Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions (Q1819147)

The article is a supplement of a result by \textit{F. Gesztesy} and \textit{B. Simon} [J. Funct. Anal. 128, No. 1, 245-252 (1995; Zbl 0828.47009)]. They showed the existence of the strong resolvent limit \(A_{\infty,g}\) obtained from \(A_{\alpha,g}= A+\alpha\langle.,g\rangle g\) as \(\alpha\to\infty\), where \(A\) is a self-adjoint positive operator and \(g\in{\mathcal H}_{-1}\) (\({\mathcal H}_{-1}\) means a Hilbert space in \(A\) scale). The present article shows that \(A_{\infty,g}\) equals the Friedrichs extension of \(A\) restricted to \(\{f\in \text{dom}(A),\langle f,g\rangle= 0\}\). Moreover, some further consequences following from Krein's theory of self-adjoint extensions are studied.