Hyponormality and spectra of Toeplitz operators

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.