The Essential Norm of Operators on $${A^p_\alpha(\mathbb{B}_n)}$$

Mitkovski, Mishko; Suárez, Daniel; Wick, Brett D.

doi:10.1007/s00020-012-2025-1

Cited by 42 publications

(39 citation statements)

References 15 publications

(21 reference statements)

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“…A version of this result in the classical Bergman space setting was first proved by Suárez in [31]. Related results were later given in [4,20,22,25]. …”

Section: Reproducing Kernel Thesis For Compactnessmentioning

confidence: 74%

“…For the proof see [21]. Related results can be found in [4,8,22,31] where it is shown that nice domains, such as the unit ball, polydisc, or C n have this property.…”

Section: Projection Operators On Bergman-type Spacesmentioning

confidence: 84%

“…This is not a trivial assumption and it is (at this point) not known whether this holds for all Bergman-type spaces (see [21]). It does hold in the standard Bergman spaces on the ball and polydisc and also on the Fock space see [4,20,22,31]. Thus, the following theorem can be viewed as an extension of the main theorems in [4,20,22,31] to the ℓ 2 -valued setting.…”

Section: Reproducing Kernel Thesis For Compactnessmentioning

confidence: 91%

“…However, results of this type are well known for the classical Bergman spaces on the ball. See for example [4,8,21,22,31].…”

Section: ℓ 2 -Valued Bergman-type Spacesmentioning

confidence: 99%

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The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

Rahm

2015

Complex Anal. Oper. Theory

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Abstract. In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on ℓ 2 -valued Bergman-type spaces. This paper generalizes many well-known results about classical function spaces to their ℓ 2 -valued versions. In particular, the results in this paper apply to the weighted ℓ 2 -valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

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“…A version of this result in the classical Bergman space setting was first proved by Suárez in [31]. Related results were later given in [4,20,22,25]. …”

Section: Reproducing Kernel Thesis For Compactnessmentioning

confidence: 74%

“…For the proof see [21]. Related results can be found in [4,8,22,31] where it is shown that nice domains, such as the unit ball, polydisc, or C n have this property.…”

Section: Projection Operators On Bergman-type Spacesmentioning

confidence: 84%

Section: Reproducing Kernel Thesis For Compactnessmentioning

confidence: 91%

“…However, results of this type are well known for the classical Bergman spaces on the ball. See for example [4,8,21,22,31].…”

Section: ℓ 2 -Valued Bergman-type Spacesmentioning

confidence: 99%

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The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

Rahm

2015

Complex Anal. Oper. Theory

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“…The behaviour of these operators on the Hardy spaces, Bergman spaces, and Fock spaces has been studied widely and a lot of results are available in the literature. The characterization of compactness has been studied in [2][3][4][5][6][7][8] just to cite a few. Given Ω ⊂ R , a measurable function : Ω → [1, ∞) will be called a variable exponent.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Agbor

2018

International Journal of Mathematics and Mathematical Sciences

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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 1 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 is a finite sum of finite products of Toeplitz operators with symbols from class , then is compact if and only if the Berezin transform of vanishes on the boundary of the unit disc.

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Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

Bauer¹,

Robert²

2020

Operator Algebras, Toeplitz Operators and Related Topics

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We consider various classes of bounded operators on the Fock space F 2 of Gaussian square integrable entire functions over the complex plane. These include Toeplitz (type) operators, weighted composition operators, singular integral operators, Volterra-type operators and Hausdorff operators and range from classical objects in harmonic analysis to more recently introduced classes. As a leading problem and closely linked to well-known compactness characterizations we pursue the question of when these operators are contained in the Toeplitz algebra. This paper combines a (certainly in-complete) survey of the classical and more recent literature including new ideas for proofs from the perspective of quantum harmonic analysis (QHA). Moreover, we have added a number of new theorems and links between known results.

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The Essential Norm of Operators on $${A^p_\alpha(\mathbb{B}_n)}$$

Abstract: In this paper we characterize the compact operators on A p α (B n ) when 1 < p < ∞ and α > −1. The main result shows that an operator on A p α (B n ) is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.

Cited by 42 publications

References 15 publications

The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

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

Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

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