Properties of one-sided generalized Drazin inverses in Banach algebras

We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a,b ? Ad. If ab = ?a?bab? for a nonzero complex number ?, then a + b ? Ad. The explicit representation of (a + b)d is presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.

In this paper, we investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a,b ? A.

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra \(\mathcal{A}\) under different conditions. Let \(x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}\) relative to the idempotent \(p\in \mathcal{A}\) and \(a\in p\mathcal{A}p\) be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement \(s=d-ca^db\) is generalized Drazin invertible, which covers the cases that \(s\) is Drazin invertible, \(s\) is group invertible, or \(s\) is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting.

Additive results for the generalized Drazin inverse in a Banach algebra

In this paper, we study a new class of generalized inverses in Banach algebras, which is termed extended pseudo Drazin inverse. We derive several characterizations and elementary properties of the extended pseudo Drazin inverse. Moreover, we obtain the generalized versions of Jacobson?s lemma for extended Drazin inverse, extended generalized Drazin inverse and extended pseudo Drazin inverse.

On the generalized Drazin inverse in Banach algebras in terms of the generalized Schur complement

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C *-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.

We establish some triple reverse order laws for the generalized Drazin inverse and then obtain from one of them various expressions of the generalized Drazin inverse of a block matrix in a Banach algebra.

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

Based on the conditions a b 2 = 0 and b π ( a b ) ∈ A d , we derive that ( a b ) n , ( b a ) n , and a b + b a are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A . By using these results, some results on the symmetry representations for the generalized Drazin inverse of a b + b a are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b .

In this paper, we introduce and investigate the weighted pseudo Drazin inverse of elements in associative rings and Banach algebras. Some equivalent conditions for the existence of the w-pseudo Drazin inverse of a + b are given. Using the Pierce decomposition, the representations for the w-pseudo Drazin inverse are given in Banach algebras.

Reverse order laws for the generalized Drazin inverse in Banach algebras

Inthis paper, we present an extended Jacobson?s lemma for g-Drazin inverse in Banach algebras. Let A be a Banach algebra, and let a, b, c, d ? A satisfying (ac)2 a = acdba = dbaca = (db)2 a; (ac)2 d = acdbd = dbacd = (db)2 d. Then 1-ac ? Ad if and only if 1-bd ? Ad. Related generalized Jacobson?s lemma for Drazin, core and p-core inverses in a Banach algebra are thereby obtained.

In this paper, we give expressions for the generalized Drazin inverse of a (2,2,0) block matrix over a Banach algebra under certain circumstances, utilizing which we derive the generalized Drazin inverse of a 2x2 block matrix in a Banach algebra under weaker restrictions. Our results generalize and unify several results in the literature.

In this article, we obtain new additive results on the generalized Drazin inverse of a sum of two elements in a Banach algebra. Applying these additive results, we also give explicit formulas for the generalized Drazin inverse of a block matrix in a Banach algebra.