Operator equations AX + YB = C and AXA⁎ + BYB⁎ = C in Hilbert C⁎-modules

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