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 $M = \frac 54$.References
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Additional Information
- Kenneth R. Davidson
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 55000
- ORCID: 0000-0002-5247-5548
- Email: krdavids@uwaterloo.ca
- Vern I. Paulsen
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 137010
- ORCID: 0000-0002-2361-852X
- Email: vpaulsen@uwaterloo.ca
- Hugo J. Woerdeman
- Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
- MR Author ID: 183930
- Email: hugo@math.drexel.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1189-1195
- MSC (2010): Primary 47A12, 47A25, 15A60
- DOI: https://doi.org/10.1090/proc/13801
- MathSciNet review: 3750231