Loewner positive entrywise functions, and classification of measurable solutions of Cauchy's functional equations

TL;DR

This paper characterizes functions that preserve Loewner positivity on rank 1 matrices, linking the problem to Cauchy's functional equations and showing that under mild measurability, such functions are smooth and fully classifiable.

Contribution

It provides a complete classification of entrywise functions preserving positivity on rank 1 matrices, connecting to classical functional equations and measurability assumptions.

Findings

Functions preserving positivity on rank 1 matrices are smooth under mild measurability.

Such functions can be fully characterized, extending to complex Hermitian matrices.

The study links positivity preservation to solutions of Cauchy's functional equations.

Abstract

Entrywise functions preserving Loewner positivity have been studied by many authors, most notably Schoenberg and Rudin. Following their work, it is known that functions preserving positivity when applied entrywise to positive semidefinite matrices of all dimensions are necessarily analytic with nonnegative Taylor coefficients. When the dimension is fixed, it has been shown by Vasudeva and Horn that such functions are automatically continuous and sufficiently differentiable. A natural refinement of the aforementioned problem consists of characterizing functions preserving positivity under rank constraints. In this paper, we begin this study by characterizing entrywise functions which preserve the cone of positive semidefinite real matrices of rank $1$ with entries in a general interval. Classifying such functions is intimately connected to the classical problem of solving Cauchy's functional equations, which have non-measurable solutions. We demonstrate that under mild local measurability assumptions, such functions are automatically smooth and can be completely characterized. We then extend our results by classifying functions preserving positivity on rank $1$ Hermitian complex matrices.