Operator equations $AX+YB=C$ and $AXA^*+BYB^*=C$ in Hilbert $C^*$-modules

• DOI: 10.48550/arxiv.1612.03857
• Publication Date: 2016-01-01
• Authors: Zahrasadat Mousavi; Rasoul Eskandari; Mohammad Sal Mosl...; F. Mirzapour
• Keywords: Operator (biology)

Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving suitable version of the celebrated Douglas theorem in context $C^*$-modules, we present general solution equation $AX+YB=C$ when ranges are not necessarily closed. We examine result Fillmore Williams setting $C^*$-modules. Moreover, obtain some necessary sufficient conditions for existence $AXA^*+BYB^*=C$. Finally, deduce that there exist nonzero $X, Y\geq 0$ $Z$ such $AXA^*+BYB^*=CZ$, $A, B$ given subject to conditions.