New additive results for the g-Drazin inverse

Abstract

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

Representations for the Generalized Drazin Inverse of the Sum in a Banach Algebra and Its Application for Some Operator Matrices

“…First we start the following result which is proved in [8] for matrices, extended in [9] for a bounded linear operator and in [10] for arbitrary elements in a Banach algebra.…”

Weak core and central weak core inverse in a proper $*$-ring

“…The major strength of these classes is that it can be applied easily to C * -algebra (see Koliha et al [12] for the Moore-Penrose inverse). On the other hand, the problem of the sum of two generalized invertible elements in * -ring has generated a tremendous amount of interest in the algebraic structure of ring theory [16][17][18]. In this context, Moore [19] first discussed the invertible elements in a complex matrix ring.…”

“…In this context, Moore [19] first discussed the invertible elements in a complex matrix ring. Since then, many researchers studied the additive properties for various classes of generalized inverses in [16,[20][21][22]. In the paper, we derive an explicit expression for weak core and central weak core invertible element in proper * -ring.…”

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

“…. , p n } be a total system of idempotents in A if p 2 i = p i , for all i, p i p j = 0 if i = j, and p 1 + · · · + p n = 1, as in [22]. If a ∈ A d , then…”