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Rates of decay in the classical Katznelson-Tzafriri theorem

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Abstract

The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ǁT n(I − T)ǁ → 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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  1. St. John’s College, University of Oxford, Oxford, OX2 6GG, UK

    David Seifert

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  1. David Seifert
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Seifert, D. Rates of decay in the classical Katznelson-Tzafriri theorem. JAMA 130, 329–354 (2016). https://doi.org/10.1007/s11854-016-0039-3

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  • DOI: https://doi.org/10.1007/s11854-016-0039-3

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