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Automorphic Nelson Dilations for Contractions and Invariant Subspace Tracking
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Automorphic Nelson Dilations for Contractions and Invariant Subspace Tracking

Kelly Bickel, J. E Pascoe and Ryan Tully-Doyle
ArXiv.org
15 Jul 2026
DOI: https://doi.org/10.48550/arxiv.2607.14372
url
https://doi.org/10.48550/arXiv.2607.14372View
Preprint (Author's original) Open arXiv.org - Non-exclusive license to distribute

Abstract

Mathematics - Classical Analysis and ODEs Mathematics - Functional Analysis
Given ann × nstrictly contractive matrixT , an (automorphic) Nelson dilation\widehat{T}{}{o}fTis a certain type of analytic matrix-valued function on the unit disk with\widehat{T}{(}{0}) = T . Its construction gives a method for lifting a matrix to a matrix-valued function with nice boundary behavior, a trick that has proved useful in recent operator theoretic developments. In this paper, we show that Nelson dilations give a quick way to obtain the minimal isometric and unitary dilations ofTand thus, connect naturally to the classical Sz.-Nagy dilation theory. We then initiate the study of the automorphic Nelson dilations as a fundamental object in their own right and prove that everyThas Nelson dilations\widehat{T}{}{w}ith particularly useful/interesting properties; for example, they either have strongly entangled eigenvalue functions or have reducing subspaces that are independent ofz . Along the way, we examine when the product of an invertible matrix and a diagonal matrix has distinct eigenvalues.

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