The Banach algebra generated by a $ {C}_{0}$-semigroup

Mustafayev, Heybetkulu

doi:10.1016/j.crma.2006.02.017

Functional Analysis

The Banach algebra generated by a

C_{0}

-semigroup
[L'algebre de Banach engendrée par un

C_{0}

-semigroupe]

Présenté par : Gilles Pisier

Mustafayev, Heybetkulu ¹

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

Comptes Rendus. Mathématique, Tome 342 (2006) no. 8, pp. 575-578.

Résumé
Abstract

Soit $T = {T (t)}_{t ⩾ 0}$ un $C_{0}$ -semigroupe borné dans un espace de Banach par générateur A. Nous définissons $A_{T}$ comme la clotûre par rapport à la topologie de la norme opérateur de l'ensemble ${\hat{f} (T) : f \in L^{1} (R_{+})}$ , où $\hat{f} (T) = \int_{0}^{\infty} f (t) T (t) d t$ est la transformée de Laplace de $f \in L^{1} (R_{+})$ par rapport au semigroupe T. Alors $A_{T}$ est une algèbre de Banach commutative. Dans cet article il est montré que, si la spectre unitaire $σ (A) \cap i R$ de A est au plus dénombrable, alors la transformée de Gelfand de $S \in A_{T}$ s'annule sur $σ (A) \cap i R$ si et seulement si $\lim_{t \to \infty} ‖ T (t) S ‖ = 0$ . Nous donnons aussi quelques applications de la semisimplicité du problème.

Let $T = {T (t)}_{t ⩾ 0}$ be a bounded $C_{0}$ -semigroup on a Banach space with generator A. We define $A_{T}$ as the closure with respect to the operator-norm topology of the set ${\hat{f} (T) : f \in L^{1} (R_{+})}$ , where $\hat{f} (T) = \int_{0}^{\infty} f (t) T (t) d t$ is the Laplace transform of $f \in L^{1} (R_{+})$ with respect to the semigroup T. Then $A_{T}$ is a commutative Banach algebra. It is shown that if the unitary spectrum $σ (A) \cap i R$ of A is at most countable, then the Gelfand transform of $S \in A_{T}$ vanishes on $σ (A) \cap i R$ if and only if, $\lim_{t \to \infty} ‖ T (t) S ‖ = 0$ . Some applications to the semisimplicity problem are given.

Reçu le : 2005-11-10
Accepté le : 2006-02-07
Publié le : 2006-03-09

DOI : 10.1016/j.crma.2006.02.017

Affiliations des auteurs :

Mustafayev, Heybetkulu ¹

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

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     title = {The {Banach} algebra generated by a $ {C}_{0}$-semigroup},
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Mustafayev, Heybetkulu. The Banach algebra generated by a $ {C}_{0}$-semigroup. Comptes Rendus. Mathématique, Tome 342 (2006) no. 8, pp. 575-578. doi : 10.1016/j.crma.2006.02.017. http://www.numdam.org/articles/10.1016/j.crma.2006.02.017/

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