African Diaspora Journal of Mathematics

Bounded and Compact Operators on the Bergman Space $L^{1}_{a}$ in the Unit Disk of $\mathbb{C}$

D. Agbor, D. Békollé, and E. Tchoundja

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Abstract

We characterize boundedness and compactness of the Toeplitz operator $T_{\mu}$, on the Bergman space $L_{a}^{1}(\Delta)$, where the symbols, $\mu$, are complex Borel measures on the unit disk of the complex plane, $\Delta$. The case of Toeplitz operators whose symbols are anti-analytic integrable functions is settled. Our results are related to the reproducing kernel thesis. We also study the case of symbols which are positive measures and the case of radial symbols. Moreover, we give a characterization of compactness for general bounded operators on $L^1_a.$

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2 (2011), 1-29.

Dates
First available in Project Euclid: 6 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1323180283

Mathematical Reviews number (MathSciNet)
MR2862562

Zentralblatt MATH identifier
1262.47041

Subjects
Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Keywords
Toeplitz operator Compact operator Carleson measure

Citation

Agbor, D.; Békollé, D.; Tchoundja, E. Bounded and Compact Operators on the Bergman Space $L^{1}_{a}$ in the Unit Disk of $\mathbb{C}$. Afr. Diaspora J. Math. (N.S.) 11 (2011), no. 2, 1--29. https://projecteuclid.org/euclid.adjm/1323180283


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