The Detectable Subspace for the Friedrichs Model

This paper discusses how much information on a Friedrichs model operator can be detected from ‘measurements on the boundary’. We use the framework of boundary triples to introduce the generalised Titchmarsh–Weyl M -function and the detectable subspaces which are associated with the part of the opera...

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Bibliographic Details
Published in:Integral equations and operator theory 2019-10, Vol.91 (5), p.1-26, Article 49
Main Authors: Brown, B. M., Marletta, M., Naboko, S., Wood, I. G.
Format: Article
Language:English
Subjects:
Analysis
Independent variables
Mathematics
Mathematics and Statistics
Multiplication
Perturbation methods
Subspaces
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Summary:This paper discusses how much information on a Friedrichs model operator can be detected from ‘measurements on the boundary’. We use the framework of boundary triples to introduce the generalised Titchmarsh–Weyl M -function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. The Friedrichs model, a finite rank perturbation of the operator of multiplication by the independent variable, is a toy model that is used frequently in the study of perturbation problems. We view the Friedrichs model as a key example for the development of the theory of detectable subspaces, because it is sufficiently simple to allow a precise description of the structure of the detectable subspace in many cases, while still exhibiting a variety of behaviours. The results also demonstrate an interesting interplay between modern complex analysis, such as the theory of Hankel operators, and operator theory.
ISSN:0378-620X
1420-8989
DOI:10.1007/s00020-019-2548-9