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Hyponormal Toeplitz operators on H2(T) with polynomial symbols

Published online by Cambridge University Press:  22 January 2016

Dahai Yu
Dahai Yu*
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, China 610064
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Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 144 , December 1996 , pp. 179 - 182
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1996

References

[ 1 ] Cowen, C., Hyponormal and subnormal Toeplitz operators, in Surveys of Some Recent Results in Operator Theory (Conway, J. B. and Morrel, B. B., editors), Pitman Research Notes in Math., 171 (1988), 155–167.Google Scholar
[ 2 ] Nakazi, T. and Takahashi, K., Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Tran. Amer. Math. Soc, 338 (1993), 753–766.CrossRefGoogle Scholar
[ 3 ] Zhu, Kehe, Hyponormal Toeplitz operators with polynomial symbols, Integr. Equat Oper. Th., 21 (1995), 376–381.Google Scholar
[ 4 ] Power, S. C., Hankel operators on Hilbert space, Research Notes in Math,. 64 (Jeffrey, A., Douglas, R. G.).Google Scholar