Dilation Theory for Scaled $q$-Numerical Ranges

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 $${\rho }_q=\max \{1,\frac{2\mathrm{\mid }q\mathrm{\mid }}{1+\sqrt{1-\mathrm{\mid }q{\mathrm{\mid }}^2}}\}\mathrm{.}$$ As a corollary we show that every disk containing ${\mathrm{\Omega }}_q(A)$ 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 $A$.