  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.
