Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures
Abstract
Let (,) be a measurable space with -finite continuous measure, ()=. A linear operator T:L1()+L()L1()+L() is called the Dunford-Schwartz operator if ||T(f)||1||f||1 (respectively, ||T(f)||||f||) for all fL1() (respectively, fL()). {Tt}t0is a strongly continuous in L1() semigroup of Dunford-Schwartz operators, then each operator At(f)=1t∫0tTs(f)ds∈L1(Ω){{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}} has a unique extension to the Dunford-Schwartz operator, which is also denoted by At, t0. It is proved that in the completely symmetric space of measurable functions on (,) the means At converge strongly as t+ for each strongly continuous in L1() semigroup {Tt}t0 of Dunford-Schwartz operators if and only if the norm ||.||E() is order continuous.