A sharp bound for the functional calculus of $\rho$-contractions

Felix L. Schwenninger; Jens de Vries

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arXiv · 2412.02326

A sharp bound for the functional calculus of $\rho$-contractions

Felix L. Schwenninger·Jens de Vries

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Abstract

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.

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BibTeX RIS

Felix L. Schwenninger, Jens de Vries. 2024-12-03. A sharp bound for the functional calculus of $\rho$-contractions. https://arxiv.org/abs/2412.02326

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