Properties (B)‎ and (gB)‎ for Bounded Linear Operators

Abstract EN

We shall consider properties which are related to Weyl type theorem for bounded linear operators T∈ℒ(?), defined on a complex Banach space ?.

These properties, that we call property (B), means that the set of all poles of the resolvent of T of finite rank in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-Fredholm with index less than or equal to 0 and we call property (gB), means that the set of all poles of the resolvent of T in the usual spectrum are exactly those points λ of the spectrum for which T-λ is an upper semi-B-Fredholm with index less than or equal to 0.

Properties (B) and (gB) are related to a strong variants of classical Weyl’s theorem, the so-called property (b) and property (gb) We shall characterize properties (B) and (gB) in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems.

Our main tool is localized version of the single valued extension property.

Also, we consider the properties (B) and (gB) in the frame of polaroid type operators.