Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T08:34:24.208Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Amerio, L. and Prouse, G., Almost periodic functions and functional equations (Van Nostrand, New York, 1971).CrossRefGoogle Scholar
[2]Argabright, L. N., ‘On the mean value of weakly almost periodic functions’, Proc. Amer. Math. Soc. 36 (1972), 315–316.Google Scholar
[3]Basit, B., ‘Some problems concerning different types of vector-valued almost periodic type functions’, Dissertationes Math. (Rozprawy Mat.) 338 (1995).Google Scholar
[4]Basit, B., ‘Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem’, Semigroup Forum 54 (1997), 58–74.CrossRefGoogle Scholar
[5]Basit, B. and Pryde, A. J., ‘Polynomials and functions with finite spectra on locally compact abelian groups’, Bull. Austral. Math. Soc. 51 (1994), 33–42.CrossRefGoogle Scholar
[6]Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on Semigroups: Function spaces, compactifications, representations (Wiley-Interscience, New York, 1989).Google Scholar
[7]Datry, C. and Muraz, G., ‘Analyse harmonique dans les modules de Banach II: presque-périodicité et ergodicité’, Bull. Science Math. (2) 120 (1996), 493–536.Google Scholar
[8]Dunford, N. and Schwartz, J. T., Linear operators, Parts I, II (Interscience, New York, 1958, 1963).Google Scholar
[9]Eberlein, W. F., ‘Abstract ergodic theorems and weak almost periodic functions’, Trans. Amer. Math. Soc. 69 (1949), 217–240.CrossRefGoogle Scholar
[10]Goldstein, J. A., ‘Application of operator semigroups to Fourier analysis’, Semigroup Forum 52 (1996), 37–47.CrossRefGoogle Scholar
[11]Iseki, K., ‘Vector valued functions on semigroups, I-III’, Proc. Japan Acad. Ser. A Math. Sci. (1955), 16–19, 152–155 and 699–702.Google Scholar
[12]Katznelson, Y. and Tzafriri, L., ‘On power bounded operators’, J. Funct. Anal. 68 (1986), 313–328.CrossRefGoogle Scholar
[13]Levitan, B. M., ‘Integration of almost periodic functions with values in Banach spaces’, Math. USSR-Izv. 30 (1966), 1101–1110 (in Russian).Google Scholar
[14]Reiter, H., Classical Fourier analysis on locally compact groups (Oxford University Press, 1968).Google Scholar
[15]Rudin, W., Harmonic analysis on groups (Interscience, New York, 1963).Google Scholar
[16]Ruess, W. M. and Phóng, V. Q., ‘Asymptotically almost periodic solutions of evolution equations in Banach spaces’, J. Differential Equations 122 (1995), 282–301.Google Scholar
[17]Ruess, W. M. and Summers, W. H., ‘Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups’, Israel J. Math. 64 (1988), 139–157.CrossRefGoogle Scholar
[18]Ruess, W. M. and Summers, W. H., ‘Integration of asymptotically almost periodic functions and weak almost periodicity’, Dissertationes Math. (Rozprawy Mat.) 279 (1989).Google Scholar
[19]Ruess, W. M. and Summers, W. H., ‘Ergodicity theorems for semigroups of operators’, Proc. Amer. Math. Soc. 114 (1992), 423–432.CrossRefGoogle Scholar
[20]Yosida, K., Functional Analysis (Springer, Berlin, 1966).Google Scholar
[21]Zhang, C., ‘Vector-valued means and their applications in some vector-valued function spaces’, Dissertationes Math. (Rocprawy Mat.) 334 (1994).Google Scholar