| Summary: | The Katznelson-Tzafriri Theorem states that, given a powerbounded operator T , T n(I − T ) → 0 as n → ∞ if and only if the spectrum σ(T ) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T ) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(eiθ , T ) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.
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