A Schwarz Lemma and Composition Operators

Abstract

Abstract. We give an alternative description of the Carathéodory pseudo-distance 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 ∞ (∆) in [12] and for H ∞ (B), B the unit ball of a Hilbert space or commutative C * -algebra in [2].

The identity is isolated among composition operators

“…First, does Theorem 1.1 hold when E is a JB * -triple? A positive answer to this question has been given in [7]. The second conjecture is that Theorem 1.2 holds for every Banach space E. We give a positive answer and a simple proof in this paper.…”