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

The aim of this paper is to develop the conditions for a symmetric relation in a Hilbert space ？ to have self-adjoint extensions in terms of defect indices and discuss some spectral theory of such linear relation. 1. 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 [1–4]. 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 be a Hilbert space over and denote by the Hilbert space . A linear relation on is a subspace of . 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 and is a 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]. and are respectively defined as the domain and range of the relation . denotes the inverse relation. The adjoint of on is a closed linear relation defined by A linear relation is called symmetric if and self-adjoint if . From now on, we write relation to mean linear relation. A relation is called isometry if and is unitary if it is isometry and . Let and . It is clear to see that . 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. 2. Defect Indices and Self-Adjoint Extension Let be a relation on a Hilbert space . The set is defined as the regularity domain of and is defined as the Spectral