Pointwise limit theorem for a class of unbounded operators in L^r-spaces

Abstract

Abstract. We distinguish a class of unbounded operators in L r , r ≥ 1, related to the self-adjoint operators in L 2 . For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in L r -spaces are applied.1. Introduction. The paper is an attempt of extension of the pointwise ergodic theory in L r (µ)-spaces [10] to the case of some unbounded operators. Some questions are maybe touched here for the first time and perhaps new techniques will have to be developed. We start by trying to follow some ideas and methods known in the case of L 2 (µ). That is why we confine ourselves to a class of linear maps in L r related to unbounded self-adjoint operators in L 2 . To discuss the asymptotic properties of unbounded operators we pass from Cesàro averages to Borel summability. Let us recall the definition.We say that a sequence x = {ξ k } of numbers (or vectors in a Banach soace) is summable to ξ by the Borel method if