## Revista Matemática Iberoamericana

### Toeplitz operators on Bergman spaces with locally integrable symbols

#### 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

https://projecteuclid.org/euclid.rmi/1275671316

Mathematical Reviews number (MathSciNet)
MR2677012

Zentralblatt MATH identifier
1204.47040

#### 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

#### References

• Axler, S. and Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47 (1998), no. 2, 387-400.
• Karapetyants, A.: The space $\rm BMO^p_\lambda (\mathbbD)$, compact Toeplitz operators with $\rm BMO^1_\lambda (\mathbbD)$ symbols on weighted Bergman spaces, and the Berezin transform. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2006, no. 8, 76-79; translation in Russian Math. (Iz. VUZ) 50 (2006), no. 8, 71-74 (2007).
• Luecking, D.H.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73, no. 2, (1987) 345-368.
• Murphy, G.J.: C*-algebras and operator theory. Academic Press, Boston, 1990.
• Suárez, D.: The essential norm of operators in the Toeplitz algebra on $A^p(\mathbbB_n)$. Indiana Univ. Math. J. 56, no. 5, (2007) 2185-2232.
• Taskinen, J. and Virtanen, J.A.: Spectral theory of Toeplitz and Hankel operators on the Bergman space $A^1$. New York J. Math. 14 (2008), 305-323.
• Vasilevski, N.: Commutative algebras of Toeplitz operators on the Bergman space. Operator Theory: Advances and Applications 185. Birkhäuser Verlag, Basel, 2008.
• Zhu, K.: Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Operator Theory 20 (1988), no. 2, 329-357.
• Zhu, K.: BMO and Hankel operators on Bergman spaces. Pacific J. Math. 155 (1992), no. 2, 377-395.
• Zhu, K.: Operator theory in function spaces. Second edition. Mathematical Surveys and Monographs 138. American Mathematical Society, Providence, RI, 2007.
• Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Sci. 2003, no. 46, 2929-2945.