Mean ergodicity of positive operators in KB-spaces

Abstract

We prove that any positive power bounded operator T in a KB-space E which satisfieswhere 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.

Stability and lower-bound functions of C0-Markov semigroups on KB-spaces

“…In the following theorem, we will establish the asymptotic properties of C 0 Markov semigroups on KB-spaces. For the proof, we refer to [1] and [3] Theorem 2.2. Let E be a KB-space with a quasi-interior point e, T = (T t ) t2R be a C 0 Markov semigroup on E and A T t be the Cesaro averages of T .…”

Lotz–Räbiger's nets of Markov operators in L1-spaces

“…(1) each A λ is a linear operator in X , (2) for each x ∈ X and all λ, A λ ∈ coT x, (3) operators A λ are equi-continuous, and…”

“…1], had been generalized in[2] to Cesàro averages of a positive power bounded operator in arbitrary K B-space (see [2, Thm. 1, Thm.…”

Mean ergodicity vs weak almost periodicity

“…From this we conclude in Theorem 3.3 that every countably order complete Banach lattice on which mean ergodicity and weak almost periodicity are equivalent for positive and power-bounded operators is necessarily a KB-space. We also refer to [2] for a characterization of KB-spaces by mean ergodicity of certain positive operators. In the final Section 4 we prove that all powers of a mean ergodic positive operator T are also mean ergodic in case that the mean ergodic projection of T equals 0.…”