On the weighted contraharmonic means

TL;DR

This paper introduces a new weighted contraharmonic mean for positive definite elements in a unital C*-algebra, providing a characterization and exploring its properties.

Contribution

It defines the $ u$-weighted contraharmonic mean in a C*-algebra setting and derives a variational characterization for it.

Findings

Provides a new definition of the weighted contraharmonic mean in C*-algebras.

Establishes a variational formula involving a maximization over elements summing to the identity.

Explores properties and potential applications of this new mean.

Abstract

Let $\mathscr{A}$ be a unital $C^*$-algebra with unit $e$ and let $\nu\in(0, 1)$. We introduce the concept of the $\nu$-weighted contraharmonic of two positive definite elements $a$ and $b$ of $\mathscr{A}$ by \begin{align*} {C}_{\nu}(a, b):= (1-\nu)\nu^{-1}b + \nu (1-\nu)^{-1}a - \left((1-\nu)a^{-1}+\nu b^{-1}\right)^{-1}. \end{align*} We show that \begin{align*} {C}_{\nu}(a, b)= \displaystyle{\max_{x+y=e}}\left\{(1-\nu)^{-1}\left(\nu a - x^*ax\right) + \nu^{-1}\left((1-\nu)b - y^*by\right)\right\}, \end{align*} and then apply it to present some properties of this weighted mean.