Spectral Sets: Numerical Range and Beyond

Abstract

We extend the proof in [M. Crouzeix and C. Palencia, The numerical range is a (1 + √ 2)spectral set, SIAM Jour. Matrix Anal. Appl., 38 (2017), pp. 649-655] to show that other regions in the complex plane are K-spectral sets. In particular, we show that various annular regions are (1 + √ 2)-spectral sets and that a more general convex region with a circular hole or cutout is a (3 + 2 √ 3)-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylov subspace methods for approximating f (A)b, where A is a square matrix, b is a given vector, and f is a function that can be uniformly approximated on such a region by rational functions with poles outside the region.2000 Mathematical subject classifications : 47A25 ; 47A30

Some uses of the field of values in numerical analysis

“…Nevertheless, in this case it may still be possible to identify a C-spectral set, i.e., a subset S of the complex plane satisfying Λ(A) ⊂ S ⊂ W( A), not containing 0 (or, more generally, any singularities of the function f ), and such that g(A) ≤ C sup z∈S |g(z)| for all rational functions g bounded on S, where C is a universal constant. We refer to [16] for some examples illustrating this technique. It is, however, too early to say if this approach can be successfully applied to prove convergence bounds for the preconditioned GMRES method in realistic applications.…”

Polygonal functional calculus for operators with finite peripheral spectrum

“…For T as above, ( 6) is valid for any λ in the resolvent set of A j . Equivalently, for any z in the resolvent set of T , we have (7) (…”

“…In Hilbertian operator theory, several important topics are related to von Neumann's inequality and to the search for inequalities of the form (1). This includes the study of polynomial boundedness, K-spectral sets and similarity problems, for which we refer to [4,5,7,9,22,23] and the references therein. We recall that T : X → X is called polynomially bounded if there exists a constant K ≥ 1 such that (1) holds true with Ω = D. In this paper we are interested in the case when the open set Ω ⊂ C in (1) is a polygon.…”

Geometric Dilations and Operator Annuli

“…See Section 2. Later in this introduction we discuss connections between the quantum annulus and other operator annuli [BY+,CG,Ts+,Ts22].…”