On the Drazin index of regular elements

Abstract

Abstract:It is known that the existence of the group inverse # of a ring element is equivalent to the invertibility of 2 − + 1 − − , independently of the choice of the von Neumann inverse − of . In this paper, we relate the Drazin index of to the Drazin index of 2 − + 1 − − . We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks. MSC:15A09, 16A14, 16A30

Generalized Inverses of a Sum in Rings

“…The following theorem is an answer to a question raised by Patricio and Veloso in [8] about the equivalence between the conditions ind(a 2 a − + 1 − aa − ) = k and ind(a + 1 − aa − ) = k, and provides a new characterization of the Drazin index. THEOREM 3.1.…”

On Jacobson’s lemma and Drazin invertibility

“…4) In this note we wish, in answer to a question of Patricio and da Costa [3], to note that Jacobson's lemma extends to Drazin invertibility, preserving also the ''Drazin index''. We extend Jacobson's lemma to various abstract versions of ''Drazin invertibility''.…”

On a new generalized inverse for matrices of an arbitrary index

“…Some properties for all these generalized inverses can be found in [2,3,4,7,8]. All of these generalized inverses are known to be used in important applications.…”