On High-power Operator Inequalities and Spectral Radii of Operators

Abstract

For some different types of operators on a Hilbert space, we present new high-power operator inequalities, and their corresponding operator inequalities involving spectral radii of operators. In particular, we show that Halmos’ two operator inequalities, Reid’s inequality and many others hold easily. In applications we obtain a new generalized classical Löwner inequality; and a slightly generalized Löwner–Heinz inequality is given.