The Detectable Subspace for the Friedrichs Model

• 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
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-019-2548-9

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.