# BOUNDED, COMPACT AND SCHATTEN CLASS WEIGHTED COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

Wolf, Elke

• Published : 2011.07.31
• 34 13

#### Abstract

An analytic self-map ${\phi}$ of the open unit disk $\mathbb{D}$ in the complex plane and an analytic map ${\psi}$ on $\mathbb{D}$ induce the so-called weighted composition operator $C_{{\phi},{\psi}}$: $H(\mathbb{D})\;{\rightarrow}\;H(\mathbb{D})$, $f{\mapsto} \;{\psi}\;(f\;o\;{\phi})$, where H($\mathbb{D}$) denotes the set of all analytic functions on $\mathbb{D}$. We study when such an operator acting between different weighted Bergman spaces is bounded, compact and Schatten class.

#### Keywords

weighted Bergman space;composition operator

#### References

1. J. Bonet, P. Domanski, and M. Lindstrom, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 42 (1999), no. 2, 139-148. https://doi.org/10.4153/CMB-1999-016-x
2. J. Bonet, P. Domanski, M. Lindstrom, and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), no. 1, 101-118. https://doi.org/10.1017/S1446788700001336
3. J. Bonet, M. Lindstrom, and E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Austral. Math. Soc. 84 (2008), no. 1, 9-20.
4. M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 69 (2000), no. 1, 41-60. https://doi.org/10.1017/S144678870000183X
5. C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
6. Z. Cuckovic and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. (2) 70 (2004), no. 2, 499-511. https://doi.org/10.1112/S0024610704005605
7. P. Duren and A. Schuster, Bergman Spaces, Mathematical Surveys and Monographs 100, American Mathematical Society, Providence, RI, 2004.
8. W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237-241. https://doi.org/10.1090/S0002-9939-1975-0374886-9
9. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics 199, Springer-Verlag, New York, 2000.
10. T. Kriete and B. MacCluer, Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J. 41 (1992), no. 3, 755-788. https://doi.org/10.1512/iumj.1992.41.41040
11. B. MacCluer, S. Ohno, and R. Zhao, Topological structure of the space of composition operators on $H^{\infty}$, Integral Equations Operator Theory 40 (2001), no. 4, 481-494. https://doi.org/10.1007/BF01198142
12. J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), no. 1, 70-92. https://doi.org/10.1016/j.jfa.2004.01.012
13. P. Nieminen, Compact differences of composition operators on Bloch and Lipschitz spaces, Comput. Methods Funct. Theory 7 (2007), no. 2, 325-344. https://doi.org/10.1007/BF03321648
14. N. Palmberg, Weighted composition operators with closed range, Bull. Austral. Math. Soc. 75 (2007), no. 3, 331-354. https://doi.org/10.1017/S0004972700039277
15. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.
16. E. Wolf, Weighted composition operators between weighted Bergman spaces, RACSAM Rev. R. Acad. Cienc. Exactas Fis Nat. Ser. A Mat. 103 (2009), no. 1, 11-15. https://doi.org/10.1007/BF03191830
17. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.
18. K. Zhu, Schatten class composition operators on weighted Bergman spaces of the disk, J. Operator Theory 46 (2001), no. 1, 173-181.

#### Cited by

1. ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS vol.93, pp.03, 2012, https://doi.org/10.1017/S1446788712000316