Symmetric Linear Relations Research Articles

Spectral functions of a symmetric linear relation with a directing mapping, II

SynopsisThe results of part I (see [5]) are applied to pairs of formally symmetric differential expressions, to Hermitian differential systems and to a reduced operator moment problem.

Spectral functions of a symmetric linear relation with a directing mapping, I

SynopsisFor a symmetric linear relation S with a directing mapping, the notion of a spectral function is defined by means of a Bessel–Parseval inequality, and a description of all such spectral functions is given. As an application, we describe the set of all spectral functions of a canonical regular first order differential system.

L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations

Let S be a closed linear relation with equal defect numbers n ≤ ∞ in a Hilbert space. We prove that for each n-dimensional subspaceL of generalized elements, S has anL-resolvent matrix, and we give the characteristic properties of suchL-resolvent matrices. The results are illustrated at the example of a regular canonical differential relation, and it is shown that its generalized resolvents are given by certain boundary problems, containing the eigenvalue parameter in the boundary condition.

Extensions of isometric and symmetric linear relations in a Krein space

Extensions of isometric and symmetric linear relations in a Krein space

On generalized resolvents of symmetric linear relations in a Pontrjagin space

On generalized resolvents of symmetric linear relations in a Pontrjagin space

A generalization of M. G. Krein's method of directing functional to linear relations

SynopsisM. G. Krein's method of directing functional is generalized in a straight-forward way to symmetric linear relations. Applications to Stieltjes differential boundary problems are indicated.

On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space

In some problems related to the spectral theory in Hubert space it is more natural and at the same time often less restrictive to use symmetric linear relations (in the terminology of [1], subspaces in the terminology of [2-4]) instead of symmetric operators. Hence the question arises if the theory of generalized resolvents of symmetric operators can be extended to symmetric linear relations. In [4] a description of all generalized resolvents of a symmetric linear relation was given, following the lines of A. V. Straus [5] in the operator case. It is the aim of this paper to generalize M. G. Kreϊn's formula for the generalized resolvents of a symmetric operator (see [6, 7]) to the symmetric linear relation case. This can be done rather easily by means of the Gayley transformation, using the results of [8]. However, in this connection there arise natural problems and questions: To introduce and to study the Q-function of a linear relation, to prove criteria for the selfadjoint extension of the given symmetric linear relation being an operator, to study the special case of a bounded nondensely defined operator etc. After the necessary definitions and their simple consequences in §1, the §2 is devoted to a study of the Q-f unction. From arguments similar to those in [9, 10] it follows that every function Q, whose values are bounded operators in a Hubert space and which is holomorphic in the upper half plane and has the property