Refined operator inequalities for relative operator entropies

TL;DR

This paper extends the study of relative operator entropies to broader algebraic settings and improves existing bounds, enhancing the theoretical understanding of operator inequalities.

Contribution

It generalizes operator entropy inequalities to C*-algebras, real C*-algebras, and JC-algebras, and refines bounds for the relative operator $(eta, eta)$-entropy.

Findings

Operator inequalities hold in broader algebraic settings.

Improved lower and upper bounds for relative operator $(eta, eta)$-entropy.

Extensions of previous bounds by Fujii, Kamei, and Nikoufar.

Abstract

In this paper, we investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on relative operator entropies still hold in these broader settings. In addition, we improve the lower and upper bounds of the relative operator $(\alpha, \beta)$-entropy established by Nikoufar which refined the bounds for the relative operator entropy obtained by Fujii and Kamei.