Ergodic Theory and Dynamical Systems

Modulated and subsequential ergodic theorems in Hilbert and Banach spaces

a1 Departments of Mathematics and of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail:
a2 Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail:
a3 Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA (e-mail:
a4 Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail:


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)