Asymptotically Spirallike Mappings in Reflexive Complex Banach Spaces

Abstract

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces \(X\), and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in \(\mathbb{C }^{n}\) which have parametric representation in the sense of Loewner theory. The authors introduced the notions of \(A\)-asymptotic spirallikeness and \(A\)-parametric representation on the Euclidean unit ball of \(\mathbb{C }^{n}\), where \(A\in L(\mathbb{C }^{n})\) with \(m(A)>0\). They showed that these notions are equivalent whenever \(k_+(A)<2m(A)\). In this paper we prove that if \(k_+(A)<2m(A)\) and \(f\in S(B)\) has \(A\)-parametric representation, then \(f\) is also \(A\)-asymptotically spirallike on the unit ball \(B\) of \(X\). For the converse, we need the additional assumption that \(f\) is a smooth \(A\)-asymptotically spirallike mapping, except in the finite-dimensional case \(X=\mathbb{C }^{n}\) with an arbitrary norm. The notion of asymptotic spirallikeness involves differential equations and may be regarded as giving a geometric characterization of certain domains in \(X\). That is one of the motivations for considering this notion in the case of reflexive complex Banach spaces.