A note on the Jensen inequality for self-adjoint operators. II

TL;DR

This paper explores a specific order relation for positive operators on a Hilbert space, defined via Jensen's inequality with the square-root function, and shows it is antisymmetric for invertible operators.

Contribution

It introduces and analyzes an order-like relation based on Jensen's inequality for positive operators, extending previous work and establishing antisymmetry under invertibility.

Findings

The relation is antisymmetric for invertible positive operators.

The relation is defined using Jensen's inequality with the square-root function.

Extension of previous results on operator inequalities.

Abstract

This is a continuation of our previous paper. We consider a certain order-like relation for positive operators on a Hilbert space. This relation is defined by using the Jensen inequality with respect to the square-root function. We show that this relation is antisymmetric if the operators are invertible.