Mean ergodic theorems for semigroups of linear operators
type: Item
instance of: scholarly article
title: Mean ergodic theorems for semigroups of linear operators
zbMATH Open document ID: 0608.47045
DOI: 10.1016/0022-247X(84)90136-7
author: Kazuo Kido
author: Wataru Takahashi
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: In 1938, Yosida proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and T be a linear operator of E into itself such that there exists a constant C with \(\| T^ n\| \leq C\) for \(n=1,2,3,...\), and T is weakly completely continuous, i.e., T maps the closed unit ball of E into a weakly compact subset of E. Then, the Cesàro means \(S_ nx=(1/n)\sum^{n}_{k=1}T^ kx\) converge strongly as \(n\to +\infty\) to a fixed point of T for each \(x\in E.\) In this paper, by using \textit{G. Rodé}'s method [this J. 85, 172-178 (1982; Zbl 0485.47041)], we extend Yosida's theorem to semigroups of linear operators in Banach spaces.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 3983943
zbMATH Keywords: mean ergodic theorem
MaRDI profile type: Publication
cites work: Q4064547
cites work: Amenable semigroups
cites work: Q5562332
cites work: Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space
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cites work: A Nonlinear Ergodic Theorem for an Amenable Semigroup of Nonexpansive Mappings in a Hilbert Space
description: scientific article; zbMATH DE number 3983943