The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules

Document Type: Research Paper

Authors

1 Department of Mathematics, Shahrood University of Technology, and, Mathematisches Institut, Fachbereich Mathematik und Informatik der Universitat Munster, Einsteinstrasse 62, 48149 Munster, Germany.

2 Department of‎ ‎Mathematics‎, ‎International Campus of Ferdowsi University‎, ‎Mashhad‎, ‎Iran.

Abstract

Suppose $T$ and $S$ are Moore-Penrose invertible operators between
Hilbert C*-module. Some necessary and sufficient conditions are given for the
reverse order law $(TS)^{ dag} =S^{ dag} T^{ dag}$ to hold.
In particular, we show that the equality holds if and only if $Ran(T^{*}TS) subseteq Ran(S)$ and $Ran(SS^{*}T^{*}) subseteq Ran(T^{*}),$ which was studied first by Greville [{it SIAM Rev. 8 (1966) 518--521}] for matrices.

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