Toeplitz operators on Bergman spaces with locally integrable symbols

Taskinen, J. and Virtanen, J. (2010) Toeplitz operators on Bergman spaces with locally integrable symbols. Revista Matemática Iberoamericana, 26 (2). pp. 693-706. ISSN 0213-2230 doi: 10.4171/RMI/614

Abstract/Summary

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.