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Modulated and subsequential ergodic theorems in Hilbert and Banach spaces

Published online by Cambridge University Press:  06 November 2002

D. BEREND
Affiliation:
Departments of Mathematics and of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail: berend@math.bgu.ac.il)
M. LIN
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail: lin@math.bgu.ac.il)
J. ROSENBLATT
Affiliation:
Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA (e-mail: jrsnbltt@symcom.math.uiuc.edu)
A. TEMPELMAN
Affiliation:
Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: arkady@stat.psu.edu)

Abstract

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Type
Research Article
Copyright
2002 Cambridge University Press

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