Toeplitz Operators on Bergman Spaces

1982 ◽  
Vol 34 (2) ◽  
pp. 466-483 ◽  
Author(s):  
Sheldon Axler ◽  
John B. Conway ◽  
Gerard McDonald
Keyword(s):  
Hilbert Space ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Bergman Spaces ◽  
Inner Product ◽  
Area Measure ◽  
Lebesgue Area ◽  
Continuous Complex ◽  
Complex Valued

Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.

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 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.

scholarly journals (m,λ)-Berezin Transform on the Weighted Bergman Spaces over the Polydisk

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ran Li ◽  
Yufeng Lu
Keyword(s):  
Linear Operator ◽  
Bergman Space ◽  
Bounded Linear Operator ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Main Tool ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bounded Linear

We prove that every bounded linear operator on weighted Bergman space over the polydisk can be approximated by Toeplitz operators under some conditions. The main tool here is the so-called(m,λ)-Berezin transform. In particular, our results generalized the results of K. Nam and D. C. Zheng to the case of operators acting onAλ2(Dn).

scholarly journals Commutants of Toeplitz operators on the ball and annulus

1995 ◽  
Vol 37 (3) ◽  
pp. 303-309 ◽  
Author(s):  
Željko Čučković ◽  
Dashan Fan
Keyword(s):  
Positive Integer ◽  
Bergman Space ◽  
Analytic Functions ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Volume Measure ◽  
Image Position

In this paper we study commutants of Toeplitz operators with polynomial symbols acting on Bergman spaces of various domains. For a positive integer n, let V denote the Lebesgue volume measure on ℂn. If ω is a domain in ℂn, then the Bergman space is defined to be the set of all analytic functions from ω into ℂ such that

scholarly journals Hyponormality on general Bergman spaces

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5737-5741 ◽  
Author(s):  
Houcine Sadraoui
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Toeplitz Operators ◽  
Bounded Operator ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

scholarly journals Commuting of Toeplitz operators on the Bergman spaces of the bidisc

2002 ◽  
Vol 66 (2) ◽  
pp. 345-351 ◽  
Author(s):  
Yufeng Lu
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces

In this paper we describe when two Toeplitz operators Tf and Tg on the Bergman space of the bidisc commute, where f = f1 + f̅2, g = g1 + ḡ2, fi, gi ∈ H∞(D2)(i = 1, 2).

scholarly journals Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Dieudonne Agbor
Keyword(s):  
Bergman Space ◽  
Unit Disc ◽  
Variable Exponent ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Berezin Transform ◽  
Variable Exponents ◽  
Bounded Operators ◽  
Finite Products ◽  
Finite Sum

We study the compactness of some classes of bounded operators on the Bergman space with variable exponent. We show that via extrapolation, some results on boundedness of the Toeplitz operators with general L1 symbols and compactness of bounded operators on the Bergman spaces with constant exponents can readily be extended to the variable exponent setting. In particular, if S is a finite sum of finite products of Toeplitz operators with symbols from class BT, then S is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disc.

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 Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2)

2016 ◽  
Vol 24 (1) ◽  
pp. 43-69 ◽  
Author(s):  
Stephen Bruce Sontz
Keyword(s):  
Hilbert Space ◽  
Toeplitz Operators ◽  
General Setting ◽  
Inner Product ◽  
Mathematical Structures ◽  
Planck’S Constant ◽  
Quantum Objects ◽  
Planck's Constant ◽  
Densely Defined ◽  
Creation Operators

Abstract Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SUq(2) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck’s constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck’s constant and a Hilbert space where natural, densely defined operators act.

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