An Expansion Theorem for State Space of Unitary Linear System Whose Transfer Function is a Riemann Mapping Function

Abstract

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 unitary linear system whose state space D(W)is a Hilbert space. If 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 Lowner family of injective analytic mappings of the disk. The Lowner differential equation supplies a family of Herglotz functions. Each Herglotz function is associated with a Herglotz space of functions analytic in the unit disk. There exists an associated extended Herglotz space. An application of the Lowner differential equation is an expansion theorem for the starting state space in terms of the extended Herglotz spaces of the Lowner family. A generalization of orthogonality called complementation is used in the proof.