Modulated and subsequential ergodic theorems in Hilbert and Banach spaces

Let {a_k}_k≥0 be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^-1 ∑_k=0ⁿ a_k T^kx for every contraction T on a Hilbert space H and every x ∈ H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers {k_j}, we study the problem of when n^-1 ∑_j=1ⁿ T^kjx converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.