On High-power Operator Inequalities and Spectral Radii of Operators

Abstract

For some different types of operators on a Hilbert space, we present new highpower 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.

Spectral radius inequalities for functions of operators defined by power series

“…In addition to the inequality (1.1), the most important properties of the spectral radius are the spectral radius formula properties hold. For additional properties of the spectral radius, the reader is referred to the classical book [3] and the papers [5]- [14].…”

“…For additional properties of the spectral radius, the reader is referred to the classical book [3] and the papers [5]- [14].…”

Generalizations of Reid inequality

“…The proof, which uses a standard argument (cf. [2] or [6]), is more technical in the unbounded case. Theorem 3.5.…”

On Some Inequalities for the Generalized Euclidean Operator Radius

“…The equality in (2.6) holds iff the vectors BAx and C * D * y are linearly dependent in H . For other closely related version of Kato's inequality see [2], [14], [15], [18], [19] and [22].…”