Almost weak polynomial stability of operators

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We investigate whether almost weak stability of an operator T on a Banach space X implies its almost weak polynomial stability. We show, using a modified version of the van der Corput Lemma that if X is a Hilbert space and T a contraction, then the implication holds. On the other hand, based on a TDS arising from a two dimensional ODE, we give an explicit example of a contraction on a C₀ space that is almost weakly stable, but its appropriate polynomial powers fail to converge weakly to zero along a subsequence of density 1. Finally we provide an application to convergence of polynomial multiple ergodic averages.