Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt{2})$-spectral set

Abstract

Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\sqrt{2})$-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 $(1+\sqrt{2})$ is sharp.

4 pages, few typos corrected and references added, final version to be published in SIAM Journal on Matrix Analysis and Applications

Cited by in corpus (4)

• Crouzeix's Conjecture and related problems
• Dilation theory in finite dimensions and matrix convexity
• Between the von Neumann inequality and the Crouzeix conjecture
• The algebraic numerical range as a spectral set in Banach algebras