Mean ergodicity of positive operators in KB-spaces

We prove that any positive power bounded operator T in a KB-space E which satisfies lim n→∞ dist 1 n n−1 k=0 T kx,[−g,g] + ηBE = 0 ∀x ∈ E, x 1 , (1) where BE is the unit ball of E, g ∈ E+, and 0 η < 1, is mean ergodic and its fixed space Fix(T ) is finite dimensional. This generalizes the main result of [E.Yu. Emelyanov, M.P.H. Wolff, Mean lower bounds for Markov operators, Ann. Polon. Math. 83 (2004) 11–19]. Moreover, under the assumption that E is a σ-Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T , the condition (1) implies that T is mean ergodic then E is a KB-space. © 2005 Elsevier Inc. All rights reserved.