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

Abstract

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