Löwner Expansions

Abstract It is well known that a power series W ( z ) with complex coefficients which represents a function bounded by one in the unit disk is the transfer function of a canonical conjugate isometric linear system whose state space H ( W ) is a Hilbert space. If, in addition, the power series has constant coefficient zero and coefficient of z positive, and if it represents an injective mapping of the unit disk, it appears as a factor mapping in a Löwner family of injective analytic mappings of the disk. The Löwner differential equation supplies a family of Herglotz functions. Each Herglotz function is associated with a Herglotz space of functions analytic in the unit disk. These spaces from the spectral theory of unitary transformations are related by perturbation theory to the state spaces of canonical conjugate isometric linear systems. In this paper an application of the Löwner differential equation is made to obtain an expansion theorem for the starting state space in terms of the Herglotz spaces of the Löwner family. A generalization of orthogonality called complementation is used in the proof. A localization of the expansion theorem is presented as an application of the preservation of complementation under surjective partial isometries. A strengthening of the Robertson conjecture is a proposed application of the expansion.