Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set

TL;DR

This paper provides a concise proof of a key lemma in the Crouzeix-Palencia theorem, establishing that the numerical range of a Hilbert-space operator is a sharp $(1+\sqrt{2})$-spectral set, refining understanding of spectral set bounds.

Contribution

The authors present a new, shorter proof of an essential lemma and demonstrate the sharpness of the $(1+\sqrt{2})$ constant in the spectral set bound.

Findings

New short proof of the functional-analysis lemma

Confirmation that the $(1+\sqrt{2})$ constant is optimal

Enhanced understanding of spectral set bounds

Abstract

Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\sqrt2)$-spectral set for the operator. One of the principal ingredients of their proof can be formulated as an abstract functional-analysis lemma. We give a new short proof of the lemma and show that, in the context of this lemma, the constant $(1+\sqrt2)$ is sharp.