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On mapping theorems for numerical range


Authors: Hubert Klaja, Javad Mashreghi and Thomas Ransford
Journal: Proc. Amer. Math. Soc. 144 (2016), 3009-3018
MSC (2010): Primary 47A12; Secondary 15A60
Published electronically: January 27, 2016
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Abstract: Let $ T$ be an operator on a Hilbert space $ H$ with numerical radius $ w(T)\le 1$. According to a theorem of Berger and Stampfli, if $ f$ is a function in the disk algebra such that $ f(0)=0$, then $ w(f(T))\le \Vert f\Vert _\infty $. We give a new and elementary proof of this result using finite Blaschke products.

A well-known result relating numerical radius and norm says $ \Vert T\Vert \leq 2w(T)$. We obtain a local improvement of this estimate, namely, if $ w(T)\le 1$, then

$\displaystyle \Vert Tx\Vert^2\le 2+2\sqrt {1-\vert\langle Tx,x\rangle \vert^2} \qquad (x\in H,~\Vert x\Vert\le 1). $

Using this refinement, we give a simplified proof of Drury's teardrop theorem, which extends the Berger-Stampfli theorem to the case $ f(0)\ne 0$.

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Additional Information

Hubert Klaja
Affiliation: Département de mathématiques et de statistique, Université Laval, 1045 avenue de la Médecine, Québec (QC), Canada G1V 0A6
Email: hubert.klaja@gmail.com

Javad Mashreghi
Affiliation: Département de mathématiques et de statistique, Université Laval, 1045 avenue de la Médecine, Québec (QC), Canada G1V 0A6
Email: javad.mashreghi@mat.ulaval.ca

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, 1045 avenue de la Médecine, Québec (QC), Canada G1V 0A6
Email: ransford@mat.ulaval.ca

DOI: https://doi.org/10.1090/proc/12955
Received by editor(s): April 24, 2015
Received by editor(s) in revised form: September 1, 2015
Published electronically: January 27, 2016
Additional Notes: The second author was supported by NSERC
The third author was supported by NSERC and the Canada Research Chairs Program.
Communicated by: Pamela Gorkin
Article copyright: © Copyright 2016 American Mathematical Society