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

We consider an abstract sequence {An}n=1∞ of closed symmetric operators on a separable Hilbert space H . It is assumed that all An ’s have equal deficiency indices (k, k) and thus self-adjoint extensions {Bn}n=1∞ exist and are parametrized by partial isometries {Un}n=1∞ on H according to von Neumann’s extension theory. Under two different convergence assumptions on the An ’s we give the precise connection between strong resolvent convergence of the Bn ’s and strong convergence of the Un ’s. © 2023, The Author(s).