On majorization and range inclusion of operators on Hilbert $C^{\ast }$-modules

It is proved that for adjointable operators $A$ and $B$ between Hilbert $C^{\ast }$-modules, certain majorization conditions are always equivalent without any assumptions on $\overline{\mathcal{R}(A^{\ast })}$, where $A^{\ast }$ denotes the adjoint operator of $A$ and $\overline{\mathcal{R}(A^{\ast })}$ is the norm closure of the range of $A^{\ast }$. In the case that $\overline{\mathcal{R}(A^{\ast })}$ 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.