Range Inclusion and Operator Equations

Let A and D be positive operators on a complex Hilbert space H. In this work we show that the operator equation lDXqX*AsA l)0.has a unique solution Cl satisfying certain conditions, if there exists a positive operator B such that ABq DyB.AG0. This implies several previously known results in special cases on this class of operator equations. Also, along with some definitions, a counterexample to a previous conjecture will be given