## Abstract and Applied Analysis

### A Theorem of Nehari Type on Weighted Bergman Spaces of the Unit Ball

#### Abstract

This paper shows that if $S$ is a bounded linear operator acting on the weighted Bergman spaces ${A}_{\alpha }^{2}$ on the unit ball in ${\mathbb{C}}^{n}$ such that $S{T}_{{z}_{i}}={T}_{{\overline{z}}_{i}}S\text{\,}(i=1,\ldots ,n)$, where ${T}_{{z}_{i}}={z}_{i}f$ and ${T}_{{\overline{z}}_{i}}=P({\overline{z}}_{i}f)$; and where $P$ is the weighted Bergman projection, then $S$ must be a Hankel operator.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 538573, 7 pages.

Dates
First available in Project Euclid: 10 February 2009

https://projecteuclid.org/euclid.aaa/1234298998

Digital Object Identifier
doi:10.1155/2008/538573

Mathematical Reviews number (MathSciNet)
MR2466220

Zentralblatt MATH identifier
1162.32003

#### Citation

Lu, Yufeng; Yang, Jun. A Theorem of Nehari Type on Weighted Bergman Spaces of the Unit Ball. Abstr. Appl. Anal. 2008 (2008), Article ID 538573, 7 pages. doi:10.1155/2008/538573. https://projecteuclid.org/euclid.aaa/1234298998

#### References

• K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
• K. Avetisyan and S. Stević, Equivalent conditions for Bergman space and Littlewood-Paley type inequalities,'' Journal of Computational Analysis and Applications, vol. 9, no. 1, pp. 15--28, 2007.
• F. Beatrous and J. Burbea, Holomorphic Sobolev spaces on the ball,'' Dissertationes Mathematicae, vol. 276, pp. 1--60, 1989.
• G. Benke and D.-C. Chang, A note on weighted Bergman spaces and the Cesàro operator,'' Nagoya Mathematical Journal, vol. 159, pp. 25--43, 2000.
• T. L. Kriete III and B. D. MacCluer, Composition operators on large weighted Bergman spaces,'' Indiana University Mathematics Journal, vol. 41, no. 3, pp. 755--788, 1992.
• S. Li and S. Stević, Integral type operators from mixed-norm spaces to $\alpha$-Bloch spaces,'' Integral Transforms and Special Functions, vol. 18, no. 7-8, pp. 485--493, 2007.
• L. Luo and S.-I. Ueki, Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of $\mathbbC^n$,'' Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 88--100, 2007.
• J. H. Shi, Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\mathbbC^n$,'' Transactions of the American Mathematical Society, vol. 328, no. 2, pp. 619--637, 1991.
• S. Stević, A generalization of a result of Choa on analytic functions with Hadamard gaps,'' Journal of the Korean Mathematical Society, vol. 43, no. 3, pp. 579--591, 2006.
• S. Stević, Continuity with respect to symbols of composition operators on the weighted Bergman space,'' Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1177--1188, 2007.
• S. Stević, Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,'' Utilitas Mathematica, vol. 76, pp. 59--64, 2008.
• S. Stević, On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,'' Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008.
• X. Zhu, Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces,'' Indian Journal of Mathematics, vol. 49, no. 2, pp. 139--150, 2007.
• J. Barría and P. R. Halmos, Asymptotic Toeplitz operators,'' Transactions of the American Mathematical Society, vol. 273, no. 2, pp. 621--630, 1982.
• P. Lin and R. Rochberg, Hankel operators on the weighted Bergman spaces with exponential type weights,'' Integral Equations and Operator Theory, vol. 21, no. 4, pp. 460--483, 1995.
• V. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003.
• S. C. Power, Hankel Operators on Hilbert Space, vol. 64 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1982.
• S. C. Power, $C^\ast$-algebras generated by Hankel operators and Toeplitz operators,'' Journal of Functional Analysis, vol. 31, no. 1, pp. 52--68, 1979.
• Z. Nehari, On bounded bilinear forms,'' Annals of Mathematics, vol. 65, no. 1, pp. 153--162, 1957.
• N. S. Faour, A theorem of Nehari type,'' Illinois Journal of Mathematics, vol. 35, no. 4, pp. 533--535, 1991.
• Y. Lu and S. Sun, Hankel operators on generalized $H^2$spaces,'' Integral Equations and Operator Theory, vol. 34, no. 2, pp. 227--233, 1999.
• K. H. Zhu, Duality and Hankel operators on the Bergman spaces of bounded symmetric domains,'' Journal of Functional Analysis, vol. 81, no. 2, pp. 260--278, 1988. \endthebibliography