The complete Pick property for pairs of kernels and Shimorin's factorization

TL;DR

This paper characterizes the complete Pick property for pairs of kernels, providing a strong converse to Shimorin's factorization theorem and establishing necessary conditions through Carathéodory-Fejér interpolation.

Contribution

It offers a full characterization of the complete Pick property for kernel pairs and presents new proofs and necessary conditions, extending Shimorin's work.

Findings

Strong converse to Shimorin's theorem for holomorphic kernel pairs

Full characterization of the complete Pick property for kernel pairs

Necessary conditions for the complete Pick property using Carathéodory-Fejér interpolation

Abstract

Let $(\mathcal{H}_k, \mathcal{H}_{\ell})$ be a pair of Hilbert function spaces with kernels $k, \ell$. In a 2005 paper, Shimorin showed that a certain factorization condition on $(k, \ell)$ yields a commutant lifting theorem for multipliers $\mathcal{H}_k\to\mathcal{H}_{\ell}$, thus unifying and extending previous results due to Ball-Trent-Vinnikov and Volberg-Treil. Our main result is a strong converse to Shimorin's theorem for a large class of holomorphic pairs $(k, \ell),$ which leads to a full characterization of the complete Pick property for such pairs. We also present a short alternative proof of sufficiency for Shimorin's condition. Finally, we establish necessary conditions for abstract pairs $(k, \ell)$ to satisfy the complete Pick property, further generalizing Shimorin's work with proofs that are new even in the single-kernel case $k=\ell.$ Our approach differs from Shimorin's in that we do not work with the Nevanlinna-Pick problem directly; instead, we are able to extract vital information for $(k, \ell)$ through Carath\'eodory-Fej\'er interpolation.