2016/12/17 by Kenneth R. Davidson, Vern I. Paulsen, Davidson, Kenneth R. +3
Computer Science · Mathematics · #Matrix Theory and Algorithms #Advanced Topics in Algebra #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1612.05683
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 ∈ \mathcal B(\mathcal H), then w(P(T)) ≤ M ‖P‖W(T). When W(T) = \mathbb D, we have M = \frac54.