Operators with small Kreiss constants
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917330735333376 |
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| author | Chalmoukis, Nikolaos Tsikalas, Georgios Yakubovich, Dmitry |
| author_facet | Chalmoukis, Nikolaos Tsikalas, Georgios Yakubovich, Dmitry |
| contents | 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 $1+ε(|z|)$, where the positive continuous function $ε(|z|)$ 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 $ε$ 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 $ε$ are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operators with small Kreiss constants Chalmoukis, Nikolaos Tsikalas, Georgios Yakubovich, Dmitry Functional Analysis Complex Variables Spectral Theory [2020] 47A10 (Primary), 47A63, 47A20, 47B37 (Secondary) 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 $1+ε(|z|)$, where the positive continuous function $ε(|z|)$ 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 $ε$ 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 $ε$ are also provided. |
| title | Operators with small Kreiss constants |
| topic | Functional Analysis Complex Variables Spectral Theory [2020] 47A10 (Primary), 47A63, 47A20, 47B37 (Secondary) |
| url | https://arxiv.org/abs/2512.10025 |