Conjugacy classes of automorphisms of the unit ball in a complex Hilbert space

In this article, we consider the ball model of an infinite dimensional complex hyperbolic space, i.e. the open unit ball of a complex Hilbert space centered at the origin equipped with the Caratheodory metric. We consider the group of holomorphic automorphisms of the ball and classify the conjugacy classes of automorphisms. We also compute the centralizers for elements in the group of automorphisms.