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.


Introduction
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 problem of its determination is physically motivated, as it addresses the important inverse problem of determining the part of the physical model which We thank Marco Marletta for producing Figure 1. We are also very grateful to the referees whose careful reading of our earlier draft enabled us to make substantial improvements to the presentation. SN was supported by the Grant RScF 20-11-20032 and the Knut and Alice Wallenberg Foundation.
is 'visible' from boundary measurements. Therefore, it determines what partial information on the system is available in many standard experimental settings. Roughly speaking, the detectable subspace is the reducible subspace corresponding to the part of the operator which is 'visible' from the boundary behaviour of the solutions of the formal spectral equations (see the rigorous definition below). The analysis of its structure, in particular the question of detectability (the coincidence of the detectable subspace with the whole Hilbert space), is very important in the study of various types of inverse problems.
By the nature of the problem, finding the detectable subspace is very technically involved. Therefore, consideration of particular examples, such as the Friedrichs model considered here, is relevant and important in understanding the phenomena of 'visibility' or 'cloaking' in various fields such as in quantum mechanics, wave propagation theory and questions related to metamaterials and homogenisation.
Friedrichs model operators are perturbations of the multiplication operator by an integral operator. They were introduced by Friedrichs in 1948 [15] as a first rigorous model for scattering theory and enabled L.D. Faddeev's famous results on multi-particle Schrödinger operators [14]. Via the Fourier transform, Friedrichs model operators allow the study of Schrödinger operators with socalled separable potentials, see [11] for a detailed discussion of the scattering theory for such operators. Moreover, according to L.D. Faddeev, they provide a simple model for renormalisation theory in physics.
In the previous papers [7,9,10], the analysis of Friedrichs model operators showed how complicated the structure of the detectable subspace may be, even in the seemingly simple case of a rank one perturbation. Therefore, considering various special cases for the perturbation can provide an important step in understanding the nature of the whole problem. In [10], the technique of Hankel operators was used for the analysis of the detectable subspace under special conditions on the rank one perturbation in terms of Hardy classes [20]. In the current paper, we consider another wide class of perturbations, again of rank one, where the theory of Toeplitz operators appears as the main tool. We stress that the paper does not add any new results in the theory of Toeplitz operators, but shows a new application of the theory to the detectable subspace problem for a class of operators.
The paper is organised as follows. Section 2 contains a collection of basic facts on the generalised Weyl-Titchmarsh function (or Dirichlet-to-Neumann map), the abstract boundary triple approach and the definition of the detectable subspaces. Section 3 briefly discusses the main properties of the Fried richs model, including its boundary triple set-up. The main results are presented in Sect. 4, where, using the canonical factorisation of a Toeplitz operator's symbol, the indices (the codimensions of the detectable subspaces) are explicitly calculated (Theorem 4.7). The section also contains a few examples illustrating the main theorems.

Background: Boundary Triples and the Detectable Subspace
For the convenience of the reader and to keep this article as self-contained as reasonably possible, in this section and the next, we give an introduction to concepts and notation that will be used throughout the article. We make the following basic assumptions.
(1) A, A are closed, densely defined operators in a Hilbert space H.
(2) A and A are an adjoint pair, i.e. A * ⊇ A and A * ⊇ A.
Then, see [21], there exist 'boundary spaces' H, K and 'trace operators' such that for u ∈ Dom ( A * ) and v ∈ Dom (A * ) we have an abstract Green formula The trace operators Γ 1 , Γ 2 , Γ 1 and Γ 2 are bounded with respect to the graph norm. The pair ( The collection {H ⊕ K, (Γ 1 , Γ 2 ), ( Γ 1 , Γ 2 )} is called a boundary triple for the adjoint pair A, A.
We next define Weyl M -functions associated with boundary triples (see, e.g. [5,12]). Given bounded linear operators B ∈ L(K, H) and B ∈ L(H, K), consider extensions of A and A (respectively) given by In the following, we assume the resolvent set ρ( and for λ ∈ ρ( A B ), we define For λ ∈ ρ(A B ), the linear operator S λ,B : Ran (Γ 1 − BΓ 2 ) → ker( A * − λ) given by is called the solution operator. For λ ∈ ρ( A * B ), we similarly define the linear operator S λ,B * : We are now ready to define one of the main concepts of the paper, the detectable subspaces, introduced in [7]. Fix μ 0 ∈ σ(A B ). Then, define the spaces and similarly the 'adjoint' pair of linear sets Remark 2.2. In many cases of the Friedrichs model, we will be considering, the spaces S B and T B coincide and are independent of B. This follows from [7] or [9, Proposition 2.9]. To avoid cumbersome notation, in many places we shall denote all these spaces by S. We will refer to S as the detectable subspace.
In [7,Lemma 3.4], it is shown that S is a regular invariant subspace of the resolvent of the operator A B : that is, From (2.4) and [5, Proposition 3.9], we get that the orthogonal complement of S is

Basic Properties of the Friedrichs Model
In this section, we briefly introduce the Friedrichs model and collect some results. More details and proofs of the results can be found in [7]. Let φ, ψ be in L 2 (R). We consider in L 2 (R) the operator A with domain given by the expression Observe that since the constant function 1 does not lie in L 2 (R) the domain of A is dense. The adjoint of A is given on the domain by the formula Note that Dom (A) ⊆ Dom (A * ) and that c f = 0 for f ∈ Dom (A). We introduce an operator A in which the roles of φ and ψ are exchanged: We immediately see that Dom ( A * ) = Dom (A * ) and that dx is uniquely determined, we can define trace operators Γ 1 and Γ 2 on Dom (A * ) as follows: (3.6) Note that the limit in (3.6) always exists and that which is the expression used in [7]. Then, moreover, the following Green's formula holds showing that we have constructed a boundary triple for the pair A, A.
We can now determine the M -function and the resolvent. Suppose that Here, the perturbation determinant D is the function Moreover, the Weyl-function M B (λ) is given by the scalar function For the resolvent, we have that ( (3.12) in which the coefficient c f is given by There is another approach to the Friedrichs model via the Fourier transform, which turns the perturbed multiplication operator into a rank one perturbation of a first-order differential operator with an appropriate matching condition at 0. See [10] for more details.

Spectra of Toeplitz Operators and Detectability
In our previous paper on detectable subspaces for Friedrichs model operators [10], Hankel operators played a special role in the analysis of determining the detectable subspace. However, for another class of examples of the Friedrichs model, the theory of Toeplitz operators is the main instrument of our analysis. We will discuss this type of examples and the related detectability problem here. We first introduce some notation. Let H + p (C + ) and H − p (C − ), 1 ≤ p ≤ ∞, denote the Hardy spaces of p-integrable functions, analytic in the upper and lower half-plane, respectively, where the norm is given by Functions in the Hardy spaces can be identified with their boundary values on the real line which lie in L p (R) (see [20] for more details). In what follows we will use this identification without further comment and denote the Hardy spaces simply by H + p and H − p . The operators P ± : L 2 (R) → H ± 2 given by are the Riesz projections where the limit is to be understood in L 2 (R) (see [20]). We next give a characterisation of the space S, or, more precisely, its orthogonal complement for the Friedrichs model which is taken from [9, Proposition 7.2]. The proof is based on the definition of S using (2.5) and the results from Sect. 3. Proposition 4.1. Let P ± be the Riesz projections defined in (4.1) and D(λ) be as in (3.10). Denote by D ± (λ) its restriction to C ± and by D ± the boundary values of these functions on R (which exist a.e., cf. [20,24]).
if and only if any of the following three equivalent conditions holds:
Throughout this section, we will make the following assumptions. (2) From (i), it follows from the Uniqueness Theorem [20,24] that both φ and ψ are nonzero a.e. on R. (3) Under the above assumptions, we have D + (λ) ≡ 1 and P + φ = φ, P − φ = 0. In particular, from (3.12), we get that the corresponding Friedrichs model operator has no spectrum in C + . (4) The majority of functions a ∈ H − 1 can be decomposed as in assumption (i). To see this, choose φ = (x − i) − 1 2 −ε for some ε > 0 and a suitable choice of the branch cut. Then, we have φ ∈ H − 2 . To satisfy the first assumption, we then require ψ(x) := a(x)(x + i) 1 2 +ε ∈ L 2 (R), or a ∈ L 2 (R; (1+x 2 ) 1 2 +ε ). Therefore, the assumption that φ ∈ H − 2 only imposes a mild extra condition on the decay of a at infinity. (5) The third assumption is only introduced for the sake of convenience and may be significantly relaxed. However, this would introduce more technical details which would obscure the main results. It means that a ∈ H − ∞ , that the operator T is a bounded operator in L 2 (R) and T a in H + 2 , in particular, the operators are defined on the whole space. (6) Under the first assumption, we can express the operator T a by T a u = P + (aP + u).
We will now analyse the spectral properties of the operators T and T a ; by σ p we denote the set of eigenvalues of an operator. The proofs of parts (1)-(4) are very similar to the corresponding proofs for Toeplitz operators, e.g. in [4].
since the first term acted on by P + is analytic in C − , in L 2 (R) and is easily seen to lie in H − 2 , while the second is in H + 2 since z 1 ∈ C − . Proof of (2): We first consider μ = 0. Choosing g = φh for h ∈ H − 2 , we get T g = φP + ψφh = 0, since ψφh ∈ H − 2 , due to a ∈ H − ∞ . Hence, all functions in φH − 2 are eigenfunctions to the eigenvalue 0. Now let μ = 0 and assume that (a − μ) is an outer function in C − (see, e.g. [20] for the definition). We use that if f ∈ H − ∞ , then f is outer if and only if the closure of the set fH − 2 is H − 2 (Beurling Theorem, see [20]) and that the functions (k − z 0 ) −1 for z 0 ∈ C + span H − 2 . Therefore, Now assume there exists g ∈ L 2 (R) \ {0} with T g = μg and set h = ψg. Then, This implies P + h = 0, and since μ = 0 we get P − h = 0, so h = 0. As ψ is nonzero a.e., we have g = 0, so μ is not an eigenvalue of T . Next let μ = 0 and assume that (a − μ) is not an outer function in C − . This implies that there − μ)h) = 0 and P + (ah) = μP + h = μh (as h ∈ H + 2 ). This implies that T (φh) = φP + (ψφh) = μφh. As φ ∈ L ∞ (R), φh ∈ L 2 (R) and it is not identically zero (as φ ≡ 0 and h is nonzero a.e. by the uniqueness theorem, see [20]), so μ ∈ σ p (T ) with the nonzero eigenfunction φh.
Proof of (3): We first note that as a ∈ H − 1 we have that μ ∈ Ran z∈C− a(z) implies μ = 0 and that inf z∈C− |a(z) − μ| > 0. We want to calculate the resolvent of T at μ. Consider (T − μ)g = v with v ∈ L 2 (R). Since ψ = 0 a.e. and (a − μ)| R is invertible we get (all equalities hold a.e.) Note that, as μ a−μ ∈ H − ∞ , the last term lies in H − 2 . Applying P + and P − to (4.7), we get Thus, Formally, we have for arbitrary v ∈ L 2 (R) that Since φ, ψ ∈ L ∞ (R), the operator defined by the r.h.s. is bounded in L 2 (R).
Checking the formal calculation of the resolvent for any v ∈ L 2 (R), Therefore, the r.h.s. of (4.8) is the right inverse of the operator (T − μ). Similarly we see that the same expression gives the left inverse, . Now, (4) follows from (1) and (3), as σ(T ) ⊆ Ran z∈C− a(z) ⊆ σ p (T ) ⊆ σ(T ), so all three sets must coincide.
Proof of (5): We again solve T g = μg. As ψ = 0 a.e., this is equivalent to aP + (ψg = μψg. Setting h = ψg this gives aP + h = μh. Note that if h ∈ L 2 (R) and μ = 0 with aP + h = μh, then h = ψ φP+h μ ∈ ψL 2 (R), so g = h/ψ ∈ L 2 (R). Thus, for nonzero μ, T g = μg is equivalent to aP + h = μh. This reduces the problem to considering Toeplitz operators: . Thus, P + h determines P − h uniquely and we only need to consider the first equation in H + 2 which shows equality of the point spectra of T and T a away from 0.
Proof of (6): The existence of such a non-trivial h is then equivalent to a not being an outer function in C − . To determine Ran z∈C− a(z), we consider a(t), t ∈ R and take the inside of the curve (Fig. 1). Let We first check that all nonzero points inside the inner curve are in the range. If y = 0 and x is small and negative (so 4 √ so, e.g. for m = 2, z = i + 1 Therefore, the boundary of the set Ran z∈C− a(z) consists of the outer curve (|t| ≤ 1) together with the isolated point 0 (corresponding to t = ∞). For these μ, the function (a − μ) is outer in C − . For μ = 0 this is clear. For all other such μ, (a − μ) takes values outside a cone. This implies that (a − μ) k is outer for some sufficiently small positive k which implies that (a − μ) is outer, since any Herglotz function (i.e. analytic functions on C + with positive imaginary part) is outer.
We finally consider the behaviour at t = ±1 and t = ±∞. As t → ±∞, 1 4 , and using symmetry of the range of a w.r.t. complex conjugation, we get a cone of angle π/2 at this point.
Note that the number of roots of B α (z) is counted with multiplicity and may be infinite.
We consider the equation pointwise and multiply by ψ. Setting h = ψg, we get h ∈ L 2 (R) and (4.11) By virtue of (4.11), the fact that a is divisible by ψ and μ α = 0, h/ψ = g ∈ L 2 (R). Now, using h = P + h + P − h, we find (a − μ α )P + h = μ α P − h. otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.
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