Complete spectral sets and numerical range

Davidson, Kenneth; Paulsen, Vern; Woerdeman, Hugo

doi:10.1090/proc/13801

Complete spectral sets and numerical range
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by Kenneth R. Davidson, Vern I. Paulsen and Hugo J. Woerdeman PDF

Proc. Amer. Math. Soc. 146 (2018), 1189-1195 Request permission

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 $M=\frac 12(C+C^{-1})$. 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 $T \in \mathcal B(\mathcal H)$, then $w(P(T)) \le M \|P\|_{W(T)}$. When $W(T) = \overline {\mathbb D}$, we have $M = \frac 54$.

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