The operator equation $THT=K$

Abstract:

Let H and K be bounded positive operators on a Hilbert 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 ${({H^{1/2}}K{H^{1/2}})^{1/2}} \leqq aH$ for some $a > 0$, and then $\left \| T \right \| \leqq a$; (iii) this condition is satisfied if H is invertible or more generally if $K \leqq {a^2}H$ for some $a > 0$; (iv) an exact formula for T is given when H is invertible.