Define $ A $ a unbounded self-adjoint operator on Hilbert space $ X $. Let $ \{ A_n \} $ be its resolvent approximation sequence with closed range $ \mathcal{ R}(A_n) (n \in \mathrm{ N}) $, that is, $ A_n (n \in \mathrm{ N}) $ are all self-adjoint on Hilbert space $ X $ and \begin{ equation*} \hbox{ \raise-2mm\hbox{ $\textstyle s-\lim \atop \scriptstyle { n \to \infty}$}} R_\lambda (A_n) = R_\lambda (A)\quad (\lambda \in \mathrm{ C} \setminus \mathrm{ R}), \ \textrm{ where} \ R_ \lambda(A) := (\lambda I-A)^{ -1}. \end{ equation*} The Moore-Penrose inverse $ A^\dagger_n \in \mathcal{ B}(X) $ is a natural approximation to the Moore-Penrose inverse $ A^\dagger $. This paper shows that: $ A^\dagger $ is continuous and strongly converged by $ \{ A^\dagger_n \} $ if and only if $ \sup\limits_n \Vert A^\dagger_n \Vert < +\infty $. On the other hand, this result tells that arbitrary bounded computational scheme $ \{ A^\dagger_n \} $ induced by resolvent approximation $ \{ A_n \} $ is naturally instable (that is, $ \sup_n \Vert A^\dagger_n \Vert = \infty $) for any self-adjoint operator equation with non-closed range, for example, free Schr\"{ o}dinger operator, Schr\"{ o}dinger operator with Coulumb potential and Schr\"{ o}dinger operator in model of many particles. This implies the infeasibility to globally and approximately solve non-closed range self-ajoint operator equation by resolvent approximation.
Categories: math.FA