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Complete spectral sets and numerical range


Authors: Kenneth R. Davidson, Vern I. Paulsen and Hugo J. Woerdeman
Journal: Proc. Amer. Math. Soc. 146 (2018), 1189-1195
MSC (2010): Primary 47A12, 47A25, 15A60
DOI: https://doi.org/10.1090/proc/13801
Published electronically: October 23, 2017
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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 \Vert P\Vert _{W(T)}$. When $ W(T) = \overline {\mathbb{D}}$, we have $ M = \frac 54$.


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Additional Information

Kenneth R. Davidson
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: krdavids@uwaterloo.ca

Vern I. Paulsen
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: vpaulsen@uwaterloo.ca

Hugo J. Woerdeman
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: hugo@math.drexel.edu

DOI: https://doi.org/10.1090/proc/13801
Keywords: Complete spectral set, numerical range, Crouzeix conjecture
Received by editor(s): December 16, 2016
Received by editor(s) in revised form: April 27, 2017
Published electronically: October 23, 2017
Additional Notes: The first author was partially supported by an NSERC grant.
The second author was partially supported by an NSERC grant.
The third author was partially supported by a Simons Foundation grant and by the Institute for Quantum Computing at the University of Waterloo.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2017 American Mathematical Society

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