Continuous dependence of ergodic limits (Q909957)

Let T be a linear bounded operator on a Banach space and assume that \(\sup \{\| T^ n\|:\) \(n\in {\mathbb{N}}\}<\infty\). It is well known (mean ergodic theorem) that the sequence (1/n\(\sum^{n-1}_{k=0}T^ k:\) \(n\in {\mathbb{N}})\) converges to a limit P which is a bounded projection constructed from T. The authors have shown that, in a sense, P depends continuously on T (Theorem 1). A similar result has been obtained for continuous parameter semigroups of uniformly bounded operators: the ergodic limit depends continuously on the infinitesimal generator of the semigroup (Theorem 2). Some simple examples illustrate both results. The paper is motivated by recent questions studied in the theory of Markov chains.