Convergence of operators with deficiency indices (k, k) and of their self-adjoint extensions

We consider an abstract sequence \(\{A_n\}_{n=1}^\infty \) of closed symmetric operators on a separable Hilbert space \({\mathcal {H}}\). It is assumed that all \(A_n\)’s have equal deficiency indices (k, k) and thus self-adjoint extensions \(\{B_n\}_{n=1}^\infty \) exist and are parametrized by partial isometries \(\{U_n\}_{n=1}^\infty \) on \({\mathcal {H}}\) according to von Neumann’s extension theory. Under two different convergence assumptions on the \(A_n\)’s we give the precise connection between strong resolvent convergence of the \(B_n\)’s and strong convergence of the \(U_n\)’s.