The Furuta inequality and an operator equation for linear operators (Q1974550)

The following theorem has been proved: Let \(H\) and \(K\) denote bounded linear operators on a Hilbert space, \(H\geq K\geq 0\) and assume that \(H\) is non-singular. For \(p,r\geq 0\) and an integer \(n\geq 0\) with \((1+ 2r)(n+ 1)\geq p+ 2r\), the following (i) \(H^{{p+ 2r\over n+1}}\geq (H^r K^pH^r)^{{1\over n+1}}\), (ii) There exists a unique operator \(T\geq 0\) with \(\|T\|\leq 1\) such that \[ K^P= H^{{p- 2rn\over 2(n+ 1)}} T(H^{{p+2r\over n+1}} T)^n H^{{P- 2rn\over 2(n+ 1)}}. \] The author gives a new characterization of the Löwner-Heinz formula and some applications.