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Abelian ergodic theorems for vector-valued functions

Published online by Cambridge University Press:  18 May 2009

P. E. Kopp
P. E. Kopp
Affiliation:
Department of Pure Mathematics, University of Hull
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This note contains extensions of the Abelian ergodic theorems in [3] and [6] to functions which take their values in a Banach space. The results are based on an adaptation of Rota's maximal ergodic theorem for Abel limits [8]. Convergence theorems for continuous parameter semigroups are deduced by the approximation technique developed in [3], [6]. A direct application of the resolvent equation also enables us to deduce a convergence theorem for pseudo-resolvents.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 16 , Issue 1 , March 1975 , pp. 57 - 60
Copyright
Copyright © Glasgow Mathematical Journal Trust 1975

References

REFERENCES

1.Chacon, R. V., An ergodic theorem for operators satisfying norm conditions, J. Math. Mech. 11 (1962), 165–172.Google Scholar
2.Dunford, N. and Schwartz, J. T., Linear Operators, Part I, Interscience (New York, 1958).Google Scholar
3.Edwards, D. A., A maximal ergodic theorem for Abel means of continuous-parameter operator semigroups, J. Functional Anal. 7 (1971), 61–70.CrossRefGoogle Scholar
4.Edwards, D. A., On resolvents and general ergodic theorems, Z. Wahrscheinlichkeitstheorie undverw. Gebiete 20 (1971), 1–8.CrossRefGoogle Scholar
5.Hille, E. and Phillips, R. S., Functional Analysis and Semigroups, American Math. Soc. Colloquium Publications Vol. XXXI (Providence, R.I., 1957).Google Scholar
6.Kopp, P. E., Almost everywhere convergence for Abel means of Dunford-Schwartz operators, J. Lond. Math. Soc. (2), 6 (1973), 368–372.CrossRefGoogle Scholar
7.Neveu, J., Relations entre la théorie des martingales et la théorie ergodique, Ann. Inst. Fourier (Grenoble) 15 (1965), 31–42.CrossRefGoogle Scholar
8.Rota, G.-C., On the maximal ergodic theorem for Abel limits, Proc. American Math. Soc. 14 (1963), 722–723.CrossRefGoogle Scholar