Generalized resolvents and the boundary value problems for Hermitian operators with gaps. (English) Zbl 0748.47004

Let \(A\) be a Hermitian operator with gaps \((\alpha_ j,\beta_ j)\), \(j=1,2,\dots,m\). The main result of the paper (Section 8) is the description of all self-adjoint extensions of \(A\) putting exactly \(k_ j<+\infty\) eigenvalues into the gap \((\alpha_ j,\beta_ j)\), \(j=1,2,\dots,m\). In particular, for \(k_ j=0\), \(1,2,\dots,m\), this answered a question which was posed by M. G. Krein in 1945. The solution of the problem is given in terms of the so-called Weyl function which is an important object in the extension theory of Hermitian operators in the framework of abstract boundary conditions.
In Section 1 the Weyl function is introduced and its relation to the \(\mathcal D\)-function of M. G. Krein and the characteristic function of a Hermitian operator is clarified. Section 2 deals with the famous Krein resolvent formula for self-adjoint extensions of Hermitian operators. In Section 3 the extensions of a nonnegative operator \(A\geq 0\) are studied in detail. In particular, in terms of the Weyl function a characterization of the so-called Friedrichs and Krein extensions are given. In Section 4 the special case of one gap \((m=1)\) is investigated, i.e., extensions are studied which put exactly \(k\) eigenvalues into this gap. In order to prepare the main result of Section 8 in Section 5 new classes \(S^{\pm k}_{\mathcal H}\) of analytic functions are introduced which generalize the well-known Krein-Stieltjes classes \((S)_{\mathcal H}\) and \((S)^{-1}_{\mathcal H}\). Section 6 and 7 are devoted to the problem of generalized resolvents of Hermitian operators preserving the gap \((\alpha,\beta)\). We recall that the generalized resolvent is the compression of the resolvent of a self-adjoint extension of \(A\) in larger Hilbert space to the Hilbert space on which \(A\) acts. In Section 9 the results are used to study spectral properties of boundary value problems for some differential operators. At the end (Section 10) an application of the main theorems of Section 8 is given to the so-called Hamburger moment problem. In particular, a solvability criterion and a description of all solutions of this problem with the property that the supports are situated in \(\mathbb{R}^ 1\setminus\bigcup^ m_{j=1}(\alpha_ j,\beta_ j)\) are obtained in terms of the Nevanlinna matrix.
Reviewer: H.Neidhardt


47A20 Dilations, extensions, compressions of linear operators
47A10 Spectrum, resolvent
47A57 Linear operator methods in interpolation, moment and extension problems
Full Text: DOI


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