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$q$-Numerical Ranges and Spectral Sets

Ryan O'Loughlin, Jyoti Rani

Abstract

We study spectral constants for convex domains $Ω$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $γ$ and relating these bounds to geometric properties of $Ω$ 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.

$q$-Numerical Ranges and Spectral Sets

Abstract

We study spectral constants for convex domains containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter and relating these bounds to geometric properties of and the numerical range . We generalise the proof that the numerical range is a -spectral set to scaled -numerical ranges. We also propose a generalisation of Crouzeix's Conjecture in the context of -numerical ranges.
Paper Structure (4 sections, 8 theorems, 52 equations)

This paper contains 4 sections, 8 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Spectral Constants
  3. Geometric Bounds for Spectral Constants
  4. Scaled q-Numerical Range as a Spectral Set

Key Result

Lemma 2.1

Let $\Omega$ be a region with smooth boundary containing the spectrum of $A$ in its interior. For $f \in {\cal A} ( \Omega )$ with $\| f \|_{\Omega} = 1$, let Then $\| S \| \leq 2 + \gamma(1)$.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 5 more