$q$-Numerical Ranges and Spectral Sets

TL;DR

This paper extends spectral constant bounds for convex domains using geometric properties and generalizes the numerical range concept to $q$-numerical ranges, proposing a broader conjecture.

Contribution

It generalizes the spectral set bounds to scaled $q$-numerical ranges and introduces a new conjecture related to these ranges.

Findings

Bounds depending on a parameter $b3$ relate to geometric properties.

The numerical range is a $1+b2$-spectral set.

A generalization of Crouzeix's Conjecture is proposed for $q$-numerical ranges.

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.