Ergodic Theorems and Approximation Theorems with Rates
by
Sen-Yen Shaw
Department of Mathematics, National Central University

Let A be a closed linear operator, and let {A_\alpha} and {B_\alpha} be two nets of bounded linear operators on a Banach space X which satisfy: ||A_\alpha||=O(1); B_\alpha A subset AB_\alpha = I-A_\alpha for all \alpha; ||AA_\alpha|| = O (e(\alpha)); ||A_\alpha x|| = O(f(\alpha)) (resp. o(f(\alpha))) ===> ||B_\alpha y|| = O(\fracf(\alpha)e(\alpha)) (resp. o(\fracf(\alpha)e(\alpha))), where e and f are positive functions satisfying 0 < e(\alpha) <= f(\alpha) --> 0. We call {A_\alpha} an A-ergodic net of order O(e(\alpha)) and {B_\alpha} its companion net. A uniformly bounded net {T_\alpha} subset B(X) is called an A-regularized approximation process of order O(e(\alpha)) if there is another net {S_\alpha} of linear operators such that S_\alpha --> I strongly and S_\alpha A subset AS_\alpha = (e(\alpha))^-1(T_\alpha-I).

We shall discuss the convergence of {A_\alpha}, {B_\alpha}, and {T_\alpha}. There are strong convergence theorems, uniform convergence theorems, optimal convergence (or saturation) theorems, non-optimal convergence theorems, and theorems concerning the sharpness of non-optimal convergence. The general results provide unified approaches to investigation of convergence rates of ergodic limits and approximation of various operator families. In particular, we shall deduce some theorems for an integrated resolvent family, which contains C₀-semigroups, integrated semigroups, and integrated cosine functions as special cases.

Date received: April 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacb-24.