The Furuta Inequality and an Operator Equation for Linear Operators

Abstract

We show that a special form of the Futura inequality is equivalent to an operator equation $H^{\frac{p-2rn}{2(n+1)}}T(H^{\frac{p+2r}{n_{+}1}}T{)}^n{H}^{\frac{p-2_{r}n}{2(n+1)}}=K^p$. This result also generalizes Lemma 1 in [3] which is about the operator equation $T(H^{1/n}T{)}^n=K$. A new characterization of the Löwner–Heinz formula and some applications are given.