Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

DOI10.1016/J.JFA.2020.108799arXiv2002.06236MaRDI QIDQ6334837

David Seifert, Abraham C. S. Ng

Publication date: 14 February 2020

Abstract: The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)|o0$ if and only if $sigma(T)capmathbbTsubseteq1$. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms $|R(e^{iheta},T)|$ as $|heta|o0$ satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.

Mathematics Subject Classification ID

Ergodic theory of linear operators (47A35) Spectrum, resolvent (47A10) General (adjoints, conjugates, products, inverses, domains, ranges, etc.) (47A05)

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