Generalized bitangent Caratheodory--Nevanlinna--Pick problem, and $(j,J_0)$-inner

D. Z. Arov

Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 1-26

Résumé

This paper is a study of the problem of describing holomorphic $n\times n$ matrix-valued functions $c(z)$ on the unit disk $K$ with $\operatorname{Rec}(z)\geqslant 0$ (the Caratheodory class $\mathbf C_n$) such that $b_1^{-1}(c-c_0)b_2^{-1}\in\mathscr D_n$, where $b_1$, $b_2$, and $c_0$ are particular matrix-valued functions with $b_1$ and $b_2$ inner and $c_0$ in $\mathbf C_n$, and $\mathscr D_n$ is the Smirnov class of matrix-valued functions of bounded type on $K$. The matrix extrapolation problems of Caratheodory, Nevanlinna–Pick, and M. G. Krein reduce to this problem for special $b_1$ and $b_2$, as do even the tangent and $*$-tangent problems when there is extrapolation data for $c(z)$ and $c^*(z)$ not on the whole Euclidean space $C^n$ but only on chains of its subspaces. In the completely indeterminate case the solution set of the problem is obtained as the image of the class $B_n$ of holomorphic contractive $n\times n$ matrix-valued functions on $K$ under a linear fractional transformation with $(j,J_0)$-inner matrix-valued function $A(z)=[a_{ik}(z)]_1^2$ of coefficients on $K$. The $A(z)$ arising in this way form a class of regular $(j,J_0)$ -inner matrix-valued functions whose singularities appear to be determined by the singularities of $b_1$ and $b_2$. The general results are applied to Krein's problems of extension of helical and positive-definite matrix-valued functions from a closed interval.