Comptes Rendus
no. 8
Functional Analysis
The Banach algebra generated by a C0-semigroup
[L'algebre de Banach engendrée par un C0-semigroupe]

Présenté par : Gilles Pisier

Heybetkulu Mustafayev1
1 Yuzuncu Yil University, Faculty of Arts and Sciences, Department of Mathematics, 65080 Van, Turkey
Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 575-578.

Soit T={T(t)}t⩾0 un C0-semigroupe borné dans un espace de Banach par générateur A. Nous définissons AT comme la clotûre par rapport à la topologie de la norme opérateur de l'ensemble {fˆ(T):f∈L1(R+)}, où fˆ(T)=∫0∞f(t)T(t)dt est la transformée de Laplace de f∈L1(R+) par rapport au semigroupe T. Alors AT est une algèbre de Banach commutative. Dans cet article il est montré que, si la spectre unitaire σ(A)∩iR de A est au plus dénombrable, alors la transformée de Gelfand de S∈AT s'annule sur σ(A)∩iR si et seulement si limt→∞‖T(t)S‖=0. Nous donnons aussi quelques applications de la semisimplicité du problème.

Let T={T(t)}t⩾0 be a bounded C0-semigroup on a Banach space with generator A. We define AT as the closure with respect to the operator-norm topology of the set {fˆ(T):f∈L1(R+)}, where fˆ(T)=∫0∞f(t)T(t)dt is the Laplace transform of f∈L1(R+) with respect to the semigroup T. Then AT is a commutative Banach algebra. It is shown that if the unitary spectrum σ(A)∩iR of A is at most countable, then the Gelfand transform of S∈AT vanishes on σ(A)∩iR if and only if, limt→∞‖T(t)S‖=0. Some applications to the semisimplicity problem are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.02.017

Heybetkulu Mustafayev 1

1 Yuzuncu Yil University, Faculty of Arts and Sciences, Department of Mathematics, 65080 Van, Turkey 
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Heybetkulu Mustafayev. The Banach algebra generated by a $ {C}_{0}$-semigroup. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 575-578. doi : 10.1016/j.crma.2006.02.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.017/

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