Lowner expansions
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 conjugate isometric linear system whose state space ${\cal H}(W)$ 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. These spaces from the spectral theory of unitary transformations are related by perturbation theory to the state spaces of canonical conjugate isometric linear systems. An application of the Lowner differential equation is an expansion theorem for the starting state space in terms of the Herglotz spaces of the Lowner family. A generalization of orthogonality called complementation is used in the proof. A localization of the expansion theorem is an application of the preservation of complementation under surjective partial isometries. A strengthening of the Robertson conjecture is a proposed application of the expansion.
Degree
Ph.D.
Advisors
Branges, Purdue University.
Subject Area
Mathematics
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