The Banach algebra generated by a C-0-semigroup


Mustafayev H.

COMPTES RENDUS MATHEMATIQUE, vol.342, no.8, pp.575-578, 2006 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 342 Issue: 8
  • Publication Date: 2006
  • Doi Number: 10.1016/j.crma.2006.02.017
  • Journal Name: COMPTES RENDUS MATHEMATIQUE
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.575-578

Abstract

Let T = {T(t)}(t >= 0) be a bounded Co-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 {f(T): f epsilon L-1(R+)}, where f (T) = integral(infinity)(0) f(t)T(t) dt is the Laplace transform of f epsilon L-1 (R+) with respect to the semigroup T. Then A(T) is a commutative Banach algebra. It is shown that if the unitary spectrurn sigma(A)boolean AND iR of A is at most countable, then the Gelfand transform of S epsilon A(T) vanishes on sigma(A)boolean AND iR if and only if, lim(t ->infinity) parallel to T(t)S parallel to = 0. Some applications to the semisimplicity problem are given.