Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897781 | Linear Algebra and its Applications | 2018 | 13 Pages |
Abstract
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 show also that if t0 is a point in the domain of a continuous function f, then f is operator monotone if and only if (f(t)âf(t0)/(tât0) 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 Ïâf when f is operator convex or strongly operator convex.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Lawrence G. Brown, Mitsuru Uchiyama,