On hyponormal Toeplitz operators with polynomial and symmetric-type symbols
It is known, as Cowen's theorem, that, for a \(\varphi\in L^\infty\), the Toeplitz operator \(T_\varphi\) is hyponormal if and only if there exists a function \(k\in H^\infty\) such that \(\| k\|_\infty\leq 1\) and \(\varphi- k\overline\varphi\in H^\infty\). If \(\varphi(z)= \sum^N_{n=-N} a_nz^n\) where \(a_N\neq 0\), then, for \( k(z)= \sum^\infty_{j= 0}c_j z^j\), \(\varphi- k\overline\varphi\in H^\infty\) implies that \[ \begin{pmatrix} \overline{c_0}\\ \overline{c_1}\\ \vdots\\ \overline{c_N}\end{pmatrix}= \begin{pmatrix} a_1 & a_2 & a_3 & \cdots & a_N\\ a_2 & a_3 &&\cdot & 0\\ a_3 &&\cdot & \cdot &\vdots\\ \vdots & \cdot & \cdot && \vdots\\ a_N & 0 & \cdots & \cdots & 0\end{pmatrix}^{-1} \begin{pmatrix} \overline{a_{-1}}\\ \overline{a_{-2}}\\ \vdots\\ \overline{a_{-N}}\end{pmatrix}. \] Zhu's theorem asserts that, in this case, the Cowen's condition \(\| k\|_\infty\leq 1\) is equivalent to \(| k_n(0)|\leq 1\) for each \(n= 0,1,2,\dots, N-1\) where \(k_0(z)= k(z)\) and \[ k_{n+ 1}(z)= {k_n(z)- k_n(0)\over z(1- \overline{k_n(0)} k_n(z))},\quad| z|< 1,\quad n= 1,2,\dots. \] In this paper, the authors give necessary conditions of the hyponormality of special Toeplitz operators \(T_\varphi\) which have polynomial symbols \(\varphi(z)= \sum^N_{n= -N} a_nz^n\) with symmetric-type coefficients such as \(\overline{a_N}\begin{pmatrix} a_{-(j+ 1)}\\ a_{-(j+2)}\\ \vdots\\ a_{-N}\end{pmatrix}= a_N\begin{pmatrix} \overline{a_{j+ 1}}\\ \overline{a_{j+2}}\\ \vdots\\ \overline{a_N}\end{pmatrix}\) for some \(j\) \((0\leq j\leq N-1)\).
scientific article; zbMATH DE number 1216474