Hyponormality of trigonometric Toeplitz operators (Q2781414)

In order to characterize hyponormality of a Toeplitz operator \(T_\varphi\) for which the symbol \(\varphi\) is a trigonometric polynomial, \(\varphi(z)= \sum^N_{n=-m} a_nz^n\), a real \(2m-2\times 2m-2\) linear system of equations is considered, and to each solution a polynomial \(f\) of degree \(m\) is associated. It is shown that \(T_\varphi\) is hyponormal if and only if the linear system is solvable and for each zero \(\zeta\) of \(f\) lying outside the unit circle, the number \(\overline \zeta^{-1}\) is a zero of \(f\), too, and the multiplicity of \(\overline \zeta^{-1}\) is not less than the multiplicity of \(\zeta\). In this case, the rank of the selfcommutator is determined by the multiplicities.