It is proved that for adjointable operators $A$ and $B$ between Hilbert $C^*$-modules‎, ‎certain majorization conditions are always equivalent without any assumptions on‎ ‎$\overline{\mathcal{R}(A^*)}$‎, ‎where $A^*$ denotes the adjoint operator of $A$ and $\overline{\mathcal{R}(A^*)}$ is the norm closure of the range of $A^*$‎. ‎In the case that $\overline{{\mathcal R}(A^*)}$ is not orthogonally complemented‎, ‎it is proved that there always exists an adjointable operator $B$ whose range is contained in that of $A$‎, ‎whereas the associated equation‎ ‎$AX=B$ for adjointable operators is unsolvable‎.

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