On partial orders of Hilbert space operators

Abstract

First, an overview of partial orders defined on bounded linear operators on an infinite-dimensional Hilbert space is presented. A definition for the core inverse of operators on a Hilbert space is proposed. Extensions of the sharp and the core partial orders are considered. An explicit formula for the core inverse of matrices is obtained using a full-rank factorization. Relationships between all these partial orders and formula for generalized inverses of differences of operators, when they are related with respect to these partial orders, are investigated.

Keywords:

• Moore-Penrose inverse
• group inverse
• core inverse
• partial orders
• full-rank factorization
• generalized inverse of differences

AMS Subject Classifications:

• 47A05
• 06A06
• 15A09

Acknowledgements

The first author thanks Council of Scientific and Industrial Research (CSIR), India, for the financial support during the course of this work. The authors are grateful to the referee and the handling editor for their constructive remarks which led to an improved presentation. Part of the work was done while the second author visited Jan Hauke of the Adam Mickiewicz University, Poland.