The main result is that for J = 1 0 0 −1 every J-unitary 2 × 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 × 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [
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Abstract. In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1-36) it was shown that for a generalized Schur function sðzÞ, which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal sðzÞ À1 of the characteristic function sðzÞ and general factorization results for characteristic functions.2000 Mathematics Subject Classification: 46C20, 47A48, 30D99
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