Hyponormality of block Toeplitz operators
The authors study the hyponormality of block Toeplitz operators acting on the vector-valued Hardy space of the unit disk \(H^2(\mathbb C^n)\). The block Toeplitz operator with matrix symbol \(G\in L^{\infty}(\mathbb C^{n\times n})\) is defined by \(T_Gh=P(Gh)\), where \(P\) is the projection from \(L^2(\mathbb C^n)\) to \(H^2(\mathbb C^n)\). \textit{C.\,C.\thinspace Cowen} [Proc.\ Am.\ Math.\ Soc.\ 103, No.\,3, 809--812 (1988; Zbl 0668.47021)] characterized scalar hyponormal Toeplitz operators as follows: \(T_{\varphi}\) is hyponormal if and only if there exists a \(k\in H^{\infty}\) with \(\| k\|_{\infty}\leq 1\) satisfying \(\varphi-k\overline{\varphi}\in H^2\). The first author in [J.~Funct.\ Anal.\ 124, No.\,1, 135--148 (1994; Zbl 0815.47029)] extended Cowen's result to generalized Toeplitz operators whose symbols are operator-valued, under the extra assumption that the symbol is normal. In the present paper, it is shown that the block Toeplitz operator \(T_G\) is hyponormal if and only if \(G^*G=GG^*\) and there exists a matrix \(K(z)\in H^{\infty}(\mathbb C^{n\times n})\) such that \(\| K\|_{\infty}\leq 1\) and \(G-KG^*\in H^{\infty}(\mathbb C^{n\times n})\). It is, however, not easy to find a matrix \(K(z)\) satisfying these conditions. The authors show that if \(G(z)\) is a trigonometric matrix polynomial with an invertible leading coefficient, then finding such a \(K\) is equivalent to solving a matrix-valued Carathéodory interpolation problem.
scientific article; zbMATH DE number 5129572