\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 227--229.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{227} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Uniformly ergodic theorem] {Uniformly ergodic theorem for commuting multioperators} \author[S. Lahrech, A. Mbarki, A. Ouahab, S. Rais \hfil EJDE/Conf/14 \hfilneg] {Samir Lahrech, Abderrahim Mbarki, Abdelmalek Ouahab, Said Rais} % in alphabetical order \address{Samir Lahrech \newline D\'epartement de Math\'ematiques, Universit\'e Oujda, 60000 Oujda, Morocco} \email{lahrech@sciences.univ-oujda.ac.ma} \address{Abderrahim Mbarki \newline Current address: National school of Applied Sciences, P.O. Box 669, Oujda University, Morocco} \email{ambarki@ensa.univ-oujda.ac.ma} \address{Abdelmalek Ouahab \newline D\'epartement de Math\'ematiques, Universit\'e Oujda, 60000 Oujda, Morocco} \email{ouahab@sciences.univ-oujda.ac.ma} \address{Said Rais \newline D\'epartement de Math\'ematiques, Universit\'e Oujda, 60000 Oujda, Morocco} \email{said\_rais@yahoo.fr} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{47A35, 47A13} \keywords{Average; E-k condition; finite descent; uniform ergodicity} \begin{abstract} In this paper, we established some uniformly Ergodic theorems by using multioperators satisfying the E-k condition introduce in \cite{l1}. One consequence, is that if $I-T$ is quasi-Fredholm and satisfies E-k condition then $T$ is uniformly ergodic. Also we give some conditions for uniform ergodicity of a commuting multioperators satisfies condition E-k. These results are of interest in view of analogous results for unvalued operators (see, for example \cite{k1}) also in view of the recent developments in the ergodic theory and its applications. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction and main results} Throughout this paper, $X$ is a complex Banach space, and $L(X)$ is the algebra of linear continuous operators acting in $X$. If there is an integer $n$ for which $T^{n+1}X=T^nX,$ then we say that $T$ has finite descent and the smallest integer $d(T)$ for which equality occurs is called the descent of $T$. If there is exists an integer $m$ for which $kerT^{m+1}=kerT^m$, then $T$ is said to have finite ascent and the smallest integer $a(T)$ for this equality occurs is called ascent of $T$. If both $a(T)$ and $d(T)$ are finite, then they are equal \cite[38.3]{h1}. We say that $T$ is chain-finite and that its chain length is this common minimal value. Moreover \cite[38.4]{h1}, in this case there is a decomposition of the vector space $$ X=T^{d(T)}X\oplus \ker T^{d(T)}. $$ We now focus on the topological situation: For every $T\in L(X)$ we set \begin{equation} \label{e1.1} M_i(T)=i^{-1}(I+T+T^2+\dots +T^{i-1}), \quad i=1,2,3,\dots , \end{equation} i.e. the averages associated with $T$, where $I=id_X$ is the identity of $X$. If $T=(T_1, T_2,\dots , T_n)\in L(X)^n$ is commuting multioperator (briefly, c.m.), we also set \begin{equation} \label{e1.2} M_v(T)= M_{v_1}(T_1)M_{v_2}(T_2)\dots M_{v_n}(T_n), \quad v\in Z^{n}_{+}, v\geq e, \end{equation} where $Z^{n}_{+}$ is the family of multi-indices of length $n$ (i.e. n-tuples of nonnegative integers) and $e:=(1,1, \dots , 1)\in Z^{n}_{+}$. In other words, \eqref{e1.2} defines the averages associated with $T$. \begin{definition} \label{def1.1} \rm A commuting multioperator $T\in L(X)^n$ is said to be uniformly ergodic if the limit \begin{equation} \label{e1.3} \lim_v M_v(T) \end{equation} exists in the uniform topology of $L(X)$. \end{definition} \begin{remark} \label{rmk1.1}\rm (a) If $n=1$, then \eqref{e1.3} is automatically fulfilled, and therefore the above definition extends the usual concept of uniformly ergodic operator (see, for example \cite{k1}). (b) If $T= (I,\dots, T_j , I,\dots ,I)\in L(X)^n$, then $T$ is uniformly ergodic if and only the $\lim_{v_j} M_{v_j}(T_j)$ exists in the uniform topology of $L(X)$. \end{remark} \begin{definition}[\cite{l1}] \label{def3}\rm Let $k=(k_1, \dots k_n)\in Z^{n}_{+}$ and $T\in L(X)^n$ be a c.m. We say that $T$ satisfies condition E-k if $\lim_v(I-T_j)^{k_j} M_v(T)= 0$ for each $j\in\{1,\dots , n\}$. \end{definition} It is clear that condition E-k implies condition E-n for any $n\geq k$ Thus we see that the example $T=(T_1, I, \dots , I )\in Z^{n}_{+}$ with $$ T_1=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. $$ This shows that E-2e is strictly weaker than E-e. \begin{theorem} \label{thm1.1} Let $k \in Z^{n}_{+}$. Suppose $T\in L(X)^n$ satisfies condition E-k and $\sum_{j=1}^{n}(I-T_j)^{k_j}X$, $\sum_{j=1}^{n}(I^*-T_{j}^*)^{k_j}X^* $ are closed in $X$ and $X^*$ respectively. If $\big[\sum_{j=1}^{n}(I-T_j)^{k_j}X\big]\cap \big[\cap_{j=1}^{n}\ker(I-T_j)^{k_j}\big]=\{0\}$. Then $T$ is uniformly ergodic \end{theorem} \begin{proof} Arguing exactly as in \cite[Theorem 1]{m1}, with $\delta_T$ and $\gamma_T$ given by $$ \oplus^{n}_{j=1}x_j \to \delta_T(\oplus^{n}_{j=1}x_j) = \sum_{j=1}^{n}(I-T_j)^{k_j}x_j \ \mbox{and} \ x\to \gamma_T(x)= \oplus^{n}_{j=1}(I-T_j)^{k_j}x. $$ \end{proof} \begin{theorem} \label{thm2.1} Let $T\in L(X)$ satisfy condition E-r, and one of the following nine conditions: \begin{itemize} \item[(a)] $I-T$ has chain length at most $r$ \item[(b)] $1$ is a pole of the resolvent of order at most $r$ \item[(c)] $I-T$ is quasi-Fredholm operator \item[(d)] $(I-T)^rX$ is closed and $\ker(I-T)^r$ has a closed $T$-invariant complement \item[(e)] $(I-T)^rX \bigoplus \ker(I-T)^r=(I-T)^rX + \ker(I-T)^r$ \item[(f)] $(I-T)^mX$ is closed for all $m\geq r$ \item[(g)] $(I-T)^rX$ is closed \item[(h)] $(I-T)^mX$ is closed for some $m\geq r$ \item[(i)] $I-T$ has finite descent. \end{itemize} Then $T$ is uniformly ergodic. \end{theorem} \begin{proof} Firstly, from \cite[Theorem 6]{l1}, the above statements (a)--(i) are equivalent. Then, take $G=(T, I, \dots I) \in L(X)^n$ and $k=(r, 1,\dots ,1)\in Z^{n}_{+}$; Therefore, we have $\sum_{j=1}^{n}(I-G_j)^{k_j}X=(I-T)^rX$ is closed, it follows that $\sum_{j=1}^{n}(I^*-G_{j}^*)^{k_j}X^* =(I^*-T^*)^rX^* $ is closed, which implies since $(I-T)^rX \cap \ker(I-T)^r=\{0\}$, that $G$ is uniformly ergodic in $L(X)^n $. From Theorem 1.4 this means that $T$ is uniformly ergodic in $L(X)$ \end{proof} \begin{theorem} \label{thm3.1} Let $k \in Z^{n}_{+}$. If $T\in L(X)^n$ A c.m. satisfies condition E-k, such that $\sum_{j=1}^{n}(I-T_j)$ has chain length at most 1 and $\ker(\sum_{j=1}^{n}(I-T_j))= \cap_{j=1}^{n}\ker(I-T_j)$. Then $T$ is uniformly. \end{theorem} \begin{proof} There are two cases \noindent\textbf{Case 1: $d\big(\sum_{j=1}^{n}(I-T_j)\big)=0$.} Then $\sum_{j=1}^{n}(I-T_j)$ is bijective then $X=\sum_{j=1}^{n}(I-T_j)X\oplus \ker(\sum_{j=1}^{n}(I-T_j))$, which implies, since $\cap_{j=1}^{n}\ker(I-T_j)\subset \ker(\sum_{j=1}^{n}(I-T_j))=\{0\}$ that $X=\sum_{j=1}^{n}(I-T_j)X\bigoplus \cap_{j=1}^{n}\ker(I-T_j)$, from the \cite[Theorem 10]{l1} we obtain $T$ is uniformly ergodic. \noindent\textbf{Case 2: $d(\big(\sum_{j=1}^{n}(I-T_j)\big))=1$.} Then $\big(\sum_{j=1}^{n}(I-T_j)\big)X= \big(\sum_{j=1}^{n}(I-T_j)\big)^{2}X$, so $\big(\sum_{j=1}^{n}(I-T_j)\big)X=\big(\sum_{j=1}^{n}(I-T_j)\big)^{nr} \big(\sum_{j=1}^{n}(I-T_j)\big)X$, with $r=\max_{1\leq j\leq n}k_j$. so $\big(\sum_{j=1}^{n}(I-T_j)\big)^{nr}$ is a bijection of $\big(\sum_{j=1}^{n}(I-T_j)\big)X$ onto itself. Which implies, since $T$ satisfies condition E-k, that $M_v(T)|\big(\sum_{j=1}^{n}(I-T_j)\big)X\to 0$ and since $M_v(T)|\cap_{j=1}^{n}\ker(I-T_j)=I|\cap_{j=1}^{n}\ker(I-T_j)$, it follows that $T$ is uniformly ergodic. \end{proof} \begin{thebibliography}{0} \bibitem{h1} Heuser H. G., \emph{Functional analysis}, Wiley, Chichester, 1982. \bibitem{k1} Krengel U., \emph{Ergodic theorems}, De Gruyter, Berlin and New York, 1985. \bibitem{l1} Lahrech S., A. Azizi, A. Mbarki, A. Ouahab; \emph{Uniformly Ergodic theorem and finite chain length for multioperators} International Journal of Pure and Applied Mathematics, \textbf{22, No. 2} (2005), 167-172. \bibitem{l2} Laursen K. B. and Mbekhta M.; \emph{Operators with finite chain length and the ergodic theorem}, Proc. Amer. Math. Soc. \textbf{123} (1995), 3443-3448. \bibitem{m1} Mbekhta M. and Vasilescu F.-H., \emph{Uniformly ergodic multioperators}, Trans. Amer. Math. Soc. \textbf{347}, (1995), 1847- 1854. \end{thebibliography} \end{document}