Some additive results on Drazin inverses

Abstract

In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented.

An extension of Cline''s formula for a generalized Drazin inverse

“…It plays an important role in finding the Drazin inverse of a sum of two elements and that of a block matrix (see [8]). Generalizations of Cline's formula for generalized Drazin inverse and pseudo Drazin inverse were recently proved in [7] and [9], respectively.…”

On the Drazin inverses involving power commutativity

“…The power commutative relations between operators play an important role in the investigation of Drazin inverses. Some properties of the Drazin inverse according to such relations have been extensively studied in the mathematical literature (see, e.g., [3,5,6,8,11,14]). …”

“…In [14], P. Patrício and R.E. Hartwig investigated the existence of the Drazin inverse (P + Q ) D under the conditions:…”

The (2,2,0) Drazin inverse problem

“…We cannot expect a single "good formula" without additional the D-inverse depends on the "invertibility" of A (being a unit, group member or neither) as well as the interaction of A with e 1 and e 2 . Numerous recent papers have dealt with special cases [2,13] and special approaches, such as the use of Catalan Numbers and their recurrence relations [3,4]. Finally, we shall use rk(·) to denote rank.…”