## 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)