Mean ergodicity of positive operators in KB-spaces

Abstract

We prove that any positive power bounded operator T in a KB-space E which satisfies (1) lim n → ∞ dist ( 1 n ∑ k = 0 n − 1 T k x , [ − g , g ] + η B E ) = 0 ( ∀ x ∈ E , ‖ x ‖ ⩽ 1 ) , where B E 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.