Vol. 85, No. 2, 1979

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Vector-valued ergodic theorems for operators satisfying norm conditions

T. Yoshimoto

Vol. 85 (1979), No. 2, 485–499

A new approach is developed in the theory of pointwise ergodic theorems. Our consideration is based upon Ωμp (0 p < ), which is a linear space containing properly the linear span of Up>1Lp(X;𝒳), where (X,) is a σ-finite measure space and 𝒳 is a reflexive Banach space. Some weak and strong type inequalities are proved as vector valued generalizations of the Dunford and Schwartz’s results, and then, used to study the integrability of the ergodic maximal function. These results do make it possible to extend the Chacon’s vector valued ergodic theorem. We have analogous extensions for the case of continuous semigroups, and the local ergodic theorem is shown to hold on Ωμ0. The results include two applications to the random ergodic theorem and the “strong differentiability” theorem.

Mathematical Subject Classification 2000
Primary: 47A35
Secondary: 28D15
Received: 16 January 1979
Published: 1 December 1979
T. Yoshimoto