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

Volume 11, Number 2 (2011), 1 - 29

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

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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 $L^1_a.$