Maps on positive cones in operator algebras preserving power means

TL;DR

This paper characterizes isomorphisms that preserve power means of positive operators in Hilbert space and explores the relationship between different types of power means in operator algebras.

Contribution

It provides a detailed description of transformations preserving power means in positive cones and examines their interrelations in operator algebra contexts.

Findings

Characterization of isomorphisms preserving power means

Conditions under which different power means can be transformed into each other

Insights into the structure of positive cones in operator algebras

Abstract

In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo-Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary operations) on positive definite cones and on positive semidefinite cones in operator algebras. We also investigate the question when those two sorts of power means can be transformed into each other.