Epigraph of Operator Functions

TL;DR

This paper extends the classical convexity-epigraph relationship to operator convex and log-convex functions within the framework of $C^*$-algebras, providing new characterizations and representations involving eigenvalues.

Contribution

It introduces operator and $C^*$-analogues of convexity and log-convexity concepts, extending the classical convexity-epigraph theorem to these settings.

Findings

Characterization of operator convex functions via $C^*$-convex sets

Representation of $C^*$-convex hulls using eigenvalues

Extension of convexity-epigraph correspondence to operator functions

Abstract

It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in\mathbb{R}\times\mathbb{R};\ f(x)\leq y\},$$ the epigraph of $f$ is a convex set in $\mathbb{R}^2$. We state an extension of this result for operator convex functions and $C^*$-convex sets as well as operator log-convex functions and $C^*$-log-convex sets. Moreover, the $C^*$-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.