ON MODULATED ERGODIC THEOREMS FOR DUNFORD-SCHWARTZ OPERATORS

摘要

We investigate sequences of complex numbers a={a_k} for which the modulated averages 1 ∑sum_k=1~n 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 Lp. The limit is identified in some spacial cases, in particular when T is a contraction in a Hilbert space ,or when a={S~k φ(ζ)} for some positive Dunford-Schwartz operator S on a Lebesgue space and φ∈L_2. We also obtain necessary and sufficient conditions on a for the norm convergence of the modulated averages for every mean ergodic power bounded T, and identify limit.