By Ball J.A., Bolotnikov V.

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V. Gusev and A. Lindquist, From finite covariance windows to modeling filters: a convex optimization approach, SIAM Review 43(4) (2001), 645-675. ¨ [21] C. Carath´eodory, Uber die Winkelderivierten von beschr¨ ankten analytischen Funktionen, Sitzungber. Preuss. Akad. Wiss. 4 (1929), 1–18. [22] K. R. Davidson and D. R. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321– 337. [23] H. Dym, J contractive matrix functions, reproducing kernel spaces and interpolation, CBMS Lecture Notes, vol.

6 to a tangential problem for Schur functions. 7. The following are equivalent: 1. F belongs to Bd (E, E∗ ). 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.

9. 17) and the parameter T from the class Sd (E ⊕ ∆∗ , E∗ ⊕ (⊕d1 E) ⊕ ∆) for some auxiliary Hilbert spaces ∆ and ∆∗ . Moreover, E∗ Θ11 (z) ∈ Bd (E, ) Θ21 (z) E ⊕ ∆∗ E∗ Θ12 (z) ∈ Sd (E∗ ⊕ (⊕d1 E) ⊕ ∆, ). 15). 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.