Positive operators and the ergodic theorem

Ryōtarō Satō

Vol. 76 (1978), No. 1, 215–219

DOI: 10.2140/pjm.1978.76.215

Abstract

Let T be a positive linear operator on L₁(X,ℱ,μ) satisfying sup_n∥(1∕n)∑ _i=0ⁿ⁻¹Tⁱ∥₁ < ∞, where (X,ℱ,μ) is a finite measure space. It will be proved that the two following conditions are equivalent: (I) For every f in L_∞(X,ℱ,μ) the Cesàro averages of T^∗nf converge almost everywhere on X. (II) For every f in L₁(X,ℱ,μ) the Cesàro averages of Tⁿf converge in the norm topology of L₁(X,ℱ,μ). As an application of the result, a simple proof of a recent individual ergodic theorem of the author is given.

Mathematical Subject Classification

Primary: 28A65, 28A65

Milestones

Received: 20 July 1977

Published: 1 May 1978