Essential Spectra on Spaces with the Dunford-Pettis Property

Abstract

The purpose of this paper is to investigate the invariance, under weakly compact perturbations, of various essential spectrums of closed, densely defined linear operators acting on Banach spaces which possess the Dunford-Pettis property. Both bounded and unbounded perturbations are considered. ᮊ

Relatively compact-like perturbations, essential spectra and application

“…The extension consists principally in the possibility of considering the class of A-bounded operators which, regarded as operators in ^£{X A , X), are contained in one of the sets A&+(X), A&AX), Ajr+(X) D A^"_(X) or A J 2 " (X). Accordingly, using the same strategy as in [17], we find conditions which generalize previous ones discussed in [15,16]. In contrast to the proofs of the results obtained in [15] and [16], which use the geometric properties of Banach spaces considered, our analysis applies to all Banach spaces regardless of their specific properties and to a wide family of operators including, in particular, the sets AX{X), AW{X), A5^{X) and A Cy{X).…”

“…Among the works in this direction we quote, for example, [10,15,16,17,22,24,30] (see also the references therein). This work is a continuation of [17], where we can find a detailed treatment of the behaviour of essential spectra of such operators subjected to additive perturbations belonging to arbitrary closed two-sided ideals of -if(X) contained in the set of Riesz operators (see [17, page 281]).…”

“…This work is a continuation of [17], where we can find a detailed treatment of the behaviour of essential spectra of such operators subjected to additive perturbations belonging to arbitrary closed two-sided ideals of -if(X) contained in the set of Riesz operators (see [17, page 281]). It is inspired by the work published in [15] and [16], where A-weakly compact and A-strictly singular [5] Relatively compact-like perturbations 77…”

“…Our main objective here is to extend the results obtained in [15,Section 4] and [16, Section 2] to arbitrary Banach spaces and to fit them into a more general framework. The extension consists principally in the possibility of considering the class of A-bounded operators which, regarded as operators in ^£{X A , X), are contained in one of the sets A&+(X), A&AX), Ajr+(X) D A^"_(X) or A J 2 " (X).…”

“…On the other hand, the assumption r n (J (k -A)"') < 1 implies that A. e p(A + J) and In the next theorem we will give a sharper form of (2.2) which extends it to A-bounded perturbations contained in A&(X). To do so, we will assume that (2.3) [16] (respectively [15]) it is proved that in the case when X is an L p space (respectively has the property DP), the definition of a e5 (•) can be stated in terms of A -strictly singular (respectively A-weakly compact) perturbations. Since Ay(L p ) c A&{L P ) (1 < p < oo) and AW{X) c A&{X), if X has the property DP, then these two results are particular cases of our theorem.…”