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

Abstract

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

9 pages, accepted for publication

References in corpus (3)

• Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^{\ast }$-modules
• The product of operators with closed range in Hilbert C*-modules
• Moore-Penrose inverse of Gram operator on Hilbert C*-modules