Dilation Theory for Scaled $q$-Numerical Ranges

Abstract

We generalise Berger's dilation theorem from numerical ranges to scaled $q$-numerical ranges. Specifically, we prove that if the scaled $q$-numerical range $\frac{W_q(A)}{q}$ is contained in the closed unit disk, then $A$ admits a unitary ${\rho }_q$-dilation, where $${\rho }_q=\max \left\{1,\frac{2|q|}{1+\sqrt{1-|q{|}^2}}\right\}.$$ As a corollary we show that every disk containing ${\Omega }_q(A)$ is a complete ${\rho }_q$-spectral set for $A$. At $q=1$, this recovers the classical fact that any disk containing $W(A)$ is a complete $2$-spectral set for $A$.