Additive property of pseudo Drazin inverse of elements in Banach algebras

Abstract

We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. If ab = ba and a, b are pseudo Drazin invertible, we prove that a + b is pseudo Drazin invertible if and only if 1 + a ‡ b is pseudo Drazin invertible. Moreover, the formula of (a + b) ‡ is presented . When the commutative condition is weaken to ab = λba (λ = 0),1 Any element b ∈ A satisfying the conditions above is called a p-Drazin inverse of a, denoted by a ‡ . The set of all p-Drazin invertible elements of A is denoted by A pD .By a Π = 1 − aa ‡ we mean the strongly spectral idempotent of a. It is well known that J(A ) ⊂ A qnil and a ∈ A qnil if and only if a n 1 n → 0 (n → ∞) in a Banach algebra A . Wei and Deng [17] presented some additive results on the Drazin inverse for square complex matrices that commute. Zhuang, Chen et al. [18] extended the results in [17] to

On the weighted pseudo Drazin invertible elements in associative rings and Banach algebras

“…In following theorem ii) and iii) were presented in a Banach algebra in [13]. Here we prove that it is still true in an associative ring.…”

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].…”

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

“…In 2014, Zhu, Chen and Patrício [19] obtained a result about the p-Drazin inverse of a + b under the conditions a 2 b = aba and b 2 a = bab which are weaker than ab = ba in Banach algebras. More results on (generalized, pseudo) Drazin inverse can be found in [1,5,7,9,15,18].…”