On the pseudo drazin inverse of the sum of two elements in a Banach algebra

Abstract

In this paper, some additive properties of the pseudo Drazin inverse are obtained in a Banach algebra. In addition, we find some new conditions under which the pseudo Drazin inverse of the sum a + b can be explicitly expressed in terms of a, az, b, bz. In particular, necessary and sufficient conditions for the existence as well as the expression for the pseudo Drazin inverse of the sum a+b are obtained under certain conditions. Also, a result of Wang and Chen [Pseudo Drazin inverses in associative rings and Banach algebras, LAA 437(2012) 1332-1345] is extended.

On Jacobson's lemma and Cline's formula for Drazin inverses

“…Recall that, by [9,Corollary 4.2], if a ∈ J(R) and b ∈ R, then ab, ba ∈ J(R). For some interesting properties of the pseudo Drazin inverse see [21,23].…”

On Strongly pi-Regular Rings with Involution

“…According to [24], a ∈ R is p-Drazin invertible exactly when a is a pseudo-polar element (i.e., there exists p 2 = p ∈ comm 2 (a) such that a+p ∈ U(R) and ap ∈ J(R)). For more interesting properties of p-Drazin inverses, one may refer to [11,19,25,26] and the reference therewith.…”

Certain strongly clean matrices over local rings

“…In a ring R , evidently, { elements having pseudo Drazin inverses } ⊆ { quasipolar elements } ⊆ { strongly clean elements }. The subjects of strongly clean rings, quasipolar rings, and pseudo Drazin inverses were extensively studied in [1][2][3][4][5]7] and [10][11][12].…”