Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 253-269.

A comparison theorem for Aleksandrov-Clark measures on the torus

Authors: Linus Bergqvist
Author institution: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

Summary: Given two Aleksandrov-Clark measures $\sigma^1$ and $\sigma^2$ on $\mathbb{T}^2$, we prove a theorem relating the property that $\sigma^1 \ll \sigma^2$ to containment of a concrete function in a certain de Branges-Rovnyak space. We show that our theorem is applicable for all rational inner functions on $\mathbb{D}^2$, and we provide several examples of how the theorem can be applied to investigate Aleksandrov-Clark measures related to such functions.

DOI: http://dx.doi.org/10.7900/jot.2023nov09.2452
Keywords: Aleksandrov-Clark measures, polydiscs, de Branges-Rovnyak spaces

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