Lowner expansions

It is well known that a power series W(z) with complex coefficients which r epresents a function bounded by one in the unit disk is the transfer functi on of a canonical conjugate isometric linear system whose state space H(W) is a Hilbert space. If, in addition, the power series has constant coeffici ent zero and coefficient of z positive, and if it represents an injective m apping of the unit disk, it appears as a factor mapping in a Lowner family of injective analytic mappings of the disk. The Lowner differential equatio n supplies a family of Herglotz functions. Each Herglotz function is associ ated with a Herglotz space of functions analytic in the unit disk. These sp aces from the spectral theory of unitary transformations are related by per turbation theory to the state spaces of canonical conjugate isometric linea r systems. In this paper an application of the Lowner differential equation is made to obtain an expansion theorem for the starting state space in ter ms of the Herglotz spaces of the Lowner family. A generalization of orthogo nality called complementation is used in the proof. A localization of the e xpansion theorem is presented as an application of the preservation of comp lementation under surjective partial isometries. A strengthening of the Rob ertson conjecture is a proposed application of the expansion.