Ergodic Theory and Dynamical Systems



Modulated and subsequential ergodic theorems in Hilbert and Banach spaces


D. BEREND a1, M. LIN a2, J. ROSENBLATT a3 and A. TEMPELMAN a4
a1 Departments of Mathematics and of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail: berend@math.bgu.ac.il)
a2 Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail: lin@math.bgu.ac.il)
a3 Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA (e-mail: jrsnbltt@symcom.math.uiuc.edu)
a4 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.

(Received August 29 2000)
(Revised November 16 2001)