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

TL;DR

This paper establishes a new sharp bound for the norm of functions of $ ho$-contractions, refining previous estimates and confirming the bound's optimality, with implications for operator theory and functional calculus.

Contribution

The paper introduces a new characterization of $ ho$-contractions and derives a sharp bound for the functional calculus of these operators, improving upon earlier estimates.

Findings

Derived a new sharp bound for $ig\|f(A)ig\|$ for $ ho$-contractions.

Proved the bound is optimal and consistent with known special cases.

Refined previous estimates by Okubo--Ando and Drury.

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.