The operator equation \(T(H^{1/n}T)^ n=K\) (Q1114913)

The author's summary: Let H and K be bounded positive operators on a Hilbert space, and assume that H is nonsingular. This paper shows that if there exists a positive operator T such that \(T(H^{1/n}T)^ n=K\) for some natural number n, then, for any natural number m such that \(m\leq n\) there exists a positive operator \(T_ 1\) such that \(T_ 1(H^{1/m}T_ 1)^ m=K\). In each case, there is at most one positive solution T and \(T_ 1\) respectively.