SOME EXTENSIONS OF THE CROUZEIX-PALENCIA RESULT.

In [SIAM J. Matrix Anal. Appl., 38 (2017), pp. 649–655], Crouzeix and Palencia show that the closure of the numerical range of a square matrix or linear operator A is a (1 +√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)‖ ≤ (1 +√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 ∈ 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 not necessarily contain W(A) are K-spectral sets for a value of K that may be close to 1 +√2. We also find some special cases in which the constant (1 +√2) for W(A) can be replaced by 2, which is the value conjectured by Crouzeix. [ABSTRACT FROM AUTHOR]