On modulated ergodic theorems for Dunford-Schwartz operators

Abstract

We investigate sequences of complex numbers $\mathbf{a} = \{a_{k}\}$ for which the modulated averages $\frac{1}{n}\sum^{n}_{k=1}{a_{k}T^{k} f}$ converge in norm for every weakly almost periodic linear operator $T$ in a Banach space. For Dunford-Schwartz operators on probability spaces, we study also the a.e. convergence in $L_{p}$. The limit is identified in some special cases, in particular when $T$ is a contraction in a Hilbert space, or when $\mathbf{a} = \{S^{k}\phi(\xi)\}$ for some positive Dunford-Schwartz operator $S$ on a Lebesgue space and $\phi \in L_{2}$. We also obtain necessary and sufficient conditions on $\mathbf{a}$ for the norm convergence of the modulated averages for every mean ergodic power bounded $T$, and identify the limit.