Hadamard Powers with Loewner Chains and Nevanlinna Representations in Several Variables

Abstract

We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. Bracci, Graham, Hamada, and Kohr developed a new method to construct Loewner chains, by considering variations of certain Loewner chains. We complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We generalize to several variables the classical theorem of Nevan linna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalf-plane, there are representation formulae in terms of densely-defined self- adjoint operators on a Hilbert space.