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arXiv · 2610.02476

Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals

Ludvig Fagrell·Annemarie Luger
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Abstract

We first revisit the Nevanlinna condition for representing measures of Herglotz-Nevanlinna functions in two variables and give a general operator-theoretic criterion for extremality. We then derive an explicit formula for the representing measure of a real rational Herglotz-Nevanlinna function in two variables, and combine it with the extremality criterion to obtain a family of extreme elements in the full cone of Nevanlinna measures. The family includes examples with reducible denominators, and these yield counterexamples to the irreducibility assertion in a conjecture of Knese. We then characterize reducibility within this family both algebraically and geometrically.

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Ludvig Fagrell, Annemarie Luger. 2026-10-01. Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals. https://arxiv.org/abs/2610.02476

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