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Short Communications

On the integrability of a maximal square function in ergodic theory

V. F. Gaposhkin

Moscow State University of Railway Communications

Abstract: Necessary and sufficient conditions are established for $L^2$-summability of the maximal square function of general unitary operators. Extensions of the result to arbitrary contractions in $L^2$, multiparameter unitary groups, and power-bounded operators in $L^p$, $p>1$, are considered as well.

Keywords: unitary operators, contractive operators, homogeneous random fields, power-bounded operators, individual ergodic theorem, maximal square function.

Received: 03.03.1998

DOI: 10.4213/tvp777

 English version: Theory of Probability and its Applications, 2000, 44:2, 386–394

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