Algebras on the unit disk and Toeplitz operators on the Bergman space

2000 ◽  
Vol 37 (1) ◽  
pp. 106-123 ◽  
Author(s):  
Rahman Younis ◽  
Dechao Zheng
Keyword(s):  
Unit Disk ◽  
Bergman Space ◽  
Toeplitz Operators

Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators

2003 ◽  
Vol 55 (2) ◽  
pp. 379-400 ◽  
Author(s):  
Michael Stessin ◽  
Kehe Zhu
Keyword(s):  
Hardy Space ◽  
Unit Disk ◽  
Bergman Space ◽  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Inner Function ◽  
Factorization Theory

AbstractEvery classical inner function φ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when φ(z) = z. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

scholarly journals Hyponormality on a Weighted Bergman Space

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Houcine Sadraoui ◽  
Borhen Halouani ◽  
Mubariz T. Garayev ◽  
Adel AlShehri
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Bergman Space ◽  
Analytic Functions ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Weighted Bergman Space ◽  
Necessary Condition ◽  
Space Operator ◽  
Bounded Analytic Functions

A bounded Hilbert space operator T is hyponormal if T∗T−TT∗ is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the form f+g¯ where f and g are bounded analytic functions on the unit disk. We then find sufficient conditions when f is a monomial.

scholarly journals On the Commutativity of a Certain Class of Toeplitz Operators

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Issam Louhichi ◽  
Fanilo Randriamahaleo ◽  
Lova Zakariasy
Keyword(s):  
Toeplitz Operator ◽  
Unit Disk ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Harmonic Bergman Space

AbstractOne of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

Essentially Commuting Toeplitz Operators With Harmonic Symbols

1993 ◽  
Vol 45 (5) ◽  
pp. 1080-1093 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Disk ◽  
Bergman Space ◽  
Harmonic Functions ◽  
Toeplitz Operators ◽  
Bounded Harmonic Functions

AbstractIn this paper we characterize the bounded harmonic functions ƒ and g on the unit disk for which the Toeplitz operators Tƒ and Tg defined on the Bergman space of the unit disk are essentially commuting.

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

scholarly journals The Berezin Transform of Toeplitz Operators on the Weighted Bergman Space

2021 ◽  
Author(s):  
Cezhong Tong ◽  
Junfeng Li ◽  
Hicham Arroussi
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Kernel Estimates ◽  
Berezin Transforms

AbstractIn this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

scholarly journals Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Roberto C. Raimondo
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Berezin Transform ◽  
Planar Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Domains ◽  
Necessary And Sufficient ◽  
The Relationship

We study the problem of the boundedness and compactness of when and is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

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