On majorization and range inclusion of operators on Hilbert $C^*$-modules

TL;DR

This paper investigates the relationships between majorization conditions and range inclusion for adjointable operators on Hilbert $C^*$-modules, establishing equivalences without certain assumptions and exploring solvability of operator equations.

Contribution

It proves the equivalence of majorization conditions without assumptions on the closure of the range and characterizes the solvability of operator equations in this context.

Findings

Majorization conditions are equivalent without assumptions on $ar{ ext{Range}}(A^*)$.

Existence of an operator $B$ with range contained in $A$ when $ar{ ext{Range}}(A^*)$ is not orthogonally complemented.

The operator equation $AX=B$ can be unsolvable despite range inclusion.

Abstract

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.