Weighted and Subsequential Ergodic Theorems

1. Introduction. Let ( X , , μ ) be a probability space, T a linear operator on ℒ p ( X , , μ ), for some p , 1 ≦ p ≦ ∞. Let a n be a sequence of complex numbers, n = 0, 1, …, which we shall often refer to as weights. We shall say that the weighted pointwise ergodic theorem holds for T on ℒ p , if, for every ƒ in ℒ p , 1.1 Let a denote the sequence (a n ) . If (1.1) holds we shall say that a is Birkhoff for T on ℒ p , or, more briefly, that ( a , T ) is Birkhoff. We are also interested in ergodic theorems for subsequences. Let n(k) be a subsequence. We shall say the pointwise ergodic theorem holds for the subsequence n(k) and the operator T if, for every ƒ in ℒ p , 1.2

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