Toeplitz operators on the Bergman space of the unit ball

Raimondo, Roberto C.

doi:10.1017/s0004972700018748

Cited by 13 publications

(11 citation statements)

References 5 publications

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“…This was later extended by Engliš to the case of bounded symmetric domains in C n , see [7]. See also the proof by Raimondo, [12], in the specific case of B n .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 96%

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

Mitkovski

Suárez

Wick

2012

Integr. Equ. Oper. Theory

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“…This was later extended by Engliš to the case of bounded symmetric domains in C n , see [7]. See also the proof by Raimondo, [12], in the specific case of B n .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 96%

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

Mitkovski

Suárez

Wick

2012

Integr. Equ. Oper. Theory

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“…Results of this kind are numerous in the literature. Here we mention only a few and refer the reader to [5,13,17,18,21,23,24,29,32,37] and the references in those papers for more examples of these results. The RKT for compactness was first proved to hold for every Toeplitz operator on the classical Bergman space of the disc by Zheng [38].…”

Section: Introductionmentioning

confidence: 99%

A reproducing kernel thesis for operators on Bergman-type function spaces

Mitkovski

Wick

2014

Journal of Functional Analysis

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“…This gives the first two inequalities in (26). The remaining inequality is simply (22), which was proved in Theorem 5.2.…”

Section: Characterization Of the Essential Norm On A P α Weighted Bermentioning

confidence: 75%

The essential norm of operators on the Bergman space of vector–valued functions on the unit ball

Rahm¹,

Wick²

2015

Function Spaces in Analysis

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Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d . For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1). We show that every S ∈ T p,α can be approximated by localized operators. This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α . We then use this to characterize compact operators in A p α (B n ; C d ). The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

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Toeplitz operators on the Bergman space of the unit ball

Abstract: We prove that if an operator A is a finite sum of finite products of Toeplitz operators on the Bergman space of the unit ball B n , then A is compact if and only if its Berezin transform vanishes at the boundary. For n = 1 the result was obtained by Axler and Zheng in 1997.

Cited by 13 publications

References 5 publications

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

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

A reproducing kernel thesis for operators on Bergman-type function spaces

The essential norm of operators on the Bergman space of vector–valued functions on the unit ball

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