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Ergodic Properties of Lamperti Operators, II

Published online by Cambridge University Press:  20 November 2018

Charn-Huen Kan
Charn-Huen Kan*
Affiliation:
National University of Singapore, Kent Ridge, Singapore
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For T in our main Theorem 5, T* is called Lamperti in [11], whose terminology and notation we shall follow in the sequel. To avoid longish expressions, we shall also say that T* here is disjunctive and, dually, T = (T*)* is codisjunctive. The present work grows out of an attempt to establish a DEE for the general power bounded positive operator on Lp, in view of the success in the contraction case [1, 11], and forms a continuation of [11]. (In passing, we note that Calderon's technique [2] mentioned in [11] was anticipated in 1938 by M. Fukamiya [7], though in a variant form and for a more classical case, namely that of a positive Lp isometry induced by an invertible, measure preserving transformation on a totally finite measure space. Calderon's case does not assume invertibility nor total finiteness.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983 , pp. 577 - 588
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Akcoglu, M. A., A pointwise ergodic theorem in Lp-spaces, Can. J. Math. 27 (1975), 1075–1082.Google Scholar
2. Calderon, A. P., Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353.Google Scholar
3. Chacon, R. V. and McGrath, S. A., Estimates of positive contractions, Pacific J. Math. 30 (1969), 609–620.Google Scholar
4. Chacon, R. V. and Olsen, J., Dominated estimates of positive contractions, Proc. Amer. Math. Soc. 20 (1969), 266–271.Google Scholar
5. Doob, J. L., Stochastic processes (Wiley, New York, 1952).Google Scholar
6. Dunford, N. and Schwartz, J. T., Convergence almost everywhere of operator averages, J. Math. Mech. 5 (1966), 399–401.Google Scholar
7. Fukamiya, M., On dominated ergodic theorems in Lp (p ≧ 1), Tôhoku Math. J. 46 (1939), 150–153.Google Scholar
8. Garsia, A. M., A simple proof of E. Hopf's maximal ergodic theorem, J. Math. Mech. 14 (1965), 381–382.Google Scholar
9. Garsia, A. M., Topics in almost everywhre convergence, Lectures in advanced mathematics, vol. 4 (Markham, Chicago, 1970).Google Scholar
10. Tulcea, A. Ionescu, Ergodic properties of isometries in Lp-spaces, 1 < p < ∞, Bull. Amer. Math. Soc. 70 (1964), 366–371.Google Scholar
11. Kan, C. H., Ergodic properties of Lamperti operators, Can. J. Math. 30 (1978), 1206–1214.Google Scholar