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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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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: http://dx.doi.org/10.1090/S0002-9939-06-08699-0
PII: S 0002-9939(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.