Tauberian type theorem for operators with interpolation spectrum for Hölder classes
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- by C. Agrafeuil and K. Kellay
- Proc. Amer. Math. Soc. 136 (2008), 2477-2482
- DOI: https://doi.org/10.1090/S0002-9939-08-09273-3
- Published electronically: March 11, 2008
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Abstract:
We consider an invertible operator $T$ on a Banach space $X$ whose spectrum is an interpolating set for Hölder classes. We show that if $\|T^{n}\|=O(n^p)$, $p\geq 1$, $\|T^{-n}\|=O(w_n)$ with $n^q=o(w_n)$ $\forall q\in \mathbb {N}$ and $\sum _n 1/(n^{1-\alpha } (\log w_{n})^{1+\alpha })=+\infty$, then $\|T^{-n}\|=O(n^{p+s})$ for all $s > \tfrac {1}{2}$, assuming that $(w_n)_{n\geq 1}$ satisfies suitable regularity conditions. When $X$ is a Hilbert space and $p=0$ (i.e. $T$ is a contraction), we show that under the same assumptions, $T$ is unitary and this is sharp.References
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Bibliographic Information
- C. Agrafeuil
- Affiliation: Université Aix Marseille III, Bat Henri Poincaré Cours A, 13397 Marseille cedex 20, France
- Address at time of publication: 164, rue d’Alésia, 75014 Paris, France
- Email: cyril.agrafeuil@univ.u-3mrs.fr, cyril.agrafeuil@gmail.com
- K. Kellay
- Affiliation: LATP-CMI, Université Aix Marseille I, 39 rue F. Jolio Curie, 13347 Marseille cedex 13, France
- Email: kellay@cmi.univ-mrs.fr
- Received by editor(s): February 26, 2007
- Published electronically: March 11, 2008
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2477-2482
- MSC (2000): Primary 30H05; Secondary 30D55, 47A15
- DOI: https://doi.org/10.1090/S0002-9939-08-09273-3
- MathSciNet review: 2390516