A Bitangential Interpolation Problem on the Closed Unit Ball by Ball J.A., Bolotnikov V.

By Ball J.A., Bolotnikov V.

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3. 4) =1 on Bd , such that d d zj w ¯ Φj (z, w) − j, =1 Φjj (z, w) = D(z)D(w)∗ − B(z)B(w)∗ . 4. 3). 5. 5) and kernels Λ1 , . . , Λd subject to d (zj − w ¯j )Λj (z, w) = C(z)B(w)∗ − A(z)D(w)∗ , j=1 such that the kernel    P(z, w) =   Λ(z, w) Λ1 (z, w)∗ .. Λd (z, w)∗ Λ1 (z, w) Φ11 (z, w) .. ··· ··· Λd (z, w) Φ1d (z, w) .. 2)). 2. 1), let H(kd ) be the corresponding reproducing kernel Hilbert space. In this section we apply the preceding analysis to the operator-valued H(kd )-functions. To be more precise, let E and E∗ be two Hilbert spaces and let H(kd , E, E∗ ) denote the space of L(E, E∗ )-valued functions F (z) such that the function z → F (z)x, y E∗ belongs to H(kd )(= H(kd , C, C)) for every choice of x ∈ E and y ∈ E∗ .

Ball, Interpolation problems of Pick-Nevanlinna and Loewner type for meromorphic matrix functions, Integral Equations and Operator Theory, 6 (1983), 804-840. [12] J. 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.

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. English translation in: Journal of Soviet Mathematics, 48(2) (1990), 176–186.

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