Kernels of Hankel operators and hyponormality of Toeplitz operators

We give a formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm ker} H_{\overline{\theta_{1}}}^{\ast}H_{\overline {\theta_{2}}}$\end{document} and describe when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm ker}H_{\overline{\theta_{1}}}^{\ast}H_{\overline{\theta_{2}}}={\mathrm ker} H_{\overline{\theta_{2}}}$\end{document}. We explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator.