Generalized Drazin Invertibility Research Articles

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab = ba, we show that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a + b) D in terms of a, a D , b and b D , but also (1 + a D b) D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.

Generalized Drazin invertibility of combinations of idempotents

The paper serves as a correction to J. Math. Anal. Appl. 359 (2009) 731–738 dealing with the Drazin invertibility of combinations of idempotents p,q in a Banach algebra. As the arguments employed to do this work equally well for both the usual and generalized Drazin inverse, the latter is included in the discussion which covers the equivalence of the Drazin invertibility of 1-pq, p-q, p+q and in general of αp+βq with αβ≠0, as well as the equivalence of the Drazin invertibility of the commutator pq-qp and anticommutator pq+qp of p,q.

Some results on the generalized Drazin inverse of operator matrices

The generalized Drazin inverse M d of a 2 × 2 operator matrix is considered, where A ∈ ℬ(X) and D ∈ ℬ(Y) are generalized Drazin invertible. Expressions for the generalized Drazin inverse M d of operator matrix M in terms of the individual blocks A, B, C, D, A d and D d are derived under some conditions.