Complete Spectral Sets and Numerical Range

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 = 12 (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), then w(P (T )) ≤M‖P‖W (T ). When W (T ) = D, we have M = 5 4 . In 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.