NONOPTIMAL RATES OF ERGODIC LIMITS AND APPROXIMATE SOLUTIONS

This paper is concerned with non-optimal rates of convergence for two processes {A(alpha)} and {B-alpha}, which satisfy \\A(alpha)\\ = O(1), B(alpha)A subset of AB(alpha) = I - A(alpha),\\AA(alpha)\\ = O(e(alph a)), where A is a closed operator and e(alpha) --> 0. Under suitable c onditions, we describe, where A is a closed operator and e(alpha) --> 0. Under suitable conditions, we describe, in terms of K-functionals, those x (resp. y) for which the non-optimal convergence rare of {A(alp ha)x} (resp. {B(alpha)y}) is of the order O(f(alpha)), where f is a fu nction satisfying e(alpha) less than or equal to f(alpha) --> 0. In ca se that f(alpha)/e(alpha) --> infinity, the sharpness of the non-optim al rate of {A(alpha)x} is equivalent to that A has non-closed range. T he result provides a unified approach to dealing with non-optimal rate s for many particular mean ergodic theorems and for various methods of solving the equation Ax = y. We discuss in particular applications to alpha-times integrated semigroups, n-times integrated cosine Function s, and tensor product semigroups. (C) 1998 Academic Press.