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2008 | Book

Preliminaries Read chapter Read first chapter

Commutative Algebras of Toeplitz Operators on the Bergman Space

Author: Nikolai L. Vasilevski

Publisher: Birkhäuser Basel

Book Series : Operator Theory: Advances and Applications

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

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Table of Contents

Frontmatter
Chapter 1. Preliminaries
Abstract
The chapter contains some algebraic material which will be used substantially in the book and which can not be easily found in the standard C*-algebraic sources. Those immediately interested in the main content of the book may skip this chapter at the first reading.
Chapter 2. Prologue
Abstract
As it unfortunately happens in mathematics, some terms carry different meanings depending on the context in which they are used. The term “symbol” is not exceptional in this sense. It will be used systematically in the book and will be supplied with different adjectives clarifying its different meanings: Fredholm symbol, defining symbol, Wick symbol, anti-Wick symbol, etc. That is why we would like to comment first on its meanings and usage.
Chapter 3. Bergman and Poly-Bergman Spaces
Abstract
We start by recalling an old and well-known result. Let \( H_ + ^2 \left( \mathbb{R} \right)\left( { \subset L_2 \left( \mathbb{R} \right)} \right) \) be the Hardy space on the upper half-plane II in ℂ, which by definition consists of all functions ϕ on ℝ admitting analytic continuation in II and satisfying the condition
$$ \mathop {\sup }\limits_{v > 0} \int_{\mathbb{R} + iv} {\left| {\phi \left( {u + iv} \right)} \right|^2 du < \infty .} $$
Let \( P_\mathbb{R}^ + \) be the (orthogonal) Szegö projection of L2(ℝ) onto \( H_ + ^2 \left( \mathbb{R} \right) \). Then: the Fourier transform F gives an isometric isomorphism of the space L 2 (ℝ), under which
1.
the Hardy space\( H_ + ^2 \left( \mathbb{R} \right) \) is mapped onto L2(ℝ+),
$$ F:H_ + ^2 \left( \mathbb{R} \right) \to L_2 \left( {\mathbb{R}_ + } \right), $$
 
2.
the Szegö projection \( P_\mathbb{R}^ + \): L2(ℝ)→\( H_ + ^2 \left( \mathbb{R} \right) \) is unitary equivalent to the projection
$$ F:P_\mathbb{R}^ + F^{ - 1} - \chi + I. $$
 
Chapter 4. Bergman Type Spaces on the Unit Disk
Abstract
Let \( \mathbb{D} \) be the unit disk in ℂ. Rearranging the basis of L2(\( \mathbb{D} \)), L. Peng, R. Rochberg and Z. Wu [154] proved that the space L2(\( \mathbb{D} \)) can be decomposed onto a direct sum of the Bergman type spaces
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-7643-8726-6_4/978-3-7643-8726-6_4_Equ1_HTML.gif
(4.0.1)
where
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-7643-8726-6_4/978-3-7643-8726-6_4_Equa_HTML.gif
Chapter 5. Toeplitz Operators with Commutative Symbol Algebras
Abstract
Theorems 2.4.5, 2.8.3, 2.8.6, and 2.8.7 show a certain difference between the compactness properties of commutators and semi-commutators. In order to understand this difference we start with the following setting.
Chapter 6. Toeplitz Operators on the Unit Disk with Radial Symbols
Abstract
As follows, for example, from Theorem 2.8.3, the Toeplitz operator with radial defining symbols a(r), which is continuous at the boundary point 1, has a trivial structure, nothing but a compact perturbation of a scalar operator, T a(r) =a(1)I+K.
Chapter 7. Toeplitz Operators on the Upper Half-Plane with Homogeneous Symbols
Abstract
In this chapter we return to the upper half-plane Π, the space L2(Π) and its Bergman subspace A2(Π). Passing to polar coordinates we have
$$ L_2 \left( \prod \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {\left[ {0,\pi } \right],d\theta } \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {0,\pi } \right), $$
and
$$ \frac{\partial } {{\partial \bar z}} = \frac{{\cos \theta + i\sin \theta }} {2}\left( {\frac{\partial } {{\partial r}} + i\frac{1} {r}\frac{\partial } {{\partial \theta }}} \right) = \frac{{\cos \theta + i\sin \theta }} {{2r}}\left( {r\frac{\partial } {{\partial r}} + i\frac{\partial } {{\partial \theta }}} \right). $$
Chapter 8. Anatomy of the Algebra Generated by Toeplitz Operators with Piece-wise continuous Symbols
Abstract
In this chapter we continue the study of the C*-algebra generated by Toeplitz operators T a with piece-wise continuous defining symbols a acting on the Bergman space (\( \mathcal{A}^2 \left( \mathbb{D} \right) \)) on the unit disk \( \mathbb{D} \). Our aim here is to describe explicitly each operator from this algebra and to characterize the Toeplitz operators which belong to the algebra.
Chapter 9. Commuting Toeplitz Operators and Hyperbolic Geometry
Abstract
The C*-algebras of Toeplitz operators considered in Chapters 5, 6, and 7 are commutative. This property is impossible (except for the trivial case of scalar operators ≡ constant defining symbols) for the C*-algebras of Toeplitz operators acting on the Hardy space. The special features of the defining symbols which make this phenomenon possible were symbols depending only on the imaginary part of a variable for Toeplitz operators on the upper half-plane, radial symbols for Toeplitz operators on the unit disk, and symbols depending only on the angular part of a variable for Toeplitz operators on the upper half-plane, respectively. In this stage a natural question appears: whether there exist other classes of defining symbols which generate commutative Toeplitz operator C * -algebras, and how they can be classfied.
Chapter 10. Weighted Bergman Spaces
Abstract
We recall here some necessary facts on weighted Bergman spaces and the corresponding Bergman projections; for further details see, for example [102, 240].
Chapter 11. Commutative Algebras of Toeplitz Operators
Abstract
The commutative C*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space, has remained open.
Chapter 12. Dynamics of Properties of Toeplitz Operators with Radial Symbols
Abstract
Given a smooth defining symbol a=a(z), the family of Toeplitz operators \( T_a = \left\{ {T_a^{\left( h \right)} } \right\} \), where h∈(0, 1), was considered in the previous chapter under the Berezin quantization procedure. For a fixed h the Toeplitz operator T a (h) acts on the weighted Bergman space \( \mathcal{A}_h^2 \left( \mathbb{D} \right) \), where the parameter h characterizes the weight (10.1.5) on \( \mathcal{A}_h^2 \left( \mathbb{D} \right) \). In the sequel we will consider another form of presentation of the weighted Bergman spaces, see (10.1.1), the space \( \mathcal{A}_\lambda ^2 \left( \mathbb{D} \right) \) which is parameterized by λ∈(−1, +∞) being connected with h∈(0, 1) by the rule \( \lambda + 2 = \frac{1} {h} \), see Section 10.1.
Chapter 13. Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Parabolic Case
Abstract
Recall that by Theorems 10.4.9 and 10.5.1, the function
$$ \begin{gathered} \gamma a,\lambda \left( t \right) = \frac{{t^{\lambda + 1} }} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {2})\eta ^\lambda e^{ - t\eta } d\eta } \hfill \\ = \frac{1} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {{2t}})\eta ^\lambda e^{ - \eta } d\eta } \hfill \\ \end{gathered} $$
(13.1.1)
is responsible for the boundedness of a Toeplitz operator with symbol a=a(y). If \( a = a\left( y \right) \in L_\infty \left( {\mathbb{R}_ + } \right) \), then the operator Ta(λ) is obviously bounded on all spaces \( \mathcal{A}_\lambda ^2 \left( \Pi \right) \), where λ∈(−1, ∞), and the corresponding norms are uniformly bounded by sup z |a(z)|. That is, all spaces \( \mathcal{A}_\lambda ^2 \left( \Pi \right) \), where λ∈(−1, ∞), are natural and appropriate for Toeplitz operators with bounded symbols. One of our aims is a systematic study of unbounded symbols. To avoid unnecessary technicalities in this chapter we will always assume that λ∈[0, ∞).
Chapter 14. Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Hyperbolic Case
Abstract
Recall that by Theorems 10.4.16 and 10.5.1, the function
$$ \gamma a,\lambda \left( \xi \right) = \left( {\int_0^\pi {e^{ - 2\xi \theta } \sin ^\lambda \theta d\theta } } \right)^{ - 1} \int_0^\pi {a\left( \theta \right)e^{ - 2\xi \theta } \sin ^\lambda \theta d\theta } , \xi \in \mathbb{R} $$
(14.1.1)
is responsible for the boundedness of a Toeplitz operator with symbol a(θ) (∈L1(0, π)). If a(θ)∈L∞(0, π), then the operator T a (λ) is obviously bounded on all spaces A λ 2 (Π), for λ∈(−1, ∞), and the corresponding norms are uniformly bounded by sup z |a(z)|. That is, all spaces A λ 2 (Π), where λ∈(−1, ∞), are natural and appropriate for Toeplitz operators with bounded symbols. Studying unbounded symbols, we wish to have a sufficiently large class of them common to all admissible λ; moreover, we are especially interested in properties of Toeplitz operators for large values of λ. Thus it is convenient for us to consider λ belonging only to [0, ∞), which we will always assume in what follows.
Backmatter
Metadata
Title
Commutative Algebras of Toeplitz Operators on the Bergman Space
Author
Nikolai L. Vasilevski
Copyright Year
2008
Publisher
Birkhäuser Basel
Electronic ISBN
978-3-7643-8726-6
Print ISBN
978-3-7643-8725-9
DOI
https://doi.org/10.1007/978-3-7643-8726-6

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