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Ergodicity and differences of functions on semigroups

Part of: Abstract harmonic analysis General theory of linear operators Groups and semigroups of linear operators, their generalizations and applications Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  09 April 2009

Bolis Basit  and
A. J. Pryde
A. J. Pryde
Affiliation:
Department of Mathematics Monash University Clayton, VIC 3168Australia e-mail: bbasit(ajpryde)@vaxc.cc.monash.edu.au
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Abstract

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Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.

Keywords

ErgodicsemigroupdifferencesBeurling spectruminvariant meanssystem of invariant integrals.

MSC classification

Secondary: 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 47A10: Spectrum, resolvent 47D03: Groups and semigroups of linear operators 28B05: Vector-valued set functions, measures and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 2 , April 1998 , pp. 253 - 265
Copyright
Copyright © Australian Mathematical Society 1998

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