Invariance of the Schechter essential spectrum under polynomially compact operators perturbation (original) (raw)
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Measures of weak noncompactness have been successfully applied in topology, functional analysis and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We work with the notion of the measures of weak noncompactness in order to establish some results concerning the class of semi-Fredholm and Fredholm operators. Further, we apply the obtained results to prove, under some conditions on the perturbed operator, the invariance of the Schechter essential spectrum on Banach spaces. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator. Keywords Measures of weak noncompactness in Banach spaces • Fredholm operators • Schechter essential spectrum • Transport theory Mathematics Subject Classification 54B35 • 47A53 • 82D75 • 47H09 1 Introduction Let (X, .) be a complex Banach space. The open ball of X will be denoted by B X and its closure by B X. We denote by C(X) (resp. L(X)) the set of all closed densely
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This paper is devoted to the investigation of the stability of the Schechter essential spectrum of closed densely defined linear operators A subjected to additive perturbations K such that (λ − A − K) −1 K or K(λ − A − K) −1 belonging to arbitrary subsets of L(X) (where X denotes a Banach spaces) contained in the set J (X). Our approach consists principally in considering the class of A-closable (not necessarily bounded) which contained in the set of Aresolvent Fredholm perturbations which zero index (see Definition 3.5). They are used to describe the Schechter essential spectrum of singular neutron transport equations in bounded geometries. Definition 1.1. An operator A ∈ L(X, Y) is said to be weakly compact if A(B) is relatively weakly compact in Y for every bounded subset B ⊂ X.
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The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish a fine description of the Schechter essential spectrum of closed densely defined operators. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.
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