Relatively Strictly Singular Perturbations, Essential Spectra, and Application to Transport Operators

Abstract

The stability of essential spectra of a closed, densely defined linear operator A on L -spaces, 1 F p F ϱ, when A is subjected to a perturbation by a bounded p strictly singular operator was discussed in a previous paper by K. Latrach and A.Ž . Jeribi 1998, J. Math. Anal. Appl. 225, 461᎐485 . In the present paper we prove the invariance of the Gustafson᎐Weidmann, Wolf, Schechter, and Browder essential Ž . spectra of A under relatively strictly singular not necessarily bounded perturbations on these spaces. Further, a precise characterization of the Schechter essential Ž . spectrum is given. We show that these results are also valid on C ⌶ where ⌶ is a compact Hausdorff space. The results are applied to the one-dimensional transport equations with anisotropic scattering and abstract boundary conditions. ᮊ 2000 Academic Press

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…”

“…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.…”

“…If X has the property DP and J e AW(X), then Lemma 3.1 (i) and (iii) hold true (see [15]). Recently, it is proved that if X is an L p space (1 < p < oo) and J e AJT'(X), then statements (i) and (ii) are valid (see [16]). Note that in our case, Vladimirskii's result (see [27, Corollary 1]) writes ACy(X) c Ai f _(A r ), and therefore (i) and (iii) hold true for A-strictly cosingular perturbations.…”