On modulated ergodic theorems for Dunford-Schwartz operators
Let \((\Omega,{\mathcal F},\mu)\) be a \(\sigma\)-finite measure space. A linear operator in \(L_1\) is called contraction if \(\|T\|\leq 1\). It is called Dunford-Schwartz operator of it is, in addition, a contraction in \(L_\infty\). Let \(a= (a_k)\) be a sequence of complex numbers, and, for \(f\in L_1\), \[ L_nf= L_n(a, T)f= n^{-1} \sum^n_{k=1} a_k T^kf. \] The authors study sequences \(a\) for which \(L_nf\) converges in norm for every weakly almost periodic linear operator in a Banach space. For Dunford-Schwartz operators on probability spaces the a.e.-convergence for \(f\in L_p\) is studied. The limit is identified in some special cases. Also necessary and sufficient conditions for a sequence \(a\) are derived for the norm convergence of \(L_nf\) for every mean ergodic power bounded \(T\).
scientific article; zbMATH DE number 1345978