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

Abstract

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