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Alpay, A. Dijksma and H. 16. Let Θ(z) be a rational p × p matrix function which is J-unitary on R, and assume that Θ(z) does not have any poles in the closed upper half-plane. 17. Let z1 ∈ T (∈ R ∪ {∞}, respectively) and let Θ(z) be a normalized element in Ucz1 (in U z1 , respectively). Then Θ(z) admits a unique minimal factorization into normalized elementary factors. Proof. 15 we consider the line case and z1 ∈ R. We show that the space P(Θ) is spanned by the elements of only one chain. Suppose, on the contrary, that it is spanned by the elements of more than one chain.

0 ν0 ν1 ν2 .. 0 0 ν0 ν1 .. 0 0 0 ν0 .. ⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟. · · ·⎟ ⎠ .. 4, that is, the smallest positive integer k0 = k0 (Γ) such that for n(z) ∈ N the principal submatrix Γk0 is invertible. 9. For the function n(z) ∈ Nz1 which is not identically equal to a real constant and its Pick matrix Γ at z1 we have Im ν0 = 0 ⇐⇒ k0 (Γ) = 1; if ν0 ∈ R and k is the smallest integer ≥ 1 such that νk = 0, then k0 (Γ) = 2k. The Transformation of Issai Schur 49 Proof. The first statement follows from the formula Γ1 = γ00 = Im ν0 /Im z1 .

11. Let J be a p × p signature matrix and let Θ(z) be a rational p × p matrix function which is J-unitary on the unit circle and has no poles on the closed unit disk. Then P(Θ) = H2,J ΘH2,J . The McMillan degree is invariant under M¨ obius transformations, see [37]. 6. 12. Let Θ(z) be a rational p × p matrix function which is J-unitary on the unit circle and has a unique pole at the point 1/w∗ including, possibly, w = 0. Then deg Θ z−w , det Θ(z) = c 1 − zw∗ where c is a unimodular constant. In the sequel we shall need only the case p = 2.