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arXiv · 2512.10025

Operators with small Kreiss constants

Nikolaos Chalmoukis·Georgios Tsikalas·Dmitry Yakubovich
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Abstract

We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ We provide lower bounds for the power growth of such matrices, which complement and refine related estimates due to Nikolski and Spijker-Tracogna-Welfert. We also study operators that satisfy a variant of the above Kreiss condition where $K$ is replaced by $1+\epsilon(|z|)$, where the positive continuous function $\epsilon(|z|)$ tends to $0$ as $|z|\to 1^+.$ We show that, if the spectrum of $T$ touches the unit circle only at a single point and the resolvent of $T$ satisfies a growth restriction along the unit circle, it is possible to choose $\epsilon$ so that this Kreiss-type condition guarantees similarity to a contraction. At the core of our proof lies a positivity argument involving the double-layer potential operator. Counterexamples related to less restrictive choices of $\epsilon$ are also provided.

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Nikolaos Chalmoukis, Georgios Tsikalas, Dmitry Yakubovich. 2025-12-10. Operators with small Kreiss constants. https://arxiv.org/abs/2512.10025

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