# Generalized inverses with applications

#### by Alegra Dajic

Institution: The University of Melbourne
Date: Fri 7th March 2008
Time: 2:15 PM
Location: Room 213, Richard Berry Building, The University of Melbourne

Abstract: The seminar will give an introduction to the generalized inverses and look at some recent
developments in the theory of generalized inverses.
The generalized Drazin inverse has numerous application (for example in differential and difference equations,
Markov chains and solutions of systems of linear equations).
I will introduce the weighted generalized Drazin inverse for operators (an extension of the definition of weighted Drazin inverse) and give some results concerning this inverse.
An element $a$ of a Banach algebra is generalized Drazin invertible if and only if zero is either a point of the resolvent set of $a$ or zero is an isolated spectral point.
If $a$ has an isolated spectral set (not necessarily zero), $a$ is said to be ($\sigma$) generalized Drazin invertible. I will present some results on the properties of this inverse.
Finally, I will look at applications of the generalized inverses to the problem of
solutions of some ring and C*-algebra equations.

The results presented in this talk will be from joint work with my supervisor J. Koliha.