A normal family of operator monotone functions
Let \(A\) and \(B\) be selfadjoint operators on a complex Hilbert space such that their spectra lie in an interval \(J\). A continuous real valued function \(f\) defined on \(J\) is called operator monotone if \(A \leq B\) implies that \(f(A) \leq f(B)\), where ``\(\leq \)'' is the Löwner partial order. \(f\) is called operator convex on \(J\) if \(f(\lambda A+(1-\lambda)B) \leq \lambda f(A) +(1-\lambda)f(B)\) for all \(\lambda \in [0,1]\). For an open subset \(\Omega\) of the complex plane, let \(C(\Omega)\) denote the normed space of continuous functions on \(\Omega\) with the supremum norm. A set \(\mathcal F\)\(\subseteq C(\Omega)\) is called a normal family if every sequence in \(\mathcal F\) has a subsequence converging uniformly on each compact subset of \(\Omega\). The authors show that the family of all operator monotone functions \(f\) on \((-1,1)\) such that \(f(0)=0\) and \(f'(0)=1\) is a normal family. They provide necessary and sufficient conditions for odd operator monotone functions and even operator convex functions on \((-1,1)\).
scientific article; zbMATH DE number 6232363