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February 2018 A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras
Lawrence G. Brown
Ann. Funct. Anal. 9(1): 41-55 (February 2018). DOI: 10.1215/20088752-2017-0023


In a previous article, we proved the equivalence of six conditions on a continuous function f on an interval. These conditions determine a subset of the set of operator-convex functions whose elements are called strongly operator-convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by Akemann and Pedersen, and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on f, and the fourth is an integral representation of f, stronger than the usual integral representation for operator-convex functions. The purpose of this article is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. A treatment of other operator inequalities characterizing strong operator convexity is included.


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Lawrence G. Brown. "A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras." Ann. Funct. Anal. 9 (1) 41 - 55, February 2018.


Received: 7 December 2016; Accepted: 4 February 2017; Published: February 2018
First available in Project Euclid: 29 June 2017

zbMATH: 06841340
MathSciNet: MR3758742
Digital Object Identifier: 10.1215/20088752-2017-0023

Primary: 47A63
Secondary: 26A51‎

Keywords: ‎operator inequality , operator-convex , strongly operator-convex

Rights: Copyright © 2018 Tusi Mathematical Research Group


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Vol.9 • No. 1 • February 2018
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