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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A short proof of the Y. Katznelson's and L. Tzafriri's theorem


Author: Vũ Quôc Phóng
Journal: Proc. Amer. Math. Soc. 115 (1992), 1023-1024
MSC: Primary 47A05; Secondary 46H05, 47A10, 47A35
MathSciNet review: 1087468
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Abstract: A short proof is given to the following theorem of Y. Katznelson and L. Tzafriri: Let $ T$ be a power-bounded operator in a Banach space $ E$. Then $ {\lim _{n \to \infty }}\vert\vert{T^{n + 1}} - {T^n}\vert\vert = 0$ if and only if $ \sigma (T) \cap \{ z \in \mathbb{C}:\vert z\vert = 1\} \subset \{ 1\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1087468-0
PII: S 0002-9939(1992)1087468-0
Article copyright: © Copyright 1992 American Mathematical Society