Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities

Abstract

A special case of a fundamental result of Loewner and Horn [Trans.Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n 1, if the entrywise application of a smooth function f : (0,∞) → R preserves the set of n × n positive semidefinite matrices with positive entries, then f and its first n − 1 derivatives are non-negative on (0,∞). In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers.