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# A Class of Operators with Symbol on the Bloch Space of a Bounded Homogeneous Domain

 dc.contributor.author Allen, Robert Francis dc.creator Allen, Robert Francis dc.date 2009-04-21 dc.date.accessioned 2009-06-08T20:14:00Z dc.date.available NO_RESTRICTION en dc.date.available 2009-06-08T20:14:00Z dc.date.issued 2009-06-08T20:14:00Z dc.identifier.uri https://hdl.handle.net/1920/4541 dc.description.abstract
Let X be a Banach space of holomorphic functions on a domain D in C^n. If ψ  is a
holomorphic function on D, and ϕ is a holomorphic self-map of D, we defi ne the weighted
composition operator on X with symbols ψ and ϕby W ψ,ϕf =  (f о ϕ). This operator
is a generalization of the multiplication operator M(ψ) f = ψ f and the composition operator
Cϕf = f о ϕ, which are known as degenerate weighted composition operators.
The weighted composition operators have been an object of interest since the early 30's
with their connection to the isometries of various spaces of analytic functions on the unit
disk. The Bloch space has been of interest since the early 70's to function theorists and to
operator theorists. However, these two concepts did not meet until 2001.
Classical operator theory on spaces of holomorphic functions in several complex variables
is typically carried out on the unit ball and the unit polydisk. The respective function
theories are very di fferent. In this dissertation, we attempt to unify the operator theory on
the Bloch space on these domains and extend it further to bounded homogeneous domains
in C^n.
In this uni ed manner, we study the fundamental properties of the weighted composition
operators:
1. For what symbols ψ  and ϕ is W ψ﻿,ϕ bounded?
2. For what symbols ψ  and ϕ is W ψ﻿,ϕ compact?
3. What is an expression for ||W ψ,﻿ϕ||?
4. For what symbols ψ  and ϕ is Wψ﻿,ϕ an isometry?
5. What is the spectrum of Wψ﻿ϕ?
It is our hope that this work will mark the beginning of a paradigm shift in operator
theory research in several complex variables. This will bring in new  fields of study such as
diff erential geometry into the study of operators, thus enriching the  field.

dc.language.iso en_US en dc.subject Weighted Composition Operators en_US dc.subject Bloch Space en_US dc.subject Homogeneous Domain en_US dc.subject Operator Theory en_US dc.title A Class of Operators with Symbol on the Bloch Space of a Bounded Homogeneous Domain en dc.type Dissertation en thesis.degree.name Doctor of Philosophy in Mathematics en thesis.degree.level Doctoral en thesis.degree.discipline Mathematics en thesis.degree.grantor George Mason University en
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