International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 22, Pages 1421-1431
doi:10.1155/S0161171203011839

Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results

Khalid Latrach1 and Abdelkader Dehici1

1Département de Mathématiques, Université de Corse, Corte 20250, France
2Département des Sciences Exactes (Branche Mathématiques), Université Du 8 Mai 1945, BP 401, Guelma 24000, Algeria

Received 5 February 2001; Revised 27 July 2001

Copyright © 2003 Khalid Latrach and Abdelkader Dehici. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0>0, U(t0) is a Fredholm (resp., semi-Fredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be imbedded in a C0-group on X. Also we study semigroups which are near the identity in the sense that there exists t0>0 such that U(t0)−I∈𝒥(X), where 𝒥(X) is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where 𝒥(X) is replaced by some subsets of the set of polynomially compact perturbations.