Detectable subspaces and inverse problems for Hain–Lüst‐type operators

Abstract

ABSTRACT. We examine the extent to which a block operator matrix of Hain-Lüst type can be reconstructed from its Titchmarsh-Weyl coefficients. The detectable subspace of the operator is determined in a variety of cases and the question of unique determination of the coefficients is considered for both first and second order operators.

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

“…This paper continues the study of the so-called detectable subspace of an operator in Hilbert space started in [7][8][9][10]. The detectable subspace of a (generally) non-self-adjoint operator is defined in terms of boundary triples [5,12,21].…”

The Detectable Subspace for the Friedrichs Model

“…In this paper, we determine detectable subspaces [4,6,7] -associated with the part of the operator which is 'accessible from boundary measurements' -for the so-called Friedrichs model. The Friedrichs model is a toy model, first introduced in [9], and used frequently in the study of perturbation problems (see e.g.…”

“…It is therefore instructive to look at particular examples to see what information may be determined from the Titchmarsh-Weyl functions. In earlier articles, the authors have considered this question for certain types of matrix-differential operators [6] and looked at very simple cases of the so-called Friedrichs model [7]. Improving the result on weak equivalence in some special cases is the topic of [1,2,3,10].…”

Limit‐point/limit‐circle classification for Hain–Lüst type equations

“…Results on the essential spectrum can be found in [1,4,8,9,12,13,19,25,29,35,36,43]; see also [47]. Titchmarsh-Weyl theory for second-order differential equations depending rationally on the eigenvalue parameter was considered in [1,3,6,13,28,31,46], where apart from [3] the denominator of the rational coefficients depends linearly on the eigenvalue parameter, that is, the problem is of the form (1.2). The simplest form of such a 2 × 2 block system of differential operators is when the system itself is 2 × 2, and if such a system is additionally formally selfadjoint, then it is of the form (1.1).…”