Some Extensions of the Crouzeix-Palencia Result

TL;DR

This paper extends the Crouzeix-Palencia result by showing other complex regions are spectral sets with constants close to 1+√2, and identifies cases where the constant can be improved to 2, matching conjectures.

Contribution

It generalizes the spectral set result to regions beyond the numerical range and explores cases with improved constants, advancing understanding of spectral set bounds.

Findings

Other regions in the complex plane are K-spectral sets with K near 1+√2.

Special cases where the constant can be replaced by 2 are identified.

The results support the conjecture that the numerical range is a 2-spectral set.

Abstract

In [{\em The Numerical Range is a $(1 + \sqrt{2})$-Spectral Set}, SIAM J. Matrix Anal. Appl. 38 (2017), pp.~649-655], Crouzeix and Palencia show that the numerical range of a square matrix or linear operator $A$ is a $(1 + \sqrt{2})$-spectral set for $A$; that is, for any function $f$ analytic in the interior of the numerical range $W(A)$ and continuous on its boundary, the inequality $\| f(A) \| \leq (1 + \sqrt{2} ) \| f \|_{W(A)}$ holds, where the norm on the left is the operator 2-norm and $\| f \|_{W(A)}$ on the right denotes the supremum of $| f(z) |$ over $z \in W(A)$. In this paper, we show how the arguments in their paper can be extended to show that other regions in the complex plane that do {\em not} necessarily contain $W(A)$ are $K$-spectral sets for a value of $K$ that may be close to $1 + \sqrt{2}$. We also find some special cases in which the constant $(1 + \sqrt{2})$ for $W(A)$ can be replaced by $2$, which is the value conjectured by Crouzeix.