$q$-Numerical Ranges and Spectral Sets

Abstract

We study spectral constants for convex domains $\hat{I}\text{c◯}$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter ${\hat{I}}^3$ and relating these bounds to geometric properties of $\hat{I}\text{c◯}$ 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.