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

 

Tauberian type theorem for operators with interpolation spectrum for Hölder classes


Authors: C. Agrafeuil and K. Kellay
Journal: Proc. Amer. Math. Soc. 136 (2008), 2477-2482
MSC (2000): Primary 30H05; Secondary 30D55, 47A15.
Published electronically: March 11, 2008
MathSciNet review: 2390516
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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 $ \Vert T^{n}\Vert=O(n^p)$, $ p\geq1$, $ \Vert T^{-n}\Vert=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 $ \Vert T^{-n}\Vert=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.


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Additional 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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09273-3
PII: S 0002-9939(08)09273-3
Keywords: Interpolating set, H\"older classes, growth of the norms
Received by editor(s): February 26, 2007
Published electronically: March 11, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society