Generalized Schur Representation of Matrix-Valued Functions

The generalized Schur representation of a function matrix $\Omega ( e^{i\theta } )$ satisfying $\| \Omega \|_\infty \leqq1$ is investigated in connection with certain results concerning the extensions of block-Hankel operators acting on Hilbert spaces. Various properties of such representations are elucidated, including a parametrization of $\Omega ( e^{i\theta } )$ in terms of a double sequence of Schur parameter matrices. Special attention is paid to the way in which the representation and parametrization of the shifted function $e^{ik\theta } \Omega (e^{i\theta } )$ are related to those of $\Omega ( e^{i\theta } )$. In particular, the asymptotic behavior of the shifted representation for $k \to \pm \infty $ is studied in detail. The whole theory is developed so as to be of direct use in the analysis of half-plane block-Toeplitz systems.