Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2015 | Book

Introduction Read chapter Read first chapter

Multiplication Operators on the Bergman Space

Authors: Kunyu Guo, Hansong Huang

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This book deals with various aspects of commutants and reducing subspaces of multiplication operators on the Bergman space, along with relevant von Neumann algebras generated by these operators, which have been the focus of considerable attention from the authors and other experts in recent years. The book reviews past developments and offers insights into cutting-edge developments in the study of multiplication operators. It also provides commentary and comparisons to stimulate research in this area.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
These notes arose out of a series of research papers completed by the authors and others. This volume is devoted to recent developments in commutants, reducing subspaces and von Neumann algebras related to multiplication operators.
Kunyu Guo, Hansong Huang
Chapter 2. Some Preliminaries
Abstract
This chapter will present some basic facts from complex analysis, operator theory and von Neumann algebras. These results will be needed in the sequel.
Kunyu Guo, Hansong Huang
Chapter 3. Cowen-Thomson’s Theorem
Abstract
The root of study of reducing subspaces for multiplication operators on function spaces, as will be illustrated by subsequent chapters, lies in work on the commutants of analytic Toeplitz operators on the Hardy space \(H^{2}(\mathbb{D})\), essentially initiated by Thomson and Cowen [T1, T2, Cow1, Cow2]. In considerable detail, this chapter gives an account of Cowen-Thomson’s theorem on commutants of those operators. Also presented is Thomson’s original proof of this theorem, with some modifications. In the end of this chapter, we provide a brief review on some topics closely associated with commutants on the Hardy space, which stimulated much further work. The material of this chapter mainly comes from [T1, T2] and [Cow1].
Kunyu Guo, Hansong Huang
Chapter 4. Reducing Subspaces Associated with Finite Blaschke Products
Abstract
This chapter addresses on reducing subspaces associated with finite Blaschke products, which is the subject of current research receiving numerous attention. It was shown that for each finite Blaschke product B, there is always a nontrivial reducing subspace for M B , called the distinguished reducing subspace [GSZZ, HSXY].
Kunyu Guo, Hansong Huang
Chapter 5. Reducing Subspaces Associated with Thin Blaschke Products
Abstract
Last chapter mainly concerns with reducing subspace problem of multiplication operators M B induced by finite Blaschke products B. This chapter still focuses on the same theme, whereas the symbol B is replaced with a thin Blaschke product. In Chap. 3 it was shown that the geometric property of this symbol B is a key to the study of the abelian property of \(\mathcal{V}^{{\ast}}(B)\). However, the geometry of thin Blaschke products is far more complicated than that of finite Blaschke products.
Kunyu Guo, Hansong Huang
Chapter 6. Covering Maps and von Neumann Algebras
Abstract
Distinct classes of multiplication operators have been investigated in Chaps. 4 and 5 on the Bergman space, involving their reducing subspaces. Precisely, these multiplication operators arise from finite and thin Blaschke products. The reducing subspaces of a single multiplication operator M ϕ naturally correspond to those projections, which generate a von Neumann algebra \(\mathcal{V}^{{\ast}}(\phi )\). In the above settings, this von Neumann algebra turns out to be abelian, sometimes even trivial, and hence is of type I. However, it is not always the case if the function ϕ varies.
Kunyu Guo, Hansong Huang
Chapter 7. Similarity and Unitary Equivalence
Abstract
In this chapter, we will apply those methods developed in Chaps. 3–6 to study similarity and unitary equivalence of multiplication operators, defined on both the Hardy space and the Bergman space.
Kunyu Guo, Hansong Huang
Chapter 8. Algebraic Structure and Reducing Subspaces
Abstract
In preceding chapters, we investigated reducing subspaces of analytic multiplication operators and the related von Neumann algebras generated by these multiplication operators whose symbols range over finite Blaschke products, thin Blaschke products and covering Blaschke products. In most interesting situations, multiplication operators on function spaces are essentially normal. This chapter is firstly devoted to discussion of algebraic structure of general essentially normal operators. Then we apply these results to the study of algebraic structure and reducing subspaces of multiplication operators, and the related von Neumann algebras generated by these operators in the cases of both single variable and multi-variable.
Kunyu Guo, Hansong Huang
Backmatter
Metadata
Title
Multiplication Operators on the Bergman Space
Authors
Kunyu Guo
Hansong Huang
Copyright Year
2015
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-46845-6
Print ISBN
978-3-662-46844-9
DOI
https://doi.org/10.1007/978-3-662-46845-6

Premium Partner

    Image Credits