Univalent Subordination Chains in Reflexive Complex Banach Spaces

In this paper we consider univalent subordination chains in reflexive complex Banach spaces, allowing the chains to be normalized in terms of a positive linear operator. Related adaptations in the generalized Loewner differential equation and in the notion of parametric representation are also considered. The results in this paper are generalizations to reflexive complex Banach spaces of classical and recent results in the theory of Loewner chains and the Loewner differential equation on the unit ball in C-n. Finally, we conclude with certain open problems and conjectures.