Nested matrix disks analytically depending parameter, and theorems on the

S. A. Orlov

Izvestiya. Mathematics , Tome 10 (1976) no. 3, pp. 565-613

Résumé

In this work there is an investigation of a family of invertible (i.e. $W(b,\lambda)\not\equiv0$) analytic matrix-valued functions $W(b,\lambda)$ ($0$) which are $J$-contractive ($\Gamma(b,\lambda)\overset{\mathrm{def}}= J-W(b,\lambda)JW^*(b,\lambda)>0$, $J^*=J$, $J^2=I$) and which have monotonically increasing $J$-forms $\Gamma(b,\lambda)$ as $b\to\infty$. Invariance with respect to $\lambda$ of the rank of the matrix $R^2(\lambda)=\lim_{b\to\infty}\Gamma^{-1}(b,\lambda)$ is established, and conditions for convergence of $W(b,\lambda)$ are investigated. As a special case a theorem is obtained on the invariance of ranks of limiting radii of Weyl disks, which is fundamentally of significance in the theory of classical problems (the moment problem, the Nevanlinna–Pick problem, the Weyl problem on the number of square-integrable solutions of a system of differential equations, and so forth). Bibliography: 17 titles.