Mixture and interpolation of the parameterized ordered means

TL;DR

This paper explores the properties and relationships of parameterized ordered means of positive invertible operators, focusing on their interpolation, mixture, and inequalities within the framework of the Loewner partial order.

Contribution

It introduces a new family of parameterized ordered means, compares their mixtures, and establishes interpolation relations, extending the understanding of operator means in metric topology.

Findings

Constructed a parameterized ordered mean similar to the resolvent average.

Compared mixtures of parameterized ordered means using Loewner order.

Established relations between families of parameterized means related to power means.

Abstract

Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper we consider a family G of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of the mixture of parameterized ordered means in terms of the Loewner order. We also show the relation between two families of parameterized ordered means associated with the power mean, monotonically interpolating given two parameterized ordered means.