The Resolvent Mean and The Parametrized $\mathcal{A} \sharp \mathcal{B}$

TL;DR

This paper extends the concepts of resolvent average and weighted geometric mean from positive definite matrices to accretive matrices, establishing new inequalities and generalizations within this broader framework.

Contribution

It introduces the extension of resolvent average and weighted geometric mean to accretive matrices, generalizing known results from positive definite matrices.

Findings

Established a new inequality relating resolvent average and geometric mean for matrices.

Generalized properties of resolvent average to accretive matrices.

Provided a new relation between resolvent average and geometric mean.

Abstract

Resolvent average and weighted \(\mathcal{A}\sharp \mathcal{H}\)-mean have been defined recently for positive definite matrices. Since the class of accretive matrices provides a general framework for addressing certain known results on positive matrices, this paper extends the notions of resolvent average and the weighted \(\mathcal{A}\sharp \mathcal{H}\)-mean to accretive matrices and discusses some of their properties.\\ The obtained results happen to be legitimate generalizations of those known results on positive definite matrices.\\ Among many results, we show that if $A,B$ are positive definite matrices, and $0\leq\lambda\leq 1, \mu>0$, then \[\mathcal{R}_{\mu}(A,B,1-\lambda,\lambda)+\mu I \geq C \Big(A \sharp_{\bm{\lambda}} B +\mu I\Big),\] where $\mathcal{R}_{\lambda}$ is the resolvent average, $\sharp_{\bm{\lambda}}$ is the weighted geometric mean and $I$ is the identity matrix, for some positive constant $C$; as a new relation between the resolvent average and the geometric mean.