$q$-Numerical Ranges and Spectral Sets

We study spectral constants for convex domains $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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.