A note on majorization and range inclusion of adjointable operators on Hilbert C⁎-modules

The famous Douglas theorem contains four equivalent conditions on the solvability of the operator equation AX=B associated to Hilbert spaces, which was generalized to the Hilbert C⁎-module case by Fang et al. (2009) [5]. In this paper we prove that in the Hilbert C⁎-module case, the majorization condition and the range inclusion condition are in fact not equivalent.