Browder-type Theorems and SVEP

Abstract

We study the new properties (b) and (gb), which we had introduced in [13], for an operator having the SVEP on the complement of distinguished parts of its spectrum. Classes of operators are considered as illustrating examples. Mathematics Subject Classification (2010). 47A53, 47A10, 47A11. Keywords. property (b), property (gb), B-Fredholm operator, generalized Browder's theorem, generalized a-Browder's theorem, single valued extension property.

Property (gb) through local spectral theory

“…Property (b) for bounded linear operators on Banach spaces, has been introduced by Berkani and Zariuoh ([20] and [21]), and this property may be thought, in a sense, as stronger versions of the classical a-Browder's theorem. In this paper we consider a stronger variant of property (b), the so-called property (gb), also introduced in [20] and [21], and studied in the more recent papers [19] and [10]. In particular, we show that this property may be characterized by means of typical tools of local spectral theory.…”

Spectral properties and restrictions of bounded linear operators

“…In the next definition, we describe several spectral properties introduced recently (see [8], [9], [14], [18], [19], [20] and [21]). Definition 1.7.…”

Propertiesandfor Bounded Linear Operators

“…ese properties, that we call property ( , means that the set of all poles of the resolvent of of �nite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 (see �e�nition 3) and we call property ( , means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0 (see �e�nition 3). Properties ( and ( are related to a strong variants of classical Weyl's theorem, the so-called property ( and property ( introduced by Berkani and Zariouh [13] and more extensively studied in recent papers [12,14,20,21]. We shall characterize properties ( and ( in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems.…”