Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge extension operators on bounded symmetric domains

Abstract

ABSTRACTLet Y be a complex Banach space and let be the open unit ball of Y. In this paper we consider a generalization of the Pfaltzgraff-Suffridge extension operator on bounded symmetric domains in , and prove that if is a bounded symmetric domain in , and is an extension operator which maps normalized locally biholomorphic mappings on to locally biholomorphic mappings on , where is a certain domain with , then extends the first elements of Loewner chains from to the first elements of Loewner chains on , when , where is a constant defined by the Bergman metric on . Particular cases are also obtained. Next, we prove that normalized locally univalent I-Bloch mappings, which have finite trace order on , are mapped into R-Bloch mappings on by the operator when , where is a bounded balanced convex domain such that . In particular, the normalized bounded convex mappings on are mapped into R-Bloch mappings on under the extension operator .