Evolution Equations and Geometric Function Theory in J∗-algebras1

In a series of papers Ky Fan developed a geometric theory of holomorphic functions of proper contractions on Hilbert spaces in the sense of the functional calculus. His results are a powerful tool in the study of the discrete-time semigroups of l-analytic functions defined by iterating such a function on the open unit ball of the space of bounded linear operators on a Hilbert space. In this paper we examine the asymptotic behavior of continuous semigroups of l-analytic functions. We establish infinitesimal versions of Ky Fan’s results as well as of the classical Julia–Carathéodory and Wolff Theorems by developing the generation theory of continuous one-parameter semigroups of l-analytic functions. We then introduce a general approach to the study of geometric properties of univalent functions in Banach spaces by using the linear one-parameter semigroups defined on the space of holomorphic mappings. Applying our results on the asymptotic behavior of semigroups of l-analytic functions with no stationary point, we describe l-analytic functions which are star-like with respect to a boundary point. All our considerations are carried out in the framework of J∗-algebras with identity, which include, for example, C∗-algebras and certain Cartan factors.