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Anderson's theorem for compact operators


Authors: Hwa-Long Gau and Pei Yuan Wu
Journal: Proc. Amer. Math. Soc. 134 (2006), 3159-3162
MSC (2000): Primary 47A12; Secondary 47B07
DOI: https://doi.org/10.1090/S0002-9939-06-08699-0
Published electronically: June 5, 2006
MathSciNet review: 2231898
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Abstract: It is shown that if $ A$ is a compact operator on a Hilbert space with its numerical range $ W(A)$ contained in the closed unit disc $ \overline{\mathbb{D}}$ and with $ \overline{W(A)}$ intersecting the unit circle at infinitely many points, then $ W(A)$ is equal to $ \overline{\mathbb{D}}$. This is an infinite-dimensional analogue of a result of Anderson for finite matrices.


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

  • 1. J. Agler, Geometric and topological properties of the numerical range, Indiana Univ. Math. J. 31 (1982), 767-777. MR 0674866 (84i:47004)
  • 2. M. Dritschel and H. J. Woerdeman, Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball, Mem. Amer. Math. Soc. 129 (1997), no. 615. MR 1401492 (98b:47007)
  • 3. H.-L. Gau and P. Y. Wu, Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl. 364 (2003), 213-222. MR 1971096 (2004b:15053)
  • 4. K. E. Gustafson and D. K. M. Rao, Numerical Range, the Field of Values of Linear Operators and Matrices, Springer, New York, 1997. MR 1417493 (98b:47008)
  • 5. P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. MR 0675952 (84e:47001)
  • 6. C.-N. Hung, The Numerical Range and the Core of Hilbert-space Operators, Ph.D. dissertation, Univ. of Toronto, 2004.
  • 7. J. S. Lancaster, The boundary of the numerical range, Proc. Amer. Math. Soc. 49 (1975), 393-398. MR 0372644 (51:8851)
  • 8. F. J. Narcowich, Analytic properties of the boundary of the numerical range, Indiana Univ. Math. J. 29 (1980), 67-77. MR 0554818 (81a:47005)
  • 9. M. Radjabalipour and H. Radjavi, On the geometry of numerical ranges, Pacific J. Math. 61 (1975), 507-511. MR 0399891 (53:3732)
  • 10. F. Rellich, Perturbation Theory of Eigenvalue Problems, Gordon and Breach, New York, 1969. MR 0240668 (39:2014)
  • 11. F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955. MR 0071727 (17:175i)
  • 12. B.-S. Tam and S. Yang, On matrices whose numerical ranges have circular or weak circular symmetry, Linear Algebra Appl. 302/303 (1998), 193-221. MR 1733531 (2000m:15040)

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Additional Information

Hwa-Long Gau
Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
Email: hlgau@math.ncu.edu.tw

Pei Yuan Wu
Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
Email: pywu@math.nctu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-06-08699-0
Keywords: Numerical range, compact operator
Received by editor(s): February 4, 2005
Received by editor(s) in revised form: March 23, 2005
Published electronically: June 5, 2006
Additional Notes: This research was partially supported by the National Science Council of the Republic of China.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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