Complete spectral sets and numerical range

Abstract

Abstract. We define the complete numerical radius norm for homomorphisms from any operator algebra into B(H), and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K is a complete Cspectral set for an operator T , then it is a complete M -numerical radius set, where M = 1 2 (C + C −1 ). In particular, in view of Crouzeix's theorem, there is a universal constant M (less than 5.6) so that if P is a matrix polynomial and T ∈ B(H), thenIn 2007, Michel Crouzeix [6] proved the remarkable fact that for any operator T on a Hilbert space H, the numerical range is a complete C-spectral set for some constant with a universal bound of 11.08. Moreover in [5], he conjectures that the optimal constant is 2, which is the case for a disc. This inspired a result of Drury [7] who proved that if the numerical range of T is contained in the disc, then the numerical radius of any polynomial in T is bounded by 5 4 times the supremum norm of the polynomial over the disc. Generally all that one can say about the relationship between the norm and numerical radius is that w(X) ≤ X , with equality for many operators, so the improvement from 2 to Drury's 5 4 was unexpected. In this note, we establish a precise relationship between the completely bounded norm of a homomorphism of an arbitrary operator algebra and what we call the complete numerical radius norm of the homomorphism. When applied to the case of the disc, our relationship yields Drury's result in the matrix polynomial case, and our result also applies to the general study of C-spectral sets.2010 Mathematics Subject Classification. Primary 47A12, 47A25, 15A60.

Spectral sets and operator radii

“…We say that is a ‐ set for if for every rational function we have The analogue notion of a complete ‐ set is defined in a similar way. The case corresponds to the usual notion of ‐spectral sets while ‐ sets are the ‐numerical radius sets from . We deduce the following result from Theorem .…”

“…The following result, which is the main result of this paper, generalizes Theorem and the result from to the larger class of operator radii . Theorem We assume and and set Then, (a) if and only if for every unital homomorphism from a Banach algebra satisfying the von Neumann inequality; (b) if and only if for every unital homomorphism from an operator algebra . …”

“…The following result, showing the equivalence of ‐spectral sets and ‐numerical radius sets, is a counterpart of the result from . Theorem Let , let be an open domain in the complex plane and let be a real number.…”

“…See also for an alternate proof. This motivates the following terminology: following we say that a set is a ‐ numerical radius set for if and it is a complete ‐ numerical radius set if the same inequality holds for matrices over .…”

“…and it is a completeK-numerical radius set if the same inequality holds for matrices over R(K). Davidson, Paulsen and Woerdeman [5] recently proved that Ω is a complete K-spectral set for T if and only if Ω is a completeK-numerical radius set for T , whereK = 1 2 (K + K −1 ). The proof given in [5] uses several facts which are specific for complete spectral sets and completely bounded maps.…”