Abstract
This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that \({\rm lim}_{n \rightarrow \infty} ||T^{n}(I-T)|| = 0\) if T is a power-bounded operator on a Banach space and \(\sigma (T)\cap \mathbb{T} \subseteq\{1\}\). Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.
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Batty, C., Seifert, D. Some developments around the Katznelson–Tzafriri theorem. Acta Sci. Math. (Szeged) 88, 53–84 (2022). https://doi.org/10.1007/s44146-022-00006-1
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DOI: https://doi.org/10.1007/s44146-022-00006-1