2017/06/07 by Brown, Lawrence G.
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 47A63 #Secondary 26A51
paper · doi:10.48550/arxiv.1706.02343
We observe that if f is a continuous function on an interval I and x0 ∈ I, then f is operator monotone if and only if the function (f(x) - f(x0)/(x - x0) is strongly operator convex. Then starting with an operator monotone function f0, we construct a strongly operator convex function f1, an (ordinary) operator convex function f2, and then a new operator monotone function f3. The process can be continued to obtain an infinite sequence which cycles between the three classes of functions. We also describe two other constructions, similar in spirit. We prove two lemmas which enable a treatment of those aspects of strong operator convexity needed for this paper which is more elementary than previous treatments. And we discuss the functions phi such that the composite phi ∘ f is operator convex or strongly operator convex whenever f is strongly operator convex.