ON ERGODIC AVERAGES AND ABSORBING SETS FOR POSITIVE CONTRACTIONS IN L(1) - NOTE

[摘要] Let T: L(1)(X) --> L(1)(X) be a positive contraction, that is, a linear operator satisfying parallel to T parallel to less than or equal to 1 and T(L(1)(+)(X)) subset of or equal to L(1)(+)(X). Let S-n = Sigma(k=0)(n=1) T-k, A(n) = (1/n) S-n, and let F = {x: limsup(n-->proportional to) A(n) phi(x) = infinity}, where phi is a.e. positive. First, we prove that for every sequence (f(n)) in L(1)(+)(X) such that, for every m, (T-m f(n) - f(n))(+) tends stochastically to 0 on F, min (f(n), 1/f(n) tends stochastically to 0 on F. This implies, in particular, the Stochastic Ergodic Theorem. Second, we study the behavior of sequences of the form a(n)S(k(n)) phi(x) to obtain decompositions of the conservative part of X into six absorbing sets. (C) 1995 Academic Press, Inc.

[发布日期] 1995-09-01