Krein-Langer Factorizations via Pole Triples

Abstract

A theorem by Krein and Langer asserts existence of factorizations of special type for operator functions in a generalized Schur class, i. e., meromorphic operator functions defined on the unit disk and such that their Nevanlinna-Pick kernel has a fixed finite number of negative squares. A different view and proof of this theorem are presented, based on description of pole data of meromorphic operator functions in terms of pole pairs and pole triples. A criterion for existence, and a parametrization, of operator functions in a generalized Schur class with given pole triple is obtained. (2000). Primary 47A56. Mathematics Subject Classification

Jet functions having indefinite Carathéodory–Pick matrices

“…The following known result [27], see also [6,15] for operator valued functions, gives a characterization of meromorphic functions in S κ ( ). …”

“…Recall that a Blaschke product (all Blaschke products in this paper are assumed to be finite) is a rational function B(z) that is analytic on D and unimodular on the unit circle T : |B(z)| = 1 for |z| = 1; the degree of B(z) is the number of zeros (counted with multiplicities) of B(z) in D. See [5,6,15,22,27] for various proofs of matrix and operator-valued versions of Theorem 1.2.…”

Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

“…Krein in the 1970ies; see for instance [23,24,25,26,27]. The structure of these functions can be found in the papers [16,20] (see also [12], and for a constructive proof in the scalar case, see [11]), and the structure of the corresponding generalized Schur functions can be found in [23].…”

The Bitangential Matrix Nevanlinna–Pick Interpolation Problem Revisited

“…There is also an intrinsic characterization of matrix triples (C, A, B) which can arise as the pole triple over the unit disk for a generalized Schur class function-see [17] for details.…”