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Recent studies in operator theory have looked at results on the discrete setting of an infinite tree. This environment is widely considered as a discrete analogue of the open unit disk D in the complex plane, thought of as a metric space under the hyperbolic metric. A space of analytic functions on D which has received much attention in the last forty years is the Bloch space B. Motivated by the important role that B plays in complex analysis and more recently in operator theory, the recent study of discretizations of B and other classical spaces of analytic functions has led to numerous advances in the understanding of functional analysis on discrete structures. Another space that has recently been thoroughly studied is the Zygmund space, namely the space of analytic functions whose derivatives are in the Bloch space. A space that can be viewed as a discrete analogue of B is the Lipschitz space L on an infinite tree T. Characterizations of the bounded and the compact multiplication operators acting on L as well as between L and the space L^{∞} of bounded functions have been obtained recently. In this dissertation, we carry out several extensions of this work. We look at the space of Lipschitz functions on an infinite graph and study the discrete analogue of the Zygmund space on an infinite tree. We show that these spaces have a functional Banach space structure, and characterize the multiplication operators acting on them that are bounded, bounded below, and compact. We also determine the spectra and show that all isometries among the multiplication operators on these spaces are trivial. We establish estimates on the operator norm and the essential norm of the multiplication operators acting on the discrete Zygmund space. We also study the multiplication operators on a separable subspace of the Lipschitz space on an infinite graph we call the little Lipschitz space on the graph.
In the 1990s, Cohen and Colonna introduced the Bloch space of a homogeneous isotropic tree defined as the subspace of L whose elements are harmonic functions. We extend their results to the Bloch space of a radial tree, which is a rooted tree whose nearest-neighbor transition probability is not isotropic, but all outward probabilities are equal to a fixed constant r starting from the vertices of length 1 (the outward probabilities from the root are all equal to the reciprocal of the degree of homogeneity of the tree). In this setting the probability of an edge is not symmetric. Specifically, we show that this class of functions is a complex Banach space and study in detail the bounded functions in the Bloch space and provide extensions of some results known in the isotropic case.
Advisor: | Colonna, Flavia |
Commitee: | Allen, Robert F., Morris, Walter, Walnut, David |
School: | George Mason University |
Department: | Mathematics |
School Location: | United States -- Virginia |
Source: | DAI-B 77/10(E), Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics |
Keywords: | Bloch space, Infinite graph, Lipschitz space, Multiplication operators, Zygmund space |
Publication Number: | 10132074 |
ISBN: | 978-1-339-89905-3 |