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

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A(infinity,omega) for A(alpha,omega) = A + alpha(., omega)omega, alpha --> infinity where A is a self-adjoint positive operator, omega is an element of H-1 (H-s, s is an element of R-1 being the 'A-scale'). In the present note it is remarked that the operator A(infinity,omega) also appears directly as the Friedrichs extension of the symmetric operator A A := Ainverted right perpendicular{f is an element of D(A) \ [f, omega] = 0}. It is also shown that Krein's resolvents formula: (A(b,omega) - z)(-1) = (A - z)(-1) + b(z)(-1) (., n((z) over bar))n(z), with b(z) = b-(1 + z)(eta(z), eta(-1)), eta(z) = (A - z)(-1) omega defines a self-adjoint operator A(b,omega) for each omega is an element of H-2 and b is an element of R-1. Moreover it is proven that for any sequence omega(n) is an element of H-1 which goes to omega in H-2 there exists a sequence alpha(n) --> 0 such that A(alpha n,omega n) --> A(b,omega) in the strong resolvent sense.