The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used 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’. In this paper we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz-Nevanlinna factorisation of the symbol of a related Toeplitz operator.