$q$-Numerical Ranges and Spectral Sets

Abstract

We study spectral constants for convex domains $\Omega$ containing the spectrum of a matrix. 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 formulate a generalisation of Crouzeix's Conjecture in the context of $q$-numerical ranges and give an application of this result to GMRES convergence bounds.