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.