Revista Matemática Iberoamericana

Toeplitz operators on Bergman spaces with locally integrable symbols

Jari Taskinen and Jani Virtanen

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Abstract

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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 693-706.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1275671316

Mathematical Reviews number (MathSciNet)
MR2677012

Zentralblatt MATH identifier
1204.47040

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

Keywords
Toeplitz operators Bergman spaces boundedness compactness Fredholm properties

Citation

Taskinen, Jari; Virtanen, Jani. Toeplitz operators on Bergman spaces with locally integrable symbols. Rev. Mat. Iberoamericana 26 (2010), no. 2, 693--706. https://projecteuclid.org/euclid.rmi/1275671316


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