Nebojša Č. Dinčić

Core inverse and core partial order of Hilbert space operators

Abstract The core inverse of matrix is generalized inverse which is in some sense in-between the group and Moore–Penrose inverse. In this paper a generalization of core inverse and core partial order to Hilbert space operator case is presented. Some properties are generalized and some new ones are proved. Connections with other generalized inverses are obtained. The useful matrix representations of operator and its core inverse are given. It is shown that A is less than B under the core partial order if and only if they have specific kind of simultaneous diagonalization induced by appropriate decompositions of Hilbert space. The relation is also characterized by the inclusion of appropriate sets of generalized inverses. The spectral properties of core inverse are also obtained.

Hartwig’s triple reverse order law revisited

We extend the classical Hartwig’s triple reverse-order law for the Moore–Penrose inverse to closed range bounded linear operators on infinite dimensional Hilbert spaces.

Identities concerning the reverse order law for the Moore-Penrose inverse

We prove some identities related to the reverse order law for the Moore-Penrose inverse of operators on Hilbert spaces, extending some results from (Y. Tian, S. Cheng, Linear Multilinear Algebra 52 (2004)) and (R. E. Cline, SIAM Rev., Vol. 6, No. 1 (1964)) to infinite dimensional settings.