PreprintCC0arXiv
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 Wq(A)/q is contained in the closed unit disk, then A admits a unitary ρq-dilation, where ρq=max{1,2|q|/(1+√(1−|q|2))}. As a corollary we show that every disk containing Ωq(A) is a complete ρq-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.
Related
BibTeX
@misc{kaunisto2026dilation,
title = {{Dilation Theory for Scaled $q$-Numerical Ranges}},
author = {Matias Kaunisto and Ryan O'Loughlin and Jani Virtanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00443},
eprint = {2610.00443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00443},
note = {CC0},
}
RIS
TY - UNPB
TI - Dilation Theory for Scaled q-Numerical Ranges
AU - Matias Kaunisto
AU - Ryan O'Loughlin
AU - Jani Virtanen
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.00443
UR - https://arxiv.org/abs/2610.00443
AB - We generalise Berger's dilation theorem from numerical ranges to scaled q-numerical ranges. Specifically, we prove that if the scaled q-numerical range W_q(A)/q is contained in the closed unit disk, then A admits a unitary ρ_q-dilation, where ρ_q=max{1,2|q|/(1+√(1−|q|^2))}. 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.
ER -