An identification of ratio ergodic limits for semi-groups

Abstract

Let L 1=L 1 (X, ∞, Μ), where (X, ∞, Μ) is a σ-finite measure space and let T 1: L 1 → L 1, t≧0, be a strongly continuous semi-group of positive linear contractions and U t :L ∞→L ∞ be the dual of T t. The purpose of this paper is to give an identification of the ratio ergodic limit $$(f/g) = \mathop {\lim }\limits_{s \to \infty } \left( {{{\int\limits_0^s {T_t f} dt} \mathord{\left/ {\vphantom {{\int\limits_0^s {T_t f} dt} {\int\limits_0^s {T_t g} dt}}} \right. \kern-\nulldelimiterspace} {\int\limits_0^s {T_t g} dt}}} \right)$$ where f and g are in L 1 and g>0. We construct a sub-Banach algebra A of L ∞ that contains þ={f∈L ∞¦U t f=f all t≧0} and define a transformation: μA→A With multiplication defined by fg=π(fg), þ becomes a B *-algebra which is isometrically isomorphic under a mapping σ to C(K), the space of complex valued continuous functions on the maximal ideal space K of þ. Let M(K) denote the space of finite complex Baire measures on K. Define Τ: A→C(K)} where Τ=σπ and λ: L 1→ M(K) where, for f in L 1, ∫ fh dΜ=∫σf dλ f for every h in þ Then our identification for (f/g) in L ∞ is Τ(f/g)=dλ f/dλg.