By Ball J.A., Bolotnikov V.

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2, 374–420. [27] A. Kheifets, The abstract interpolation problem and applications, in: Holomorphic spaces (Ed. D. Sarason, S. Axler, J. McCarthy), pages 351–379, Cambridge Univ. Press, Cambridge, 1998. [28] A. Kheifets, Parseval’s equality in the abstract interpolation problem and the coupling of open systems. , Teor. Funktsii, Funktsional. Anal. , 49 (1988), 112– 120, 126; English Translation: J. Soviet Math. 49 (1990), 1114–1120. [29] I. V. Kovalishina, Carath´ eodory–Julia theorem for matrix–functions, Teoriya Funktsii, Funktsianal’nyi Analiz i Ikh Prilozheniya, 43 (1985), 70–82.

2. The kernel KF defined below is positive on Bd : IE∗ − F (z)F (w)∗ 0 (z, w ∈ Bd ). 11) 156 Ball and Bolotnikov IEOT 3. 12) for some Schur function S ∈ Sd (E, E∗ ⊕ (⊕d1 E)): S(z) = E∗ S0 (z) : E→ , S1 (z) ⊕d1 E Z(z) = z1 IE ... zd IE . 13) Proof: The equivalence (1 ⇔ 2) follows from a more general fact that F is a contractive multiplier between two reproducing kernel Hilbert spaces H(K1 ) and H(K2 ) of functions analytic on a set Ω if and only if the kernel K2 (z, w) − F (z)K1 (z, w)F (w)∗ is positive on Ω.

A. Ball and V. , to appear. A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, Birkh¨ auser Verlag, Basel, 1990. A. Ball, I. Gohberg and L. Rodman, Boundary Nevanlinna-Pick interpolation for rational matrix functions, J. Math. Systems, Estimation, and Control 1 (1991), 131164. A. W. Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix-functions: parametriztion of the set of all solutions, Integral Equations and Operator Theory 9 (1986), 155-203.