Self-Adjoint Extension and Spectral Theory of a Linear Relation in a Hilbert Space

In this paper, we discuss the theory of linear relations in a Hilbert space. These linear relations were first studied by Arnes, Coddington, Dijksma, de Snoo, and Hassi et al in [1–4]. It has also been studied extensively more recently in [5]. The theory has particular interest because, in some of the application problems, a linear operator can have multivalued part; for example, see [6, 7]. Here, we concentrate on establishing the conditions for symmetric relations to have self-adjoint extensions in terms of defect indices. Moreover, we discuss the spectral theory of such self-adjoint relations. The analogous treatment on operator theory of some of the theorems on this paper can be found in [8]. Let H be a Hilbert space over C and denote by H the Hilbert space H ⊕ H. A linear relation R = {(f, g) : f, g ∈ H} onH is a subspace ofH. The graph of an operator is an example of a linear relation but note that a relation can have multivalued part. These relations have been used in some of the eigenvalue problems in ordinary differential equations. For example, the canonical systems


Introduction
In this paper, we discuss the theory of linear relations in a Hilbert space. These linear relations were first studied by Arnes, Coddington, Dijksma, de Snoo, and Hassi et al in [1][2][3][4]. It has also been studied extensively more recently in [5]. The theory has particular interest because, in some of the application problems, a linear operator can have multivalued part; for example, see [6,7]. Here, we concentrate on establishing the conditions for symmetric relations to have self-adjoint extensions in terms of defect indices. Moreover, we discuss the spectral theory of such self-adjoint relations. The analogous treatment on operator theory of some of the theorems on this paper can be found in [8].
Let H be a Hilbert space over C and denote by H 2 the Hilbert space H ⊕ H. A linear relation R = {( , ) : , ∈ H} on H is a subspace of H 2 . The graph of an operator is an example of a linear relation but note that a relation can have multivalued part.
These relations have been used in some of the eigenvalue problems in ordinary differential equations. For example, the canonical systems where = ( 0 −1 1 0 ) and ( ) is a 2 × 2 positive semidefinite matrix whose entries are locally integrable, induce a multivalued linear relation. For instance, we may think of writing the systems in the form and consider it as an operator on a Hilbert space. But ( ) is not invertible in general therefore can not be considered as an eigenvalue equation of an operator. Instead, the system induces a linear relation that may have a multivalued part. This is one of the main motivations for our work in this paper. The boundary value problem of such canonical systems has been studied by using linear relations; see [6,7,9].
A linear relation S is called symmetric if S ⊂ S * and selfadjoint if S = S * . From now on, we write relation to mean linear relation.
A relation R is called isometry if and It is clear to see that (R * , ) = R ⊥ . In Section 2, we establish the condition for a symmetric relation to have self-adjoint extensions in terms of defect indices and in Section 3 we discuss spectral theory. (ii) if R is symmetric, then C − R ⊂ Γ(R);

Defect Indices and Self-Adjoint Extension
The subspace R ⊥ is called the defect space of R and . The cardinal number (R, ) = dim R ⊥ is called the defect index of R and .

Theorem 1. The defect index (R, ) is constant on each connected subset of Γ(R). If R is symmetric, then the defect index is constant on the upper and lower half-planes.
Proof. Let denote the orthogonal projection onto R . We for all ( , ) ∈ R. For | − 0 | < 1/(2 ( 0 )) and all ( , ) ∈ R, we have For ℎ ∈ ( Similarly, for ℎ ∈ ( ) ⊥ = R ⊥ , It follows that Let denote an orthogonal projection onto R ⊥ ; then Hence, if we choose 0 < < 1/(2 ( 0 )), then ‖ − 0 ‖ < 1, If R is symmetric, then the upper and lower half-planes are connected subsets of Γ(R); therefore, the defect index is constant there.
Letting R be a symmetric relation on a Hilbert space, for ∈ C + , the defect index = (R, ) and, for ∈ C − , the defect index = (R, ) are written as a pair ( , ) and are called the defect indices of R.
The Cayley transform of a symmetric relation R on H is defined by the relation Then clearly (V) = (R + ) and (V) = (R − ).

Theorem 2. If R is a symmetric relation on H and V is the Cayley transform of R, then,
(1) V is isometry; (2) It follows from the definition of Cayley transform.

Theorem 4. A symmetric relation R is self-adjoint if and only if V is unitary.
Proof. We show that the R is self-adjoint if and only if Since R is symmetric, we always have R ⊂ R * . Let ( , ) ∈ R * ; then − ∈ H and (R − ) = H imply that there is (ℎ, ) ∈ R such that − ℎ = − . So ( + ℎ) = + , so that ( + ℎ, ( + ℎ)) ∈ R * . That is This implies = −ℎ ∈ (R). Hence, R is self-adjoint. Conversely, suppose that R is self-adjoint. Let    Proof. (1) Suppose that V has the given form. Then V is isometric relation, since for any ( + ℎ, + ) ∈ V , we have Hence, we can define a symmetric extension R such that V is its (25) Theorem 6. Let R be a closed symmetric relation on a Hilbert space with defect indices ( , ). One has the following.
(1) R is a symmetric extension of R if and only if the following holds. There are closed subspaces + of ( + R) ⊥ and − of ( − R) ⊥ and an isometric mappingV of + onto − such that Proof.
(1) Let V and V be the Cayley transforms of the closed symmetric relation R and its symmetric extension R , respectively. By Theorem 5, there exist closed subspaces − of (R − ) ⊥ and + of (R + ) ⊥ and an isometric relationṼ on + ⊕ − for which

ISRN Mathematical Analysis
Then by definition of the Cayley transform, we see that The converse is similar. ( and T is given by

Spectral Theory of a Linear Relation
Definition 8. Let R be a closed relation on a Hilbert space H. We define (R) = { ∈ C : ∃ ∈ (H) , to be the resolvent set and (R) = C − (R) to be the spectrum of R.
Remark 9. When a relation R is an operator on H, then A complex number ∈ C is called an eigenvalue of a relation R if there exists ∈ H, ̸ = 0 such that ( , ) ∈ R. The set of all eigenvalues of R is called the point spectrum of R and is denoted by (R).