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

‎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 $(TS)^{ \dag} =S^{ \dag} T^{ \dag}$ to hold‎. ‎In particular‎, ‎we show that the equality holds‎ ‎if and only if $Ran(T^{\ast}TS) \subseteq Ran(S)$ and‎ ‎$Ran(SS^{\ast}T^{\ast}) \subseteq Ran(T^{\ast}),$ which was studied first by Greville‎ ‎[{\it SIAM Rev‎. ‎8 (1966) 518--521}] for matrices.