An Individual Ergodic Theorem

An individual ergodic theorem is proved for a linear operator T on LI of a finite measure space which satisfies certain norm conditions. Derriennic and Lin (51 showed by an example that given an E > 0 there exists a positive linear operator T on L1 of a finite measure space, with TI = 1 and 11 Tn1, = 1 + e for all n > 1, and a functionf in L1 such that the individual ergodic limit nlim 2 Tyf(x) n ni=O does not exist almost everywhere on a certain measurable subset of positive measure. In this paper, however, we shall prove the following individual ergodic theorem. THEOREM. Let (X, 6F, yt) be a finite measure space and Lp = Lp(X, '6Y, At), 1 < p < oo, the usual Banach spaces. Let T be a bounded linear operator on LI and X its linear modulus in the sense of Chacon and Krengel [4]. Assume the conditions: (1) sup 2 7i < , n n i0O (2) Sup 2 T'i <00 n in Then, for any f E L 00, the ergodic limit nlim n 2 Tf(x) exists for almost all x E X. PROOF. Let T* and T* denote the adjoint operators of T and T, respectively. Since IT*fl < T*jfl (cf. [1]) and f Tr*fl dtt = f (Tl)lfl dtL < ( IIrIIIfill for all Received by the editors November 17, 1976 and, in revised form, January 31, 1977. AMS (MOS) subject classifications (1970). Primary 47A35; Secondary 28A65.