Characterization of the Hilbert ball by its Automorphisms

We show in this paper that every domain in a separable Hilbert space, say $\text{cH}$, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of $\text{cH}$. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [KIK] and subsequent improvement of Byun/Gaussier/Kim [BGK] in the infinite dimensions.