Relatively compact-like perturbations, essential spectra and application

The purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. IfAdenotes a closed densely defined linear operator on a Banach spaceX, our approach consists principally in considering the class ofA-closable operators which, regarded as operators in ℒ(XA,X) (whereXAdenotes the domain ofAequipped with the graph norm), are contained in the set ofA-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

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

Let(U(t))t≥0be aC0-semigroup of bounded linear operators on a Banach spaceX. In this paper, we establish that if, for somet0>0,U(t0)is a Fredholm (resp., semi-Fredholm) operator, then(U(t))t≥0is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that(U(t))t≥0can be imbedded in aC0-group onX. Also we study semigroups which are near the identity in the sense that there existst0>0such thatU(t0)−I∈𝒥(X), where𝒥(X)is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where𝒥(X)is replaced by some subsets of the set of polynomially compact perturbations.

Semi-Fredholm perturbations and commutators

The Laffey–West theorem concerning finite rank perturbations of bounded Fredholm operators is extended to closed densely defined operators on Banach Spaces.