Some Aspects of Operator Theory and the Theory of Analytic Functions

Abstract

This chapter consists of four sections. Some aspects of the theory of closed Hermi-tian operators in a Hubert space and their self-adjoint extensions within the given space, as well as with exit to a larger one, which are needed in what follows, are presented (without proof) in Section 1. Section 2 is devoted to the foundations of the representation theory of Hermitian operators whose deficiency index is (1,1). The main result consists of the isomorphic identification of the space where the operator acts, with a certain space of functions meromorphic inside the upper and lower half-planes. The operator itself is transformed into multiplication by the independent variable under this isomorphism. The results of Section 3 concern the structure of a spectrum of self-adjoint extensions within the original space of a simple Hermitian operator with defect numbers equal to 1. A special class of analytic functions which is of great importance for constructing the theory of entire operators is investigated in Section 4. This is the class of the so-called N -functions. The necessary and sufficient conditions for a function analytic in a disk or in the upper (lower) half-plane to be an N -function are given. The most important result of the section is the criterion for an entire function to belong to the class of N -functions in the upper (lower) half-plane. It is also established that the indicator diagram of such a function coincides with a certain interval of the imaginary axis. The asymptotic distribution of its zeroes is found on this basis.