Some generalizations of Szegö's first limit theorem
The authors consider a sequence \(\{C_ j: -\infty <j<\infty \}\) of \(r\times r\) matrices and the corresponding truncated block Toeplitz matrices \(T_ n\). The assumption made throughout is that \(\Delta_ n=\det T_ n\neq 0\), \(n=0,1,\ldots\). The sequence \(\{C_ j\}\) is said to be \(\alpha\)-sectorial if for every \(\xi \in \mathbb{C}^{r(n+1)}\), \(| Im \xi T_ n\xi^*\!_ n| \leq Re \xi T_ n\xi^*\tan \alpha\), \(\alpha\in [0,\pi /2)\). A generalization is obtained, in this block matrix case and for \(\alpha\)-sectorial sequences, of the original theorem of Szegö concerning the limits of the ratios \(\Delta_{n+1}/\Delta_ n\) when \(\alpha =0\), \(r=1\). The generalization also includes previous results on the limits of these ratios for \(\alpha >0\), \(r=1\). A good bibliography of earlier work is given.
scientific article; zbMATH DE number 3931187