Complete spectral sets and numerical range

Abstract

We define the complete numerical radius norm for homomorphisms from any operator algebra into $\mathcal{B}(\mathcal{H})$, and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if $K$ is a complete $C$-spectral set for an operator $T$, then it is a complete $M$-numerical radius set, where $M=\frac{1}{2}(C+C^{-1})$. 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 $T\in \mathcal{B}(\mathcal{H})$, then $w(P(T))\le M\Vert P{\Vert }_{W(T)}$. When $W(T)=\overline{\mathbb{D}}$, we have $M=\frac{5}{4}$.