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
Mustafayev, Heybetkulu1
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

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.

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.

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

Mustafayev, Heybetkulu 1

1 Yuzuncu Yil University, Faculty of Arts and Sciences, Department of Mathematics, 65080 Van, Turkey 
@article{CRMATH_2006__342_8_575_0,
     author = {Mustafayev, Heybetkulu},
     title = {The {Banach} algebra generated by a $ {C}_{0}$-semigroup},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {575--578},
     year = {2006},
     publisher = {Elsevier},
     volume = {342},
     number = {8},
     doi = {10.1016/j.crma.2006.02.017},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2006.02.017/}
}
TY  - JOUR
AU  - Mustafayev, Heybetkulu
TI  - The Banach algebra generated by a $ {C}_{0}$-semigroup
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 575
EP  - 578
VL  - 342
IS  - 8
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2006.02.017/
DO  - 10.1016/j.crma.2006.02.017
LA  - en
ID  - CRMATH_2006__342_8_575_0
ER  - 
%0 Journal Article
%A Mustafayev, Heybetkulu
%T The Banach algebra generated by a $ {C}_{0}$-semigroup
%J Comptes Rendus. Mathématique
%D 2006
%P 575-578
%V 342
%N 8
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2006.02.017/
%R 10.1016/j.crma.2006.02.017
%G en
%F CRMATH_2006__342_8_575_0
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

[1] Arveson, W. The harmonic analysis of automorphism groups of operator algebras, Proc. Sympos. Pure Math., Volume 38 (1982), pp. 199-269

[2] Evans, D.E. On the spectrum of a one-parameter strongly continuous representation, Math. Scand., Volume 39 (1976), pp. 80-82

[3] Katznelson, Y.; Tzafriri, L. On power bounded operators, J. Funct. Anal., Volume 68 (1986), pp. 313-328

[4] Loomis, L.H. The spectral characterization of a class of almost periodic functions, Ann. Math., Volume 72 (1960), pp. 362-368

[5] van Neerven, J.M.A.M. The Asymptotic Behavior of Semigroups of Linear Operators, Birkhäuser, 1996

[6] Phong, V.Q. Theorems of Katznelson–Tzafriri type for semigroups of operators, J. Funct. Anal., Volume 103 (1992), pp. 74-84

Cité par Sources :