Epigraph of operator functions

Abstract

It is known that a real function f is convex if and only if the set E(f) = {(x, y) ∈ ℝ × ℝ; f (x) ≤ y}, the epigraph of f is a convex set in ℝ². 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.