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arXiv · 2603.15536

$q$-Numerical Ranges and Spectral Sets

Ryan O'Loughlin·Jyoti Rani
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Abstract

We study spectral constants for convex domains $\Omega$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $\gamma$ and relating these bounds to geometric properties of $\Omega$ and the numerical range $W(A)$. We generalise the proof that the numerical range is a $1+\sqrt{2}$-spectral set to scaled $q$-numerical ranges. We also propose a generalisation of Crouzeix's Conjecture in the context of $q$-numerical ranges.

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Ryan O'Loughlin, Jyoti Rani. 2026-03-16. $q$-Numerical Ranges and Spectral Sets. https://arxiv.org/abs/2603.15536

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