The extension theory of Hermitian operators and the moment problem. (English) Zbl 0848.47004

Derkach, V. A.; Malamud, M. M.

J. Math. Sci., New York 73, No. 2, 141-242 (1995).

This paper deals with the extension theory of a nondensely defined Hermitian operator in a Hilbert space. Generalized resolvents, preresolvent and resolvent matrices of such an operator are investigated. Necessary and sufficient conditions for a holomorphic operator-valued function to be a preresolvent or resolvent matrix of a Hermitian operator are found. A criterion for an \(R\)-function to be the Weyl function of a Hermitian operator is given. Applications to describe solutions of the truncated Hamburger, Stieltjes and Hausdorff moment problems are given. The results of this paper were partially announced in previous papers of the authors [Dokl. Akad. Nauk Ukr., Ser. A 11, 34-39 (1991); Dokl. Akad Nauk 323, No. 5, 816-822 (1992; Zbl 0820.47007); Dokl. Akad. Nauk 326, No. 1, 12-18 (1992)].

Reviewer: Wu Jingbo (Tianjin)

MSC:

47A20	Dilations, extensions, compressions of linear operators
47A57	Linear operator methods in interpolation, moment and extension problems
47A56	Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

generalized resolvents; extension theory; nondensely defined Hermitian operator; preresolvent; resolvent matrices; holomorphic operator-valued function; resolvent matrix of a Hermitian operator; \(R\)-function; Weyl function; truncated Hamburger, Stieltjes and Hausdorff moment problems

Citations:

Zbl 0820.47007

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