Vol. 76, No. 1, 1978

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Positive operators and the ergodic theorem

Ryōtarō Satō

Vol. 76 (1978), No. 1, 215–219

Let T be a positive linear operator on L1(X,) satisfying supn(1∕n) i=0n1Ti1 < , where (X,) is a finite measure space. It will be proved that the two following conditions are equivalent: (I) For every f in L(X,) the Cesàro averages of Tnf converge almost everywhere on X. (II) For every f in L1(X,) the Cesàro averages of Tnf converge in the norm topology of L1(X,). 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
Received: 20 July 1977
Published: 1 May 1978
Ryōtarō Satō