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De Branges–Pontryagin spaces B (E) with negative index κ of entire p × 1 vector valued functions based on an entire p × 2 p entire matrix valued function E (λ) (called the de Branges matrix) are studied. An explicit description of these spaces and an explicit formula for the indefinite inner product are presented. A characterization of those spaces B (E) that are invariant under the generalized backward shift operator that extends known results when κ = 0 is given. The theory of rigged de Branges–Pontryagin spaces is developed and then applied to obtain an embedding of de Branges matrices with negative squares in generalized J -inner matrices and selfadjoint extensions of the multiplication operator in B (E). A formula for factoring an arbitrary generalized J -inner matrix valued function into the product of a singular factor and a perfect factor is found analogous to the known factorization formulas for J -inner matrix valued functions. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a Schur transformation for generalized Nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational J ℓ-unitary 2×2 matrix functions into elementary factors from the same class. [Copyright &y& Elsevier]

The main result is that forJ=100−1every 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 [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1–7] and studied in [Pisot and Salem Numbers, Birkha¨user Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1–54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkha¨user Verlag, Basel, 2000, p. 1; Monatshefte fu¨r Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions. [Copyright &y& Elsevier]

Certain meromorphic matrix valued functions on C&z.drule;R, the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S, and a boundary mapping for S. Under some simple assumptions the boundary coefficients also satisfy a minimality condition. It is shown that these assumptions hold if and only if for S a generalized von Neumann equality is valid. [Copyright &y& Elsevier]