A sharp bound for the functional calculus of $\rho$-contractions
| Autor: | Schwenninger, Felix L., de Vries, Jens |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $A$ be a $\rho$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $\rho$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \frac{\rho}{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{\rho^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $\rho=2$, is consistent with a result by Drury. Comment: 9 pages, 1 figure |
| Databáze: | arXiv |
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