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arXiv · 2405.16319

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

Scott McCullough·Georgios Tsikalas
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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 Carathéodory-Fejér interpolation.

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Scott McCullough, Georgios Tsikalas. 2025-08-26. The complete Pick property for pairs of kernels and Shimorin's factorization. https://arxiv.org/abs/2405.16319

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