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

Abstract

Suppose $T$ and $S$ are Moore-Penrose invertible operators between<br />Hilbert C*-module. Some necessary and sufficient conditions are given for the<br />reverse order law $(TS)^{ dag} =S^{ dag} T^{ dag}$ to hold.<br />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.