A Schwarz lemma and composition operators

We give an alternative description of the Carathéodory pseudodistance on a domain D in an arbitrary complex Banach space. This gives a Schwarz lemma for holomorphic maps of the domain.We specialise to the case of a bounded symmetric domain and obtain some applications. In particular, we give the connected components of the space of composition operators with symbol in a bounded symmetric domain. This generalises results for the space of composition operators on H∞(Δ) in [12] and for H∞(B) , B the unit ball of a Hilbert space or commutative C*-algebra in [2].