Analytic lifts of operator concave functions

TL;DR

This paper develops analytic methods to extend and characterize operator concave functions, with applications to operator monotonicity, operator means, and their generalizations in tensor product spaces.

Contribution

It introduces new Schur complement based formulas for operator concavity and extends operator means to broader domains in tensor product spaces.

Findings

Provides analytic extension formulas for operator means.

Characterizes operator concavity in general tensor product domains.

Links operator concavity with operator monotonicity in non-matrix convex settings.

Abstract

The motivation behind this paper is threefold. Firstly, to study, characterize and realize operator concavity along with its applications to operator monotonicity of free functions on operator domains that are not assumed to be matrix convex. Secondly, to use the obtained Schur complement based representation formulas to analytically extend operator means of probability measures and to emphasize their study through random variables. Thirdly, to obtain these results in a decent generality. That is, for domains in arbitrary tensor product spaces of the form $\mathcal{A}\otimes\mathcal{B}(E)$, where $\mathcal{A}$ is a Banach space and $\mathcal{B}(E)$ denotes the bounded linear operators over a Hilbert space $E$. Our arguments also apply when $\mathcal{A}$ is merely a locally convex space.