Geodesic
Generalized bitangent Caratheodory--Nevanlinna--Pick problem, and $(j,J_0)$-inner
Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 1-26.

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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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D. Z. Arov. Generalized bitangent Caratheodory--Nevanlinna--Pick problem, and $(j,J_0)$-inner. Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 1-26. http://geodesic.mathdoc.fr/item/IM2_1994_42_1_a0/

[1] Arov D. Z., “Generalized Bitangent Caratheodory–Nevanlinna–Pick Problem and $(j,J_0)$-inner Matrix Functions”, International Workshop on Algorithms and Parallel VLSI Architectures, Part B (10–16 June 1990, Abbaye des Prémontrés. Pont-á-Mousson. France), 179–183 | MR

[2] Arov D. Z., “O regulyarnykh i singulyarnykh $j$-vnutrennikh matritsakh-funktsiyakh i sootvetstvuyuschikh zadachakh ekstrapolyatsii”, Funkts. analiz i ego prilozh., 22:1 (1988), 57–59 | MR | Zbl

[3] Arov D. Z., Ch. I, II, III, Teoriya funktsii, funkts. analiz i ikh primeneniya, no. 51, Kharkov, 1989 | MR | Zbl

[4] Arov D. Z., “Regular $j$-inner matrix-functions and related continuation problems”, Operator Theory: advances and applications, 43 (1990), 63–87 | MR | Zbl

[5] Potapov V. P., “Multiplikativnaya struktura $j$-nerastyagivayuschikh matrits-funktsii”, Tr. mosk. matem. ob-va, 4, 1955, 125–236 | MR | Zbl

[6] Adamyan V. M., Arov D. Z., Krein M. G., “Beskonechnye gankelevy matritsy i svyazannye s nimi problemy prodolzheniya”, Izv. AN ArmSSR. Ser. matem., VI:2,3 (1971), 87–112

[7] Arov D. Z., “Realizatsiya matrits-funktsii po Darlingtonu”, Izv. AN SSSR. Ser. matem., 37:6 (1973), 1299–1331 | MR | Zbl

[8] Ginzburg Yu. L., “O delitelyakh i minorantakh operator-funktsii ogranichennogo vida”, Matem. issledovaniya, 2:4 (1967), 47–72, Kishinev | MR

[9] Simakova L. A., “O plyus-matritsakh-funktsiyakh ogranichennoi kharakteristiki”, Matem. issledovaniya, 9, no. 2, Kishinev, 1974, 149–171 | MR | Zbl

[10] Simakova L. A., “O meromorfnykh plyus-matritsakh-funktsiyakh”, Matem. issledovaniya, 10, no. 1, Kishinev, 1975, 287–292 | MR | Zbl

[11] Arov D. Z., “Realizatsiya kanonicheskoi sistemy s dissipativnym granichnym usloviem na odnom kontse segmenta po koeffitsientu dinamicheskoi podatlivosti”, Sib. matem. zhurn., 16:3 (1975), 540–563 | MR

[12] Krein M. G., “O logarifme bezgranichno razlozhimoi ermitovo-polozhitelnoi funktsii”, DAN SSSR, 45:C (1944), 99–102

[13] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov, Nauka, M., 1966 | Zbl

[14] Akhiezer N. I., Klassicheskaya problema momentov, Fizmatgiz, M., 1961

[15] Krein M. G., “O probleme prodolzheniya ermitovo-polozhitelnykh nepreryvnykh funktsii”, DAN SSSR, 26:1 (1940), 17–22 | MR | Zbl

[16] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Nauk. dumka, Kiev, 1965 | MR

[17] Krein M. G., “Pro epmitovi operatori z napryamnimi funktsionalami”, 3bipnik prats Institutu matematiki AN USSR, 1948, no. 10, 83–106 | MR

[18] Friedrich J., Klotz L., “On extensions of positive definite operator-valued functions”, Reports of Math. Physics, 26:1 (1988), 45–65 | DOI | MR | Zbl

[19] Brodskii M. S., Treugolnye i zhordanovy predstavleniya lineinykh operatorov, Nauka, M., 1969 | MR