Beurling-Lax type theorems and Cuntz relations

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 633; pp. 152 - 212
Main Authors: Alpay, Daniel, Colombo, Fabrizio, Sabadini, Irene, Schneider, Baruch
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 15.01.2022
American Elsevier Company, Inc
Subjects:
Analytic functions
Backward-shift operator
Beurling-Lax theorem
Cuntz relations
de Branges-Rovnyak spaces
Linear algebra
Mathematical analysis
Multiplication
Operators (mathematics)
Rational functions
Structure theorems
Theorems
30C10
Beurling-Lax theorem
Structure theorems
Cuntz relations
Rational functions
47B32
de Branges-Rovnyak spaces
Backward-shift operator
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
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ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2021.10.008