International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 22, Pages 1421-1431
Remarks on embeddable semigroups in groups and a generalization
of some Cuthbert's results
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.
Let be a -semigroup of bounded linear operators on a Banach space . In this paper, we establish that if, for some , is a Fredholm (resp., semi-Fredholm) operator, then is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary
and sufficient condition guaranteeing that can be imbedded in a -group on . Also we study semigroups
which are near the identity in the sense that there exists
such that , where is an arbitrary closed two-sided ideal contained in the set
of Fredholm perturbations. We close this paper by discussing the
case where is replaced by some subsets of the set
of polynomially compact perturbations.