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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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. [30] S. McCullough, The local de Branges-Rovnyak construction and complete Nevanlinna-Pick kernels, in Algebraic methods in operator theory (Ed. R. Curto and P. E. T. Jorgensen), Birkh¨ auser–Verlag, Boston, 1994, pp. 15–24. [31] G. Popescu, I nterpolation problems in several variables, J.

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. To show that (2 ⇔ 3), we represent KF as KF (z, w) = IE − F (z)F (w)∗ + F (z)Z(z)Z(w)∗ F (w)∗ , 1 − z, w or, equivalently, as KF (z, w) = A(z)A(w)∗ − B(z)B(w)∗ , 1 − z, w where A(z) = IE F (z)Z(z) and B(z) = F (z).

12), leads to −1 F = Σ011 + Σ012 (I − T Σ22 ) −1 −1 I − Z(Σ111 + Σ112 (I − T Σ22 ) T Σ21 T Σ21 ) . 6. 16). 22) I − Z(z)Σ111 (z) −1 Z(z)Σ112 (z). 24) and that −1 (I − T Σ22 ) T Σ21 (I − ZΣ111 )−1 ZΣ112 (I − T Θ22 ) −1 = (I − T Θ22 ) − (I − T Σ22 ) −1 −1 . 16). 20) of Σ that the function Σ21 (z) is a Schur function. 21) belongs to Bd (E, ). 22) can be considered as the Vol. 46 (2003) Bitangential Interpolation 159 Redheffer transform of the Schur function Z(z). 20). 7. Appendix: a numerical example In conclusion we illustrate the general construction done in Sections 4 and 5 by a simple numerical example.

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