The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

Abstract

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

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

“…These calculations imply (3) and show that g j (z, iy) → −f j (z) in L(N) as y ↑ ∞. Moreover, they imply…”

“…The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle

“…In Section 2 we define the generalized Schur transformation for functions of the class S which are not constants of modulus one. The difference with the corresponding definitions in the papers [2,3] (taken from [11]) lies in the fact that here we define the transformation for all functions of S, and it becomes a little simpler since poles at zero are allowed. In Section 3 we recall the definition of reproducing kernel Pontryagin spaces and some related facts needed in the sequel.…”

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

“…Azizov, A. Dijksma, H. Langer and G. Wanjala (see [30][31][32][33][34]). Now we will sketch some recent developments on multivariable analogues of the Schur class in the unit disk.…”