In [The Numerical Range is a (1 + √ 2)-Spectral Set, 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 inequalitywhere 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.

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Let A be a square matrix or a linear operator on a Hilbert space H. A dilation of A is a linear operator M on a larger space K ⊃ H such that A = P H M | H , where P H is orthogonal projection onto H. Often it is required additionally that M k be a dilation of A k for all or a range of positive integer powers k. While much work has been aimed at proving existence of dilations with various properties, there has been little study of the behavior of functions of these dilations and how it compares to that of the original operator. Is the original operator a major part of the dilated one or is it an insignificant piece? Does the larger operator represent some physical process, where the original operator might be an important component for certain times but not for others? In this paper we construct near normal dilations of nonnormal matrices, with the spectrum of the dilated operator around the boundary of the numerical range of the matrix. We compare the behavior of e tA and e tM , for t > 0. We find that the dilated operator takes on a life of its own, representing a wave that grows or decays but eventually dominates the part corresponding to the original operator. We derive other near normal dilations in which this behavior is less pronounced.A reason for studying dilations is that the operator M may have some nice properties that A does not have. For example, if A is highly nonnormal, then it is wellknown that the spectrum σ(A) may give little information about the 2-norm behavior of functions of A. If M is a normal or near normal power dilation of A, however, then p(A) ≤ p(M ) for any polynomial p (since p(A) is a block of p(M )), and if · denotes the 2-norm (as it will throughout this paper), then p(M ) is determined or approximately determined by σ(M ). That is, if M is similar to a normal operator N via a similarity transformation with a moderate condition number Q (M