On hyponormal Toeplitz operators with polynomial and circulant-type symbols
A Toeplitz operator \(T_\varphi\) with symbol \(\varphi\in L^\infty(\mathbb{T})\) -- \(\mathbb{T}\) denotes the unit circle -- is known to be hyponormal if and only if there exists a function \(k\in H^\infty(\mathbb{T})\) such that \(\| k\|_\infty\leq 1\) and \(\varphi- k\overline{\varphi}\in H^\infty(\mathbb{T})\). This is used to investigate the case, when \(\varphi\) is a trigonometric polynomial. If the positive and negative coefficients of \(\varphi\) are composed by a certain ``circular'' condition from given complex numbers \(a_0,\dots, a_N\), then for the hyponormality of \(T_\varphi\) the zeroes of the analytic polynomial generated by the ``essential'' ones among the numbers \(a_0,\dots, a_N\) are to examine. They determine the above function \(k\), which is unique in this case, and the rank of the selfcommutator of \(T_\varphi\) can be computed from the number of zeroes inside and outside the unit circle.
scientific article; zbMATH DE number 1113519