
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 is convex if and only if the set the epigraph of is a convex set in . We state an extension of this result for operator convex functions and -convex sets as well as operator log-convex functions and -log-convex sets. Moreover, the -convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Mathematical Inequalities and Applications · Matrix Theory and Algorithms
