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.
Highlights
In recent articles [6, 8, 9] the authors have considered forward and inverse problems for operators in the boundary triples setting
We have been interested in the detectable subspaces (see (12) below) related to the Titchmarsh-Weyl functions M (λ), M (λ) associated with a formally adjoint pair, which determine upper bounds on the spaces in which the operators can be reconstructed, to some extent, from the information about boundary measurements contained in the Titchmarsh-Weyl functions
If the underlying operator is not symmetric, but the detectable subspace is the whole Hilbert space, the Titchmarsh-Weyl function determines the operators of an adjoint pair up to weak equivalence [24]
Summary
In recent articles [6, 8, 9] the authors have considered forward and inverse problems for operators in the boundary triples setting. We consider the case when the coefficients in the Hain-Lust-type operator are analytic In this case, some properties of the coefficients are uniquely determined by the Titchmarsh-Weyl coefficients (Theorem 4.1). Our considerations of first order Hain-Lust-type operators in Section 6 show that, in this simpler case, the operator is not uniquely determined by its Titchmarsh-Weyl coefficient. In terms of the detectable subspace, our results show that the first and second order results are very similar, so it seems plausible (Conjecture 4.3) that in the second order case with analytic coefficients, the Titchmarsh-Weyl coefficient does not uniquely determine the coefficients Further results can be found, e.g. in [7, 14, 16, 18,19,20,21, 27, 28]
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