Abstract. Let T be a positive contraction on L1 of a σ-finite measure space. Necessary and sufficient conditions are given in order that for any f in L1 the averages 1 Pn−1 n i=0 T if converge in the norm of L1. 1. Introduction. Let (Ω, A, µ) be a σ-finite measure space and let L1 = L1(Ω, A, µ) denote de usual Banach space of all real-valued integrable functions on Ω. A linear operator T: L1 → L1 is called positive if f ≥ 0 implies T f ≥ 0, and a contraction if ‖T ‖1 ≤ 1, with ‖T ‖1 denoting the operator norm of T on L1. We say that th
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