On majorization and range inclusion of operators on Hilbert <i>C</i>*-modules

Abstract

It is proved that for adjointable operators A and B between Hilbert -modules, certain majorization conditions are always equivalent without any assumptions on , where denotes the adjoint operator of A and is the norm closure of the range of . In the case that 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 for adjointable operators is unsolvable.