The detectable subspace for the Friedrichs model

TL;DR

This paper explores the concept of detectable subspaces in the Friedrichs model using boundary measurements, linking complex analysis and operator theory to understand what information can be retrieved from boundary data.

Contribution

It introduces the generalised Titchmarsh-Weyl M-function and characterizes detectable subspaces for the Friedrichs model, advancing the theory of boundary measurement analysis.

Findings

Explicit description of detectable subspaces in many cases

Demonstration of the interplay between complex analysis and operator theory

Development of a framework applicable to perturbation problems

Abstract

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.