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

We consider an abstract sequence $\{A_n{\}}_{n=1}^{\mathrm{\infty }}$ 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 $\{B_n{\}}_{n=1}^{\mathrm{\infty }}$ exist and are parametrized by partial isometries $\{U_n{\}}_{n=1}^{\mathrm{\infty }}$ 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.