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 Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 1, Pages 3–32 (Mi izv884)

Generalized bitangent Caratheodory–Nevanlinna–Pick problem, and $(j,J_0)$-inner matrix-valued functions

D. Z. Arov

Abstract: 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.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:1, 1–26

Bibliographic databases:

UDC: 517.5
MSC: Primary 30E05, 30D05, 30D50; Secondary 47A56, 47A57, 15A22

Citation: D. Z. Arov, “Generalized bitangent Caratheodory–Nevanlinna–Pick problem, and $(j,J_0)$-inner matrix-valued functions”, Izv. RAN. Ser. Mat., 57:1 (1993), 3–32; Russian Acad. Sci. Izv. Math., 42:1 (1994), 1–26

Citation in format AMSBIB
\Bibitem{Aro93} \by D.~Z.~Arov \paper Generalized bitangent Caratheodory--Nevanlinna--Pick problem, and $(j,J_0)$-inner matrix-valued functions \jour Izv. RAN. Ser. Mat. \yr 1993 \vol 57 \issue 1 \pages 3--32 \mathnet{http://mi.mathnet.ru/izv884} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1220579} \zmath{https://zbmath.org/?q=an:0797.30030} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994IzMat..42....1A} \transl \jour Russian Acad. Sci. Izv. Math. \yr 1994 \vol 42 \issue 1 \pages 1--26 \crossref{https://doi.org/10.1070/IM1994v042n01ABEH001525} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994NH32100001} 

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