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Hyponormality and spectra of
Toeplitz operators

Authors: Douglas R. Farenick and Woo Young Lee
Journal: Trans. Amer. Math. Soc. 348 (1996), 4153-4174
MSC (1991): Primary 47B20, 47B35; Secondary 47A10
MathSciNet review: 1363943
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Abstract: This paper concerns algebraic and spectral properties of Toeplitz operators $T_{\varphi }$, on the Hardy space $H^{2}({\mathbb {T}})$, under certain assumptions concerning the symbols $\varphi \in L^{\infty }({\mathbb {T}})$. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at $T_{\varphi }$, for each quasicontinuous $\varphi $. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

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Additional Information

Douglas R. Farenick
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

Woo Young Lee
Affiliation: Department of Mathematics, Sung Kyun Kwan University, Suwon 440-746, Korea

Keywords: Toeplitz operators, hyponormality, spectrum
Received by editor(s): May 23, 1995
Additional Notes: The work of the first author is supported in part by a grant from The Natural Sciences and Engineering Research Council of Canada. \endgraf The work of the second author is supported in part by The Basic Science Research Institute Program, Ministry of Education, 1995, Project No. BSRI-95-1420, KOSEF 94-0701-02-3 and the GARC-KOSEF, 1995.
Article copyright: © Copyright 1996 American Mathematical Society

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