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Kernels of Hankel operators and hyponormality of Toeplitz operators. (English) Zbl 0987.47016

It was shown by C. C. Cowen [Proc. Am. Math. Soc. 103, No. 3, 809-812 (1988; Zbl 0668.47021)] that a Toeplitz operator \(T_\varphi\) on the Hardy space \(H^2\) is hyponormal if and only if there exists \(k\in H^\infty\), \(\|k\|_\infty\leq 1\), such that \(\overline\varphi_- -k\overline\varphi_+\in H^2\), where \(\varphi_+=P\varphi\) and \(\overline\varphi_-=(I-P)\varphi\) are the analytic and the anti-analytic parts of \(\varphi\), respectively. In the paper under review, the authors try to give more explicit conditions for the hyponormality of \(T_\varphi\). First, they give a description of the kernel of a product \(H^*_{\overline\theta_1}H_{\overline\theta_2}\) of two Hankel operators, with inner functions \(\theta_1,\theta_2\), and show that \(\ker H^*_{\overline\theta_1}H_{\overline\theta_2}=\ker H_{\overline\theta_2}\) if and only if \(\theta_1 H^2 \cap(H^2\ominus \theta_2 H^2)=\{0\}\). This leads to conditions for hyponormality of Toeplitz operators with circulant type symbols, and to a necessary and sufficient condition for the hyponormality of \(T_{f+\overline\theta f}\) with \(\theta\) inner and \(f\in H^2\ominus\theta H^2\). Next, for \(\varphi\) of bounded type, the hyponormality of \(T_\varphi\) is shown to be equivalent to the norm of a certain Hankel operator being at most 1, and estimates for the latter norm are then used to give further examples of hyponormal Toeplitz operators, whose symbols satisfy certain symmetry conditions. Finally, some results on the rank of the self-commutator \([T^*_\varphi,T_\varphi]\) of a hyponormal Toeplitz operator \(T_\varphi\) are given.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B20 Subnormal operators, hyponormal operators, etc.

Citations:

Zbl 0668.47021
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