The operator equation $THT=K$
Gert Kjaergaard Pedersen, Masamichi Takesaki · Proceedings of the American Mathematical Society · 1972
Let H and K be bounded positive operators on a Hubert space, and assume that H is nonsingular.Then (i) there is at most one bounded positive operator T such that THT=K; (ii) a necessary and sufficient condition for the existence of such T is that (W-KHVtyi^aH for some a>0, and then ||r||á«; («') this condition is satisfied if H is invertible or more generally if K^arH for some a>0; (iv) an exact formula for T is given when H is invertible.