Extension operators and subordination chains

Since the work of Roper and Suffridge in 1995, there has been considerable interest in constructing holomorphic mappings of the unit ball in C n with various geometric properties by using lower dimensional mappings with similar properties. Such properties include convexity, starlikeness, and spirallikeness. It is also of interest to extend subordination chains (which may be related to geometric properties such as the above). Recently M. Elin has introduced an approach to extension operators on Banach spaces based on semigroups which leads to an identification of some of the essential properties of such operators. In this paper we adopt Elinʼs point of view and prove a theorem about the extension of subordination chains which extends and unifies existing results. We also consider applications in finite dimensions to the study of extreme and support points under the Loewner variation and of reachable sets in control theory. Certain examples will also be provided.