Division subspaces and integrable kernels

TL;DR

This paper proves that Hilbert spaces with the division property have reproducing kernels that are integrable and locally trace class, and such spaces consist of holomorphic functions, revealing structural insights into their kernels.

Contribution

It establishes that division property Hilbert spaces possess integrable, trace class kernels and are spaces of holomorphic functions, linking kernel properties to function space structure.

Findings

Reproducing kernels are integrable for division property spaces.

Such kernels are locally of trace class.

Hilbert spaces with division property are spaces of holomorphic functions.

Abstract

In this note we prove that the reproducing kernel of a Hilbert space satisfying the division property has integrable form, is locally of trace class, and the Hilbert space itself is a Hilbert space of holomorphic functions.