Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set
arXiv:1708.0863318 citationsh-index: 26
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For mathematicians studying spectral sets and numerical ranges, this is an incremental refinement of a known proof technique.
The authors provide a new short proof of a key lemma from Crouzeix and Palencia's result that the numerical range is a $(1+\\sqrt2)$-spectral set, and show the constant is sharp.
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.