Some Extensions of the Crouzeix--Palencia Result

Abstract

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.

Spectral Sets: Numerical Range and Beyond

“…In particular, if Ω is the disk {z ∈ C : |z−ω| < R}, with the notations of Lemma 3 we obtain δ ≤ −1. It is shown in [4][section 6.1] that in this case K ≤ max(1, S+γI) ) whence, from this lemma, we get K = 1. This is just the famous von Neumann inequality: Ω is a spectral set for A.…”

“…Then we use the following result from [4] Lemma 3. Assume that f is a rational function satisfying |f | ≤ 1 in Ω and that Sp(A) ⊂ Ω, then it holds…”

“…In section 2 we provide a proof of Lemma 1; the less obvious part is the continuity which can be found in the literature but under stronger smoothness assumptions on the boundary and in a more general context, see for instance [8] or Carl Neumann [10] for the original proof. Section 3 contains a proof of Theorem 2 using a technique of Schwenninger [11] that was also incorporated in [4] to improve upon the original result there. Section 4 gives some estimates of λ min (µ(σ, A)), and these are used in sections 5 and 6 to show that various annular-like regions and regions with circular cutouts are K-spectral sets and to bound the value of K. Finally, in section 7, we show how these new theoretical results provide better bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of the rational Arnoldi algorithm for approximating f (A)b, where A is a square matrix, b is a given vector, and f is a uniform limit of bounded rational functions on a region discussed in section 5 or 6.…”

Remarks on the Crouzeix--Palencia Proof that the Numerical Range is a (1+\sqrt2)-Spectral Set

“…The problem is how to get from (2), which is an estimate on f (T ) + (Cf )(T ) * , to an estimate on f (T ) alone. In [4,5,6], this is achieved in three different ways.…”

“…(There are now several proofs of this last inequality, originally due to Okubo and Ando. A particularly short one, obtained as a simple consequence of (2), can be found in the recent preprint of Caldwell, Greenbaum and Li [2].) Crouzeix's proof of (3) is technical and requires a lot of work.…”