Summary
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
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.
Article PDF
Similar content being viewed by others
References
Akcoglu, M. A.: An ergodic lemma. Proc. Amer. math. Soc. 16, 388–392 (1965).
—: Pointwise ergodic theorems. Trans. Amer. math. Soc. 125, 296–309 (1966).
- Cunsolo, J.: An ergodic theory for semi-groups, to appear.
—, Sharpe, R. W.: Ergodic theory and boundaries. Trans. Amer. math. Soc. 132, 447–460 (1968).
Berk, K. N.: Ergodic theory with recurrent weights. Ann. math. Statistics 39, 1107–1114 (1968).
Brunel, A.: Sur un lemme ergodique voisin du lemme du Hopf. C. r. Acad. Sci. Paris 256, 5481–5484 (1963).
Chacon, R. V.: Identification of the limit of operator averages. J. Math. Mech. 11, 961–968 (1962).
—, Ornstein, D. S.: A general ergodic theorem. Illinois J. Math. 4, 153–160 (1960).
Author information
Authors and Affiliations
Additional information
This research is supported in part by N.R.C. Grant A-3974.
Rights and permissions
About this article
Cite this article
Akcoglu, M.A., Cunsolo, J. An identification of ratio ergodic limits for semi-groups. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 219–229 (1970). https://doi.org/10.1007/BF00534919
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00534919