A sharp bound for the functional calculus of $\ddot{I}$-contractions

Abstract

Let $A$ be a $\ddot{I}$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $\ddot{I}$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{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 $\ddot{I}=2$, is consistent with a result by Drury.

9 pages, 1 figure