Asymptotic behavior of polynomially bounded operators

Heybetkulu S. Mustafayev

doi:10.1016/j.crma.2010.04.003

Mathematical Analysis

Asymptotic behavior of polynomially bounded operators
[Comportement asymptotique des opérateurs polynomialement bornés]

Présenté par : Gilles Pisier

Heybetkulu S. Mustafayev ¹

¹ Yuzuncu Yıl University, Faculty of Arts and Sciences, Department of Mathematics, 65080, Van, Turkey

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 517-520.

Résumé
Abstract

Soit T un opérateur polynomialement borné sur un espace de Banach et soit $A_{T}$ la plus petite algèbre de Banach uniformement fermé contenant T et l'identité. Il est montré dans cet article que pour tout $S \in A_{T}$ ,

\lim_{n \to \infty} ‖ T^{n} S ‖ = \sup_{ξ \in σ_{u} (T)} | \hat{S} (ξ) |,

où

\hat{S}

est la transformée de Gelfand et

σ_{u} (T) : = σ (T) \cap Γ

est la spectre unitaire de T ;

Γ : = {z \in C : | z | = 1}

.

Let T be a polynomially bounded operator on a complex Banach space and let $A_{T}$ be the smallest uniformly closed (Banach) algebra that contains T and the identity operator. It is shown that for every $S \in A_{T}$ ,

\lim_{n \to \infty} ‖ T^{n} S ‖ = \sup_{ξ \in σ_{u} (T)} | \hat{S} (ξ) |,

where

\hat{S}

is the Gelfand transform of S and

σ_{u} (T) : = σ (T) \cap Γ

is the unitary spectrum of T;

Γ = {z \in C : | z | = 1}

.

Reçu le : 2010-01-26
Accepté le : 2010-04-08
Publié le : 2010-04-24

DOI : 10.1016/j.crma.2010.04.003

Affiliations des auteurs :

Heybetkulu S. Mustafayev ¹

¹ Yuzuncu Yıl University, Faculty of Arts and Sciences, Department of Mathematics, 65080, Van, Turkey

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     title = {Asymptotic behavior of polynomially bounded operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {517--520},
     publisher = {Elsevier},
     volume = {348},
     number = {9-10},
     year = {2010},
     doi = {10.1016/j.crma.2010.04.003},
     language = {en},
}

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Heybetkulu S. Mustafayev. Asymptotic behavior of polynomially bounded operators. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 517-520. doi : 10.1016/j.crma.2010.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.003/

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