A Gr\"uss type operator inequality

TL;DR

This paper refines a Gr"uss type operator inequality involving the trace functional and provides a positive answer to an open problem posed by Renaud, advancing the theoretical understanding of operator inequalities.

Contribution

The paper offers a refined version of Renaud's operator inequality and resolves his open problem, contributing new insights to matrix and operator inequality theory.

Findings

Refined Gr"uss type operator inequality established

Positive resolution of Renaud's open problem

Enhanced understanding of trace functional inequalities

Abstract

In [P. Renaud, "A matrix formulation of Gr\"uss inequality", Linear Algebra Appl. 335 (2001), 95--100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we answer positively the Renaud's open problem.