On the convergence of ergodic ratios for positive operators

Abstract

Let (X, G?, m) be a a-finite measure space, and L, = L,(X, O!, m) the usual Banach spaces. Recently, M. A. Akcoglu constructed examples of positive contractions on L, , 1 1. We say that the ratio ergodic theorem holds for T on a set A if f~ L and g EL+ imply that lim,,, R,(f, g) exists and is finite a.e. on A n {cz Tig > O}. IfL = L, , T* denotes the adjoint operator acting on L, (where l/q + l/p = 1). A u-finite measure v < m is called subinvariant forTifjITfjdv<JIf/dvforeveryfEL.