Complete spectral sets and numerical range

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 $M=\frac12(C+C^{-1})$. 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 $w(P(T)) \le M \|P\|_{W(T)}$. When $W(T) = \overline{\mathbb D}$, we have $M = \frac54$.

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