An Alternate Proof of De Branges Theorem on Canonical Systems

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on C. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H(x)=1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.


Introduction
This paper deals with the canonical systems of the following form: Here = ( 0 −1 1 0 ) and ( ) is a 2 × 2 positive semidefinite matrix whose entries are locally integrable. For fixed ∈ C, a function (⋅, ) : [0, ] → C 2 is called a solution if is absolutely continuous and satisfies (1). Consider the Hilbert space as follows: provided with an inner product ⟨ , ⟩ = ∫ ∞ 0 ( ) * ( ) ( ) . The canonical systems (1) on 2 ( , R + ) have been studied by Hassi et al., Winkler, and Remling in [1][2][3][4] in various contexts. The Jacobi and Schrödinger equations can be written into canonical systems with appropriate choice of ( ). In addition, canonical systems are closely connected with the theory of the De Branges spaces and the inverse spectral theory of one-dimensional Schrödinger operators; see [3]. We believe that the extensions of the theories from these equations to the canonical systems are to be of general interest.
If the system (1) can be written in the form of then we may consider this as an eigenvalue equation of an operator on 2 ( , + ). But ( ) is not invertible in general. Instead, the system (1) induces a linear relation that may have a multivalued part. Therefore, we consider this as an eigenvalue problem of a linear relation induced by (1) on 2 ( , + ).
For some ∈ C, if the canonical system (1) has all solutions in 2 ( , R + ), we say that the system is in the limit circle case, and if the system has unique solution in 2 ( , R + ), we say that the system is in limit point case. The basic results in this paper are the following theorems. Theorem 1. In the limit circle case, the defect index (R 0 ) of the symmetric relation R 0 , induced by (1), is constant on C.
The immediate consequence of the Theorem 1 is the following theorem.
Theorem 2 (De Branges). The canonical systems with tr ≡ 1 prevail the limit point case.
Theorem 2 has been proved in [5] by function theoretic approach. However the proof was not easily readable to me and we thought of providing an alternate and simple proof of the theorem.
In order to prove the main theorems we use the results from the papers [1,3,4] and we use the spectral theory of a linear relation from [6].
A linear relation S is called symmetric if S ⊂ S * and selfadjoint if S = S * . The theory of such relations can be found in [5][6][7][8]. The regularity domain of R is the following set: The following theorem has been derived from [6].
The defect index (R, ) is the dimension of defect space: It has been shown in [6] that the defect index (R, ) is constant on each connected subset of Γ(R). Moreover, if R is symmetric, then the defect index is constant in the upper and lower half-planes. In addition, it is worth mentioning here the following theorem from [6] which provides us with the condition for a symmetric relation on a Hilbert space to have self-adjoint extension. The resolvent set for a closed relation R is the following set: and the spectrum of R is (R) = C − (R). We call (R) = C − Γ(R) the spectral kernel of R. The following theorem from [6] shows the relation between the spectral kernel and spectrum of a self-adjoint relation. In the next section we discuss the linear relation induced by a canonical system and prove our main theorems.

Relation Induced by a Canonical System on 2 ( ,R + ) and Proof of the Main Theorems
Consider that a relation R in the Hilbert space 2 ( , R + ) is induced by (1) as and is called the maximal relation. This relation is made up of pairs of equivalence classes ( , ), such that there exists a locally absolutely continuous representative of again denoted by and a representative of , again denoted by , such that = a.e. on R + . The adjoint relation R 0 = R * is defined by and is called the minimal relation. It has been shown in [1] that R 0 is close and symmetric. Moreover, R 0 ⊂ R and (R 0 ) * = R.
Lemma 7 (see [1]). Let ( , ), (ℎ, ) ∈ R. Then the following limit exists: Lemma 8. The minimal relation R 0 is given by Proof. By Lemma 7, we get On the other hand, let ( , ) ∈ 0 . By Lemma 6 for any ∈ C 2 there exists ( , ) ∈ R such that has compact support and (0+) = . So This implies that (0+) = 0. This would also force the following: Note that the dimension of the solution space of the system (1) is two.
Remark 9. The defect index (R 0 ) of the minimal relation R 0 is equal to the number of linearly independent solutions of the system (1) whose class lies in 2 ( , R + ). Therefore, in the limit circle case, the defect indices of R 0 are (2, 2).
Since R 0 has equal defect indices, by Theorem 4, it has self-adjoint extensions say T. Consider a relation as follows: on a compact interval [0, ].
Lemma 10. T , is a self-adjoint relation.
Proof. Clearly T , is a symmetric relation because of the boundary conditions at 0 and . We will show that T , is a 2-dimensional extension of R 0 . Then by Theorem 4, T , is a self-adjoint relation. By Lemma 6, for = ( − cos sin ) and = ( For ∈ + there is a unique ( ) such that ( , ) = ( , )+ ( )V( , ) satisfying the following: This is well defined because does not satisfy the boundary condition at ; otherwise, will be an eigenvalue of some self-adjoint relation T , . Next, we describe the spectrum of T , . Let and define It is not hard to see that.
Lemma 11. Using the notation above one has is a bounded linear operator and is defined by where Proof. Let ( , ) = ∫ 0 ( , , ) ( )ℎ( ) . We show that ( , ) solves the inhomogeneous equation as follows: for a.e. > 0. Here The kernel is square integrable since So L is a Hilbert-Schmidt operator and thus compact. Notice that ( , ) = * ( , ). This implies that L is self-adjoint.
This means that solves Conversely suppose ∈ 2 ( , [0, ]) and solves That is, ( , ) ∈ (T , − ) is unitarily equivalent to L⇂ ( ) ; that is, has only discrete spectrum consisting of only eigenvalues. Then, by Theorem 5, T , has only discrete spectrum. By Lemma 14, the spectrum of T , consists only of eigenvalues. Hence we have We would like to extend this idea over the half line R + . First note that we are considering the limit circle case of the system (1). That implies that for any ∈ C + the defect indices of R 0 are (2, 2). Suppose ∈ (R) \ (R 0 ) such that lim → ∞ ( ) * ( ) = 0. Such function clearly exists.
Consider the following relation: Hence T , is a self-adjoint relation.
Define an integral operator L on 2 ( , R + ) by Then as before the kernel is square integrable. This means that Hence L is a Hilbert Schmidt operator and so it is a compact operator. The following two lemmas are extended from the bounded interval [0, ] to R + and the proofs are exactly the same as the proofs of Lemmas 13 and 14.
(1) L = −1 . With these theories in hand, we are now ready to prove the main theorems.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.