Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators $T_{\varphi }$ with trigonometric polynomial symbols $\varphi$. Our criterion involves the zeros of an analytic polynomial $f$ induced by the Fourier coefficients of $\varphi$. Moreover the rank of the selfcommutator of $T_{\varphi }$ is computed from the number of zeros of $f$ in the open unit disk $\mathbb{D}$ and in $\mathbb{C}\setminus \overline{\mathbb{D}}$ counting multiplicity.