A short proof of the Y. Katznelson’s and L. Tzafriri’s theorem
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- by Vũ Quôc Phóng PDF
- Proc. Amer. Math. Soc. 115 (1992), 1023-1024 Request permission
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 ${\lim _{n \to \infty }}||{T^{n + 1}} - {T^n}|| = 0$ if and only if $\sigma (T) \cap \{ z \in \mathbb {C}:|z| = 1\} \subset \{ 1\}$.References
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313–328. MR 859138, DOI 10.1016/0022-1236(86)90101-1
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 1023-1024
- MSC: Primary 47A05; Secondary 46H05, 47A10, 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1992-1087468-0
- MathSciNet review: 1087468