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June, 2010 Toeplitz operators on Bergman spaces with locally integrable symbols
Jari Taskinen , Jani Virtanen
Rev. Mat. Iberoamericana 26(2): 693-706 (June, 2010).


We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.


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Jari Taskinen . Jani Virtanen . "Toeplitz operators on Bergman spaces with locally integrable symbols." Rev. Mat. Iberoamericana 26 (2) 693 - 706, June, 2010.


Published: June, 2010
First available in Project Euclid: 4 June 2010

zbMATH: 1204.47040
MathSciNet: MR2677012

Primary: 47B35

Keywords: Bergman spaces , boundedness , compactness , Fredholm properties , Toeplitz operators

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 2 • June, 2010
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