Determinacy theory for the Livšic moments problem

Abstract

The Livtc moments problem as formulated by Gil de Lamadrid [6] is presented in terms of a given vector space F of complex functions, called test functions, defined on R = (-00, + co) and a given pseudo-inner product (nonnegative sesquilinear form) on F. The problem is to find a positive measure p on R which gives an integral representation of the pseudo-inner product. This problem includes the classical Hamburger moments problem and the Bochner theorem on positive definite functions on R. Livgic [IO] and other Russian authors [l, 81 have limited their discussion of this problem to the existence of a solution. On the other hand, even for the Hamburger problem, the solution is not in general unique. This has led to the classical determinacy theory for the Hamburger problem [l, 17, 181. This theory is concerned with the classification of solutions, the study of the structure of the set of solutions, the characterization of those problems which are determinate (have a unique solution), and the study of the supports of solutions. Recently, Gil de Lamadrid [6] initiated the study of the determinacy theory for the LivSic problem. His method, as that of LivZic, was based on Naimark’s theory of transcendental extensions (extensions to larger spaces) of a symmetric operator [12, 13, 14, 151. U 1 aimark proved the existence of transcendental extensions of a symmetric operator and analyzed the set of all such extensions. He pointed to the application of his theory to the moments problem by establishing a one-to-one correspondence between the set of solutions of the Hamburger problem and the set of equivalence classes of minimal self-adjoint extensions of the operator of multiplication by the independent variable which appears naturally in the problem. Later, LivEic used Naimark’s existence theorem to establish the existence of a solution of a