\item
if $B\in\RINV(A,\la)$ and $B'\in\BB(X)$, then \
$$
B'\in\RINV(A,\la)\ \bl{\Longleftrightarrow} \  \Ran (B'-B)\subset\Ker(A-\la I).
$$
\eite

For $\la\in\sipgw(A)$ \bl{denote:}
$$
\rgw(A,\la):=\inf\{\|B\|:\,B\in\RINV(A,\la)\}.
$$

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\begin{frame}  


\Propo
If $A\in\CC(X)$, then $\sipgw(A)$ is an open subset of $\C$, moreover  if   $\la_0\in \sipgw(A)$, then 
$$
D(\la_0, {\rgw(A,\la_0)}^{-1})\subset\sipgw(A),   \ \ \ 
$$

\bl{where:} $D(\la_0,r):=\{\la\in\C:\, |\la-\la_0|<r\} $.

\nic{%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\Quest
If $A\in\BB(X)$, then $\rgw(A,\la)>0$. Can it happen that $\rgw(A,\la)=0$ for an unbounded $A\in\CC(X)$? 
(if YES, then we define\ $\frac10:=+\infty$\  above).
}%%%do \nic%%%%%%%%%%%%%%%%%%%%%%

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\begin{frame}  

\Examp \bl{Rolewicz type operator} \ \ 
 $A:=\mu T_{-1}$,\\ where $T_{-1}$ is the left-shift 
$$
T_{-1}(x_0, x_1, x_2, \ldots):=(x_1,  x_2, x_3, \ldots)
$$ 
 in $X=\ell^p(\N)$, $1\leq p\leq+\infty$ 
%($(T_{-1}f)_n=f_{n+1}$)
 and  $\mu\in\C\setminus\{0\}$. \ We have:
\bite
\item
$0\in\sipgw(A)$ and  $B:=\mu^{-1} T_{+1}\in\RINV(A, 0)$, where    $T_{+1}$ is  the   forward shift operator
$$
T_{+1}(x_0, x_1, x_2, \ldots):=(0, x_0,  x_1,  \ldots),
$$ 
\item
$\rgw(A,0)=|\mu|^{-1}$ (the ``$\inf$'' is reached by $B$ as above)
\item 
$\sipgw(A)=\sip(A)=D(\la_0, |\mu|^{-1})$.
\eite
  Note that in the   case $p=2$ and $|\mu|>1$ this operator is known as {\bf Rolewicz operator} \gree{[S. Rolewicz ``On orbits of elements'', Studia Math. 1969, vol. 32].}
 
 
 

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\begin{frame}  
 
 
 
\Examp  \bl{weighted left shift}   \  $A:=T_{-1, w}$ in  $X:=\ell^p(\N)$, \\ where  $w=\senj{w_n}$ --- ``the weight'' --- a complex 
sequence with all $w_n\not=0$ and $s:=\inf_{n\geq1}|w_n|>0$,
$$
T_{-1, w}(x_0, x_1, x_2, \ldots):=(d_1x_1, d_2x_2, d_3x_3, \ldots),
$$ 
 i.e.,  $T_{-1, w}$  is  defined by the upper-diagonal  semi-infinite matrix:
$$
\diagonalna{w_1}{w_2}
$$
\bl{Note:}  it is a generalization of the Rolewicz type oper. and  $T_{-1, w}$ can be unbounded!


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\begin{frame} 
We have:

 \bite 
\item
$0\in\sipgw(A)$ and  $B:=T_{+1, w^{-1}} \in\RINV(A, 0)$, where    $T_{+1, w^{-1}}$ is  the  
``$w^{-1}$-weighted'' forward shift operator
$$
T_{+1, w^{-1}}(x_0, x_1, x_2, \ldots):=(0, \frac{x_0}{w_1}, \frac{x_1}{w_2} ,  \ldots),
$$
\item
$\rgw(A,0)=s^{-1}$ (the ``$\inf$'' is reached by $B$ as above)
\item 
$D(\la_0, s)\subset \sipgw(A)$.
\eite
%%%\end{Exams}

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\framePod{2. R.-I. point spectrum and LSP}


Let $A\in\CC(X)$
and let $f: \Om\strz X$, where $\Emptyset\not=\Om\subset\C$.

\Defi
$f$  {\bf is a selection  of eigenvectors for $A$ (on $\Om$)} \  iff  \  
$$
f(\la)\in\Ker(A-\la I) \ \  \ \mbox{for any } \ \la\in \Om
$$
 abbreviation: \ {\bf e.v. selection} (alternative name: {\bf spanning eigenvector field}).
 
 An  e.v. selection $f$ is:
\bite
\item {\bf non-trivial} iff \ $f\not=\Zer$ ($\Zer$ ---  the constant-zero function);
\item {\bf analytic} iff \ $\Om$ is an open set  and $f$ is  an $X$-vector-valued analytic function. 
\eite
%%%\end{Def}


For $\Emptyset\not=\Om\subset\C$, $\Om$ --- open, \bl{denote}:
$$
\Ansel{\Om}{A}:=\{f\,:\ f \ \mbox{ is  an analytic  e. v. selection for $A$ on $\Om$}\}.
$$

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\begin{frame}  

We are interested   in such $A\in \CC(X)$ that  possess a non-trivial  analytic e.v. selection on a neighborhood of a given point $\la_0\in\C$. 

\Propo
Let $A\in\CC(X)$, \ $\la_0\in\sipgw(A)$,  \ $B\in\RINV(A,\la_0)$ \\ and  let $U:=D(\la_0, {\|B\|}^{-1})$
 \bl{\cgreen (recall that  $U\subset\sipgw(A)$ in such case)}. 
Then  for any $f_0\in\Ker(A-\la_0 I)\setminus\{0\}$ there exists $f\in \Ansel{U}{A}$ such that 
$f(\la_0)=f_0$ and $f(\la)\not=0$ for any $\la\in U$. Such $f$ can be given by 
 \beql{eqfla}
f(\la):=\sumnz (\la-\la_0)^nB^{n}f_0, \ \  \    \la\in U.
\eeq
%%%\end{Prop}


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\begin{frame}  
%%%\Section{Local selection property (LSP),  SVEP and  NON-SVEP}     \label{sectSVEP}

\Defi
$A\in\CC(X)$ has the  {\bf   local selection property} at $\la_0$ iff  there exists $\eps>0$ such that \  $\Ansel{D(\la_0,\eps)}{A}\not=\Zerosp$.\\
Abbreviation: {\bf  LSP} . 
%%%\end{Def}

\Propo
If   $A$ has the LSP at $\la_0$, then there exists $\eps>0$ and   such $f\in\Ansel{D(\la_0,\eps)}{A}$, that $f(\la_0)\not=0$.\\
 In particular $\la_0\in\Int\,\sip(A)$.
%%%\end{Rem}


\Coro
Defining  \bff{the LSP point spectrum} of $A$ by \\ 
$$
\silsp(A):=\{\la\in\C:\,\mbox{$A$ has the LSP at $\la$}\}
$$
we get 
\bite
\item $\sipgw(A)\subset\silsp(A)\subset\sip(A)$;
\item $\silsp(A)$ is an open subset of $\C$.
\eite


\end{frame}  

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\begin{frame}  
But recall that also: 
$\sipgw(A)\subset\sips(A)$

\Quest
Is there any inclusion between $\sips(A)$ and $\silsp(A)$?


\end{frame}  

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\framePod{3. Relations with Local Spectral Theory}

\bl{The answer} can be obtained with the usie of some basic facts from Local Spectral Theory 

Recall the main definition:


\Defi
{\bf  $A$ has the SVEP at $\la_0$} iff    \  $\Ansel{\Om}{A}=\Zerosp$ for any open connected neighborhood $\Om$ of $\la_0$.
{\bf $A$ has the non-SVEP at $\la_0$} iff   $A$ does not have the SVEP at $\la_0$.

\Coro
$A$ has the LSP at $\la_0$ iff $A$ has the NON-SVEP at $\la_0$.\\
%%(two analytic functions on a open connected set $\Om$ coincide, if they take the same values on a subset with accumulation point in $\Om$).

\vspace{2ex}


Hence, using a \bl{J.K. Finch ``non-SVEP'' result (1975)} we can get:

\Theo
If $A\in\CC(X)$, then 
$$
\sipgw(A)\subset {\cred \sips(A)\subset\silsp(A)} \subset\sip(A).
$$

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We end with  remark related to the above results, concerning some additional ``chaotic properties'' (--- {\em mixing} and {\em frequent hypercyclicity} --- see e.g., \cite{GP-LC})

\Rema\label{remsupupcha}
The dynamical system considered in Theorem \ref{thchaos} {\bf (ii)}, and thus also in Theorems \ref{thsupupcha} and \ref{thsupupcha+nu} is  mixing. Moreover, in the  discrete time case such system is frequently hypercyclic.
%%\end{Rem}


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