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Register a new userA treatment of strongly operator convex functions that does not require any knowledge of operator algebras
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Authors
Lawrence G. Brown
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In [B1, Theorem 2.36] we proved the equivalence of six conditions on a continuous function f on an interval. These conditions define 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 C. Akemann and G. Pedersen in [AP], 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 paper 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. We also provide a similar treatment of one theorem from [B1] concerning (usual) operator convex functions. And in two final sections we give a somewhat tentative treatment of some other operator inequalities for strongly operator convex functions, and we give a differential criterion for strong operator convexity.
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This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We also show that if t_0 is a point in the domain of a continuous function f, then f is operator monotone if and only if (f(t) - f(t_0))/(t - t_0) is strongly operator convex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of phi circ f when f is operator convex or strongly operator convex.
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