Some Extensions of the Crouzeix-Palencia Result

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.