Some Hilbert spaces of analytic functions. III

Abstract

This paper continues the study of Hilbert spaces of analytic functions which are involved in the structural analysis of nonself-adjoint transformations in Hilbert space. The theory of nonself-adjoint transformations originates in the quantum mechanical problems of nuclear scattering theory. Although the ordinary use of the Schriidinger equation leads to self-adjoint transformations, it is customary for physicists to subdivide the nuclear reaction in such a way that nonself-adjoint transformations occur. This observation, made by Livgic [l], caused him to found a general theory of nonself-adjoint transformations. Although LivBic’s theory is extensive and powerful, it has received very little recognition because no one, apparently, can follow his arguments. The difficulties are so great that Dolph and Penzlin [2] have attempted an independent derivation of the main results. The trouble is due not so much to logical gaps as it is to insufficient motivation of the main tool, the characteristic operator function. This quantity arises in the description of fundamental solutions of formally self-adjoint differential equations under variations of the boundary conditions. Yet the characteristic operator function is applied to transformations which have no connection with differential equations. The trouble is that LivGc has missed the meaning of the characteristic operator function, which is to be found in the construction of certain Hilbert spaces of analytic functions [3]. To explain how these spaces originate, we must go back to the basic work of Stone [4] and to our previous work with entire functions [5-81. Stone’s book has two different objectives, apart from a general formulation of concepts. The first is the study of self-adjoint transformations. It is important to note how he goes about the study of these transformations. From his point of view the structure theorem is an abstract analogue of the integral representation of functions which are analytic and have a nonnegative real part in the upper half-plane. (See, for example, Nevanlinna and Nieminen [9] for a more detailed reconstruction of the same argument.) It has