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Applications of the Kreĭn resolvent formula to the theory of self- adjoint extensions of positive symmetric operators. (English) Zbl 0561.47005

The paper is concerned with the theory of self-adjoint extensions, not bounded from below, of a symmetric positive operator \(A\geq 1\), in a Hilbert space. ”Parametrisations” of some classes of self-adjoint extensions, not bounded from below, of A are given. The basic results of the Krein-Birman theory of semi-bounded self-adjoint extensions are also recovered. The following result is a consequence of the developed theory:
Theorem. Suppose that the deficiency indices of A are (m,m), \(m<\infty\) and let \(A_ q\) be a sequence of self-adjoint extensions of A with the property that there exists \(\{a_ q\}_ 1^{\infty}\), \(\lim_{q\to \infty}a_ q=\infty\) such that \(A_ q\) has m eigenvalues (counting multiplicities) in \((-\infty,-a_ q)\). Then \(\lim_{q\to \infty}\| A_ q^{-1}-A_ F^{-1}\| =0\) where \(A_ F\) is the Friedrichs extension of A.

MSC:

47A20 Dilations, extensions, compressions of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
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