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Suppose $T$ and $S$ are bounded adjointable operators between HilbertC*-modules admitting bounded Moore-Penrose inverse operators. Some necessaryand sufficient conditions are given for the reverse order law $(TS)^{ \dag}=S^{ \dag} T^{ \dag}$ to hold. In particular, we show that the equality holdsif 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.

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