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Ergodicity and differences of functions on semigroups

  • Bolis Basit and A. J. Pryde (a1)
Abstract

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.

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Copyright
COPYRIGHT: © Australian Mathematical Society 1998
References
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[1]Amerio, L. and Prouse, G., Almost periodic functions and functional equations (Van Nostrand, New York, 1971).
[2]Argabright, L. N., ‘On the mean value of weakly almost periodic functions’, Proc. Amer. Math. Soc. 36 (1972), 315–316.
[3]Basit, B., ‘Some problems concerning different types of vector-valued almost periodic type functions’, Dissertationes Math. (Rozprawy Mat.) 338 (1995).
[4]Basit, B., ‘Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem’, Semigroup Forum 54 (1997), 58–74.
[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.
[6]Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on Semigroups: Function spaces, compactifications, representations (Wiley-Interscience, New York, 1989).
[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.
[8]Dunford, N. and Schwartz, J. T., Linear operators, Parts I, II (Interscience, New York, 1958, 1963).
[9]Eberlein, W. F., ‘Abstract ergodic theorems and weak almost periodic functions’, Trans. Amer. Math. Soc. 69 (1949), 217–240.
[10]Goldstein, J. A., ‘Application of operator semigroups to Fourier analysis’, Semigroup Forum 52 (1996), 37–47.
[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.
[12]Katznelson, Y. and Tzafriri, L., ‘On power bounded operators’, J. Funct. Anal. 68 (1986), 313–328.
[13]Levitan, B. M., ‘Integration of almost periodic functions with values in Banach spaces’, Math. USSR-Izv. 30 (1966), 1101–1110 (in Russian).
[14]Reiter, H., Classical Fourier analysis on locally compact groups (Oxford University Press, 1968).
[15]Rudin, W., Harmonic analysis on groups (Interscience, New York, 1963).
[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.
[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.
[18]Ruess, W. M. and Summers, W. H., ‘Integration of asymptotically almost periodic functions and weak almost periodicity’, Dissertationes Math. (Rozprawy Mat.) 279 (1989).
[19]Ruess, W. M. and Summers, W. H., ‘Ergodicity theorems for semigroups of operators’, Proc. Amer. Math. Soc. 114 (1992), 423–432.
[20]Yosida, K., Functional Analysis (Springer, Berlin, 1966).
[21]Zhang, C., ‘Vector-valued means and their applications in some vector-valued function spaces’, Dissertationes Math. (Rocprawy Mat.) 334 (1994).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
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