Epigraph of 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 logconvex functions and $C^*$logconvex sets. Moreover, the $C^*$convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.05529
 Bibcode:
 2015arXiv151205529K
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 to appear in "Quaestiones Mathematicae"