Properties (RB) and (gRB) for Bounded Linear Operators

Abstract

In this article, we consider pseudo invertible operators for study of the relationship between the space of relatively regular operators and some generalizations of Weyl and Browder theorems. By using the analysis and representation of pseudo invertible operators, some new properties in connection with Browder’s type theorems, were presented for bounded linear operators T ∈ B(X). These properties, which we refer to as property (RB), imply that All poles of the resolvent of T of finite rank in the typical spectrum are precisely those places of the spectrum for which a reasonably regular operator with its pseudo inverse operator is surjective. (γI – T)† ∈ SU(X), In the usual spectrum, the set of all poles of the resolvent is exactly those points of the spectrum for which we call property (gRB). γI – T is a B-relatively regular operator with its pseudo inverse operator is surjective (γI – T)† ∈ SU(X). In addition, several sufficient and necessary conditions for which properties (RB) and (gRB) hold are given.