# Noncommutative Schur-type products and their Schoenberg theorem

## TL;DR

This paper generalizes the Schoenberg theorem to a class of noncommutative Schur-type products, characterizing those that preserve positive semi-definiteness and share key properties with the classical Schur product.

## Contribution

It classifies all noncommutative products satisfying key properties of the classical Schur product and extends Schoenberg's theorem to these generalized products.

## Key findings

- Classified all products with rank-one and positive semi-definite properties.
- Extended Schoenberg's theorem to noncommutative Schur-type products.
- Provided conditions under which these products preserve positive semi-definiteness.

## Abstract

Schoenberg showed that a function $f:(-1,1)\rightarrow \mathbb{R}$ such that $C=[c_{ij}]_{i,j}$ positive semi-definite implies that $f(C)=[f(c_{ij})]_{i,j}$ is also positive semi-definite must be analytic and have Taylor series coefficients nonnegative at the origin. The Schoenberg theorem is essentially a theorem about the functional calculus arising from the Schur product, the entrywise product of matrices. Two important properties of the Schur product are that the product of two rank one matrices is rank one, and the product of two positive semi-definite matrices is positive semi-definite. We classify all products which satisfy these two properties and show that these generalized Schur products satisfy a Schoenberg type theorem.