Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/12955
Published electronically: January 27, 2016
MathSciNet review: 3487232
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Szeged) 34 (1973), 11-15. MR 0318920 (47 #7466)
  • [2] C. A. Berger, A strange dilation theorem, Abstract 625-152, Notices Amer. Math. Soc. 12 (1965), 590.
  • [3] C. A. Berger and J. G. Stampfli, Mapping theorems for the numerical range, Amer. J. Math. 89 (1967), 1047-1055. MR 0222694 (36 #5744)
  • [4] Michel Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal. 244 (2007), no. 2, 668-690. MR 2297040 (2008f:47020), https://doi.org/10.1016/j.jfa.2006.10.013
  • [5] Michael A. Dritschel and Hugo J. Woerdeman, Model theory and linear extreme points in the numerical radius unit ball, Mem. Amer. Math. Soc. 129 (1997), no. 615, viii+62. MR 1401492 (98b:47007), https://doi.org/10.1090/memo/0615
  • [6] S. W. Drury, Symbolic calculus of operators with unit numerical radius, Linear Algebra Appl. 428 (2008), no. 8-9, 2061-2069. MR 2401640 (2009c:47020), https://doi.org/10.1016/j.laa.2007.11.007
  • [7] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424 (2007e:30049)
  • [8] Karl E. Gustafson and Duggirala K. M. Rao, Numerical range. The field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, 1997. MR 1417493 (98b:47008)
  • [9] Tosio Kato, Some mapping theorems for the numerical range, Proc. Japan Acad. 41 (1965), 652-655. MR 0222693 (36 #5743)
  • [10] Carl Pearcy, An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289-291. MR 0201976 (34 #1853)
  • [11] Alexei Poltoratski and Donald Sarason, Aleksandrov-Clark measures, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 1-14. MR 2198367 (2006i:30048), https://doi.org/10.1090/conm/393/07366
  • [12] Eero Saksman, An elementary introduction to Clark measures, Topics in complex analysis and operator theory, Univ. Málaga, Málaga, 2007, pp. 85-136. MR 2394657 (2009g:47041)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47A12, 15A60

Retrieve articles in all journals with MSC (2010): 47A12, 15A60


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

American Mathematical Society