Automorphism Groups of Bounded Domains in Banach Spaces

We prove a weak Schwarz lemma in Banach space and use it to show that in Hilbert space a Siegel domain of type II is not necessarily biholomorphic to a bounded domain. We use a strong Schwarz lemma of L. Harris to find the full group of automorphisms of the infinite dimensional versions of the Cartan domains of type I. We then show that all domains of type I are holomorphically inequivalent, and are different from k-fold products of unit balls <!-- MATH $(k \geqq 2)$ --> . Other generalizations and comments are given.