Annals of Functional Analysis

A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras

Lawrence G. Brown

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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.

Article information

Ann. Funct. Anal., Volume 9, Number 1 (2018), 41-55.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 26A51: Convexity, generalizations

operator-convex strongly operator-convex operator inequality


Brown, Lawrence G. A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras. Ann. Funct. Anal. 9 (2018), no. 1, 41--55. doi:10.1215/20088752-2017-0023.

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  • [1] C. A. Akemann and G. K. Pedersen, Complications of semicontinuity in $C^{*}$-algebra theory, Duke Math. J. 40 (1973), 785–795.
  • [2] J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Amer. Math Soc. 79 (1955), 58–71.
  • [3] L. G. Brown, Semicontinuity and multipliers of $C^{*}$-algebras, Canad. J. Math. 40 (1988), no. 4, 865–988.
  • [4] L. G. Brown, Convergence of functions of self-adjoint operators, Publ. Mat. 60 (2016), no. 2, 551–564.
  • [5] L. G. Brown, Semicontinuity and closed faces of $C^{*}$-algebras, to appear in Advances in Operator Theory, preprint, arXiv:1312.3624v2 [math.OA].
  • [6] C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957), 42–44.
  • [7] C. Davis, Notions generalizing convexity for functions defined on spaces of matrices, Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence, 1963, 187–201.
  • [8] P. R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125–134.
  • [9] F. Hansen, The fast track to Löwner’s theorem, Linear Algebra Appl. 438 (2013), no. 11, 4557–4571.
  • [10] F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem, Math. Ann. 258 (1982), no. 3, 229–241.