Complete spectral sets and numerical range
Kenneth R. Davidson, Vern I. Paulsen, Hugo J. Woerdeman · Proceedings of the American Mathematical Society · 2017
We define the complete numerical radius norm for homomorphisms from any operator algebra into B ( H ) \mathcal B(\mathcal 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 K is a complete C C -spectral set for an operator T T , then it is a complete M M -numerical radius set, where M = 1 2 ( C + C − 1 ) M=\frac 12(C+C^{-1}) . In particular, in view of Crouzeix’s theorem, there is a universal constant M M (less than 5.6) so that if P P is a matrix polynomial and T ∈ B ( H ) T \in \mathcal B(\mathcal H) , then w ( P ( T ) ) ≤ M ‖ P ‖ W ( T ) w(P(T)) \le M \|P\|_{W(T)} . When W ( T )