Year: 2026
Type: preprint
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 $ρ_q$-dilation, where $$ ρ_q=\max\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\}. $$ As a corollary we show that every disk containing $Ω_q(A)$ is a complete $ρ_q$-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 $A$.
Source: arXiv (Cornell University)
Institution: University of Eastern Finland
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216592864