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On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization. (English) Zbl 0594.46022

I. Schur methods in operator theory and signal processing, Oper. Theory, Adv. Appl. 18, 89-159 (1986).
[For the entire collection see Zbl 0583.00020.]
The main theme of the first half of this paper rests upon the fact that there is a reproducing kernel Hilbert space of vector valued functions B(X) associated with each suitably restricted matrix valued analytic function X. The deep structural properties of certain classes of these spaces, and the theory of isometric and contractive inclusion of pairs of such spaces, which originates with de Branges, partially in collaboration with Rovnyak, is utilized to develop an algorithm for constructing a nested sequence B(X)\(\supset B(X_ 1)\supset..\). of such spaces, each of which is included isometrically in its predecessor. This leads to a new and pleasing viewpoint of the Schur algorithm and various matrix generalizations thereof. The same methods are used to reinterpret the factorization of rational J inner matrices and a number of related issues, from the point of view of isometric inclusion of certain associated sequences of reproducing kernel Hilbert spaces.
The second half of this paper begins by extending a part of the theory of B(X) spaces alluded to above to the more general setting of reproducing kernel Pontryagin spaces, i.e., B(X) is now permitted to have a finite number of ”negative squares”. This extended theory is then used to develop an algorithm for constructing nested sequences of reproducing kernel Pontryagin spaces which are again ordered by isometric inclusion, as well as to study the problem of factoring rational matrices which are J unitary on either the unit circle or the line.

MSC:

46E20 Hilbert spaces of continuous, differentiable or analytic functions
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)

Citations:

Zbl 0583.00020