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Ergodicity and stability of orbits of unbounded semigroup representations

Part of: General theory of linear operators Abstract harmonic analysis Commutative Banach algebras and commutative topological algebras Functional-differential and differential-difference equations 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
Bolis Basit
Affiliation:
School of Mathematical Sciences, P.O. Box 28M, Monash University, VIC 3800, Australia e-mail: bolis.basit@sci.monash.edu.aualan.pryde@sci.monash.edu.au
A. J. Pryde
Affiliation:
School of Mathematical Sciences, P.O. Box 28M, Monash University, VIC 3800, Australia e-mail: bolis.basit@sci.monash.edu.aualan.pryde@sci.monash.edu.au
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Abstract

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We develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.

Keywords

weighted ergodicityorbits of unbounded semigroup representationnon-quasianalytic weightsstabilityBeurling spectrum

MSC classification

Secondary: 46J20: Ideals, maximal ideals, boundaries 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 47A35: Ergodic theory 34K25: Asymptotic theory 28B05: Vector-valued set functions, measures and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 209 - 232
Copyright
Copyright © Australian Mathematical Society 2004

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