Characterization of the monotonicity by the inequality

TL;DR

This paper characterizes matrix and operator monotonicity using a generalized Powers-Stormer inequality involving normal states, positive functions, and operator inequalities on Hilbert space operators.

Contribution

It introduces new characterizations of monotonicity through a generalized inequality involving functions of operators, extending classical results.

Findings

Provides conditions for matrix and operator monotonicity

Establishes a generalized Powers-Stormer inequality

Links monotonicity to operator inequalities on Hilbert spaces

Abstract

Let $\varphi$ be a normal state on the algebra $B(H)$ of all bounded operators on a Hilbert space $H$, $f$ a strictly positive, continuous function on $(0, \infty)$, and let $g$ be a function on $(0, \infty)$ defined by $g(t) = \frac{t}{f(t)}$. We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\ormer inequality: $$ \varphi(A + B) - \varphi(|A - B|) \leq 2\varphi(f(A)^1/2g(B)f(A)^1/2), $$ whenever $A, B$ are positive invertible operators in $B(H).$