Abstract.
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 \( H^\infty (\Delta) \) in [12] and for \( H^\infty (B) \), B the unit ball of a Hilbert space or commutative C*-algebra in [2].
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Mackey, M., Mellon, P. A Schwarz Lemma and Composition Operators. Integr. equ. oper. theory 48, 511–524 (2004). https://doi.org/10.1007/s00020-003-1240-1
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DOI: https://doi.org/10.1007/s00020-003-1240-1