Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^*_3=C_3, A_4XA^*_4=C_4$ in Hilbert $C^*$-modules

Abstract

Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is given to show that the proposed investigation in the setting of Hilbert $C^*$-modules is different from that of Hilbert spaces.

Positive Solutions of Operator Equations AX = B, XC = D

“…Next, we consider the general representation of the positive solution X. Recall X has matrix form (7)…”

“…have been widely studied in matrix algebra [1], the operator space on the Hilbert space [2][3][4][5][6], and the adjointable operator space on Hilbert C * -modules [7][8][9][10]. There is a classical result addressing the existence of solutions for Equation (1), which is the famous Douglas range inclusion theorem [4].…”