The resolvent average on symmetric cones
Let \(V\) be an Euclidean Jordan algebra with identity \(e\), let \(\Omega \) be the interior of the set of all square elements of \(V\), let \(\Delta _{m}\) be the set of all probability vectors in \(\mathbb{R}^{m}\) with strictly positive components, and let \(\mu \geq 0\). The authors study the resolvent average of \(\mathbf{a}=(a_{1},\dots,a_{m})\in \Omega ^{m}\) with weight vector \(\omega =(w_{1},\dots,w_{m})\in \Delta _{m}\), defined by \(R_{\mu }(\omega ;\mathbf{a})=[\sum_{i=1}^{m}w_{i}(a_{i}+\mu e)^{-1}]^{-1}-\mu e\); for \(\mu =\infty\), one sets \(R_{\infty }(\omega ;\mathbf{a})=\sum_{i=1}^{m}w_{i}a_{i}\). This notion extends the corresponding one for positive semidefinite matrices introduced by \textit{H. H. Bauschke, S. M. Moffat} and \textit{X. Wang} [Linear Algebra Appl. 432, No. 7, 1757--1771 (2010; Zbl 1191.15024)].
scientific article; zbMATH DE number 6125855