Crouzeix’s Conjecture and Related Problems

Abstract

Crouzeix's conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f (A) satisfies the estimate f (A) ≤ 2 sup{| f (z)| : z ∈ W (A)}, (1) where W (A) := { Ax, x : x = 1} denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W (A). We provide a survey of recent investigations related to this conjecture and derive bounds for f (A) for specific classes of operators A. This allows us to state explicit conditions that guarantee that Crouzeix's estimate (1) holds. We describe properties of related extremal functions (Blaschke products) and associated extremal vectors. The case where A is a matrix representation of a compressed shift operator is studied in some detail.

Spectral sets, extremal functions and exceptional matrices

“…We conclude this section by remarking that the ideas behind Theorems 3.1 and 3.2 are developed further in [9]. In particular, it is shown in [9, Theorem 4.5] that, under the hypotheses of Theorem 3.2, there exists a unique probability measure µ on ∂D such that, for all continuous functions h on D that are holomorphic on D, we have…”

A note on a spectral constant associated with an annulus

“…(which is equivalent to the inclusion W (T ) ⊂ Ω) has served as a basis for establishing K-spectral estimates for W (T ); an approach originally due to Delyon and Delyon [18], it was further refined by Crouzeix in [14] and culminated in the Crouzeix-Palencia paper [13] (see also [26], [7] and the recent preprint [27] for further developments), where it was shown that W (T ) is always a (1 + √ 2)-spectral set for any T ∈ B(H). More recently, the study of K-spectral estimates through the contractivity of S Ω,2 has expanded beyond convex domains (see [8], [16]).…”