Mean ergodicity of positive operators in KB-spaces

We prove that any positive power bounded operator T in a KB-space E which satisfiesequation(1)limn→∞dist(1n∑k=0n−1Tkx,[−g,g]+ηBE)=0(∀x∈E,‖x‖⩽1), where BEBE is the unit ball of E  , g∈E+g∈E+, and 0⩽η<10⩽η<1, is mean ergodic and its fixed space Fix(T)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.