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arXiv · 1410.1297

Rates of decay in the classical Katznelson-Tzafriri theorem

David Seifert
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Abstract

Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri states that $\|T^n(I-T)\|\to0$ as $n\to\infty$ if and only if the spectrum $σ(T)$ of $T$ intersects the unit circle $\mathbb{T}$ in at most the point 1. This paper investigates the rate at which decay takes place when $σ(T)\cap\mathbb{T}=\{1\}$. 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 $R(\mathrm{e}^{\mathrm{i}θ},T)$ as $θ\to0$. 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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David Seifert. 2014-10-06. Rates of decay in the classical Katznelson-Tzafriri theorem. https://doi.org/10.1007/s11854-016-0039-3

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