On convergence of inverse functions of operators (Q1119879)

Let \(\{A_ n\}^{\infty}_{n=1}\) be a sequence of selfadjoint operators on a Hilbert space and let F be a real-valued function. The problem is whether the convergence of the sequence \(\{F(A_ n)\}\) ensures the convergence of \(\{A_ n\}\) to a selfadjoint limit. The authors give sufficient conditions imposed on a function f and on the sequence \(\{A_ n\}\) under which the existence of a norm-resolvent limit of \(\{F(A_ n\}\), \(F(A_ n)=^{def}f(| A_ n|)\), implies the existence of a selfadjoint operator A which is a norm-resolvent limit of \(\{A_ n\}\).