Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane

Abstract

Let be a holomorphic mapping on the upper half-plane and be a holomorphic self-map of . We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on , we also characterize the compactness of these operators.

1. Introduction and Preliminaries

Let be the upper half-plane, a domain in or , and the space of all holomorphic functions on . Let , and let be a holomorphic self-map of . Then by is defined a linear operator on which is called weighted composition operator. If , then becomes composition operator and is denoted by , and if , then becomes multiplication operator and is denoted by .

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in or (see, e.g., [1–22] and the references therein). There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [23, 24]). For some other operators related to weighted composition operators, see [25–30] and the references therein.

While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk , there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane (see, e.g., [2, 3, 5, 7–9, 11, 12, 16–18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19–21].

The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk . For example, there are holomorphic self-maps of which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map of induces a bounded composition operator on the Hardy and weighted Bergman spaces on . Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see [3, 5]).

For and , let denote the collection of all Lebesgue -integrable functions such that where , and .

Let . For is a Banach space with the norm defined by With this norm becomes a Banach space when , while for it is a Fréchet space with the translation invariant metric

Recall that for every the following estimate holds: where is a positive constant independent of .

Let . The weighted-type space (or growth space) on the upper half-plane consists of all such that It is easy to check that is a Banach space with the norm defined above. For weighted-type spaces on the unit disk, polydisk, or the unit ball see, for example, papers [10, 32, 33] and the references therein.

Given two Banach spaces and , we recall that a linear map is bounded if is bounded for every bounded subset of . In addition, we say that is compact if is relatively compact for every bounded set .

In this paper, we consider the boundedness and compactness of weighted composition operators acting from to the weighted-type space . Related results on the unit disk and the unit ball can be found, for example, in [6, 13, 15].

Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . Moreover, if both and hold, then one says that .

2. Main Results

The boundedness and compactness of the weighted composition operator are characterized in this section.

Theorem 2.1. Let , and let be a holomorphic self-map of . Then is bounded if and only if Moreover, if the operator is bounded then the following asymptotic relationship holds:

Proof. First suppose that (2.1) holds. Then for any and , by (1.6) we have and so by (2.1), is bounded and moreover Conversely suppose is bounded. Consider the function Then and moreover (see, e.g., Lemma 1 in [18]).
Thus the boundedness of implies that for every . In particular, if is fixed then for , we get Since is arbitrary, (2.1) follows and moreover If is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.

Corollary 2.2. Let , and be such that and . Then is bounded if and only if , where

Example 2.3. Let and be such that and . Let be a holomorphic map of defined as For and in , we have Thus if . Similarly if . By Corollary 2.2, it follows that is bounded.

Corollary 2.4. Let , and let be a holomorphic self-map of . Then is bounded if and only if

Corollary 2.5. Let be the linear fractional map Then necessary and sufficient condition that is bounded is that and .

Proof. Assume that is bounded. Then which is finite only if and .
Conversely, if and , then from (2.13) we get , and by some calculation Hence is bounded.

Corollary 2.6. Let , and be such that . Let be a holomorphic self-map of and . Then the weighted composition operator acts boundedly from to .

Proof. By Theorem 2.1, is bounded if and only if By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map of and all where the equality holds when is a Möbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator .

Corollary 2.6 enables us to show that there exist , and holomorphic maps and of the upper half-plane such that neither nor is bounded, but is bounded.

Example 2.7. Let , and be such that . Let , and . Then by Corollary 2.5, is not bounded. On the other hand, if then and so by Corollary 2.2, is not bounded. However, by Corollary 2.6, we have that is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operators and it follows from standard arguments ([4]).

Lemma 2.8. Let , and let be a holomorphic self-map of  . Then is compact if and only if, for any bounded sequence converging to zero on compacts of  , one has

Theorem 2.9. Let and be a holomorphic self-map of   . If is compact, then

Proof. Suppose is compact and (2.20) does not hold. Then there is a and a sequence such that and for all . Let , , and Then is a norm bounded sequence and on compacts of as . By Lemma 2.8 it follows that On the other hand, which is a contradiction. Hence (2.20) must hold, as claimed.

Before we formulate and prove a converse of Theorem 2.9, we define, for every such that , the following subset of :

Theorem 2.10. Let , and let be a holomorphic self-map of and be bounded. Suppose that and as within for all and . Then is compact if condition (2.20) holds.

Proof. Assume (2.20) holds. Then for each , there is an such that Let be a sequence in such that and uniformly on compact subsets of as . Thus for such that and each , we have
From estimate (1.6) we have Thus there is an such that whenever . Hence for such that and each we have If , then by the assumption there is an such that , whenever . Therefore, for each we have whenever and .
If and , then there exists some such that for all , and so Combining (2.27)–(2.32), we have that for and some independent of . Since is an arbitrary positive number, by Lemma 2.8, it follows that is compact.

Example 2.11. Let , and be such that . Let and , then and . It is easy to see that . Beside this, for , we have
Also and the set is empty. Thus and satisfy all the assumptions of Theorem 2.10, and so is compact.

Acknowledgments

The work is partially supported by the Serbian Ministry of Science, projects III 41025 and III 44006. The work of the second author is a part of the research project sponsored by National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426).

References

1. C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
2. V. Matache, “Composition operators on H^p of the upper half-plane,” Analele Universităţii din Timişoara, Seria Ştiinţe Matematice, vol. 27, no. 1, pp. 63–66, 1989. View at: Google Scholar
3. V. Matache, “Composition operators on Hardy spaces of a half-plane,” Proceedings of the American Mathematical Society, vol. 127, no. 5, pp. 1483–1491, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
4. H. J. Schwartz, Composition operators on H^p, Ph.D. thesis, University of Toledo, Ohio, USA, 1969.
5. J. H. Shapiro and W. Smith, “Hardy spaces that support no compact composition operators,” Journal of Functional Analysis, vol. 205, no. 1, pp. 62–89, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
6. A. K. Sharma and S. D. Sharma, “Weighted composition operators between Bergman-type spaces,” Communications of the Korean Mathematical Society, vol. 21, no. 3, pp. 465–474, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
7. S. D. Sharma, A. K. Sharma, and S. Ahmed, “Carleson measures in a vector-valued Bergman space,” Journal of Analysis and Applications, vol. 4, no. 1, pp. 65–76, 2006. View at: Google Scholar | Zentralblatt MATH
8. S. D. Sharma, A. K. Sharma, and S. Ahmed, “Composition operators between Hardy and Bloch-type spaces of the upper half-plane,” Bulletin of the Korean Mathematical Society, vol. 44, no. 3, pp. 475–482, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
9. S. D. Sharma, A. K. Sharma, and Z. Abbas, “Weighted composition operators on weighted vector-valued Bergman spaces,” Applied Mathematical Sciences, vol. 4, no. 41–44, pp. 2049–2063, 2010. View at: Google Scholar
10. A. L. Shields and D. L. Williams, “Bonded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287–302, 1971. View at: Google Scholar
11. R. K. Singh and S. D. Sharma, “Composition operators on a functional Hilbert space,” Bulletin of the Australian Mathematical Society, vol. 20, no. 3, pp. 377–384, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH
12. R. K. Singh and S. D. Sharma, “Noncompact composition operators,” Bulletin of the Australian Mathematical Society, vol. 21, no. 1, pp. 125–130, 1980. View at: Publisher Site | Google Scholar
13. S. Stević, “Weighted composition operators between mixed norm spaces and $H_{\alpha }^{\infty }$ spaces in the unit ball,” Journal of Inequalities and Applications, Article ID 28629, 9 pages, 2007. View at: Hα∞%20spaces%20in%20the%20unit%20ball&author=S. Stević&publication_year=2007" target="_blank">Google Scholar
14. S. Stević, “Norm of weighted composition operators from Bloch space to $H_{\mu }^{\infty }$ on the unit ball,” Ars Combinatoria, vol. 88, pp. 125–127, 2008. View at: Hμ∞%20on%20the%20unit%20ball&author=S. Stević&publication_year=2008" target="_blank">Google Scholar
15. S. Stević, “Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 499–504, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
16. S. Stević, “Composition operators from the Hardy space to Zygmund-type spaces on the upper half-plane and the unit disc,” Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 305–312, 2010. View at: Google Scholar
17. S. Stević, “Composition operators from the Hardy space to the $n-$th weighted-type space on the unit disk and the half-plane,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3950–3955, 2010. View at: Publisher Site | n-th%20weighted-type%20space%20on%20the%20unit%20disk%20and%20the%20half-plane&author=S. Stević&publication_year=2010" target="_blank">Google Scholar | Zentralblatt MATH
18. S. Stević, “Composition operators from the weighted Bergman space to the $n-$th weighted-type space on the upper half-plane,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3379–3384, 2010. View at: Publisher Site | n-th%20weighted-type%20space%20on%20the%20upper%20half-plane&author=S. Stević&publication_year=2010" target="_blank">Google Scholar | Zentralblatt MATH
19. S.-I. Ueki, “Hilbert-Schmidt weighted composition operator on the Fock space,” International Journal of Mathematical Analysis, vol. 1, no. 13–16, pp. 769–774, 2007. View at: Google Scholar | Zentralblatt MATH
20. S.-I. Ueki, “Weighted composition operators on the Bargmann-Fock space,” International Journal of Modern Mathematics, vol. 3, no. 3, pp. 231–243, 2008. View at: Google Scholar | Zentralblatt MATH
21. S.-I. Ueki, “Weighted composition operators on some function spaces of entire functions,” Bulletin of the Belgian Mathematical Society Simon Stevin, vol. 17, no. 2, pp. 343–353, 2010. View at: Google Scholar | Zentralblatt MATH
22. X. Zhu, “Weighted composition operators from area Nevanlinna spaces into Bloch spaces,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4340–4346, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
23. F. Forelli, “The isometries of H^pspaces,” Canadian Journal of Mathematics, vol. 16, pp. 721–728, 1964. View at: Publisher Site | Google Scholar
24. C. J. Kolaski, “Isometries of weighted Bergman spaces,” Canadian Journal of Mathematics, vol. 34, no. 4, pp. 910–915, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
25. S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282–1295, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
26. S. Stević, “On a new operator from $H^{\infty }$ to the Bloch-type space on the unit ball,” Utilitas Mathematica, vol. 77, pp. 257–263, 2008. View at: H∞%20to%20the%20Bloch-type%20space%20on%20the%20unit%20ball&author=S. Stević&publication_year=2008" target="_blank">Google Scholar
27. S. Stević, “On an integral operator from the Zygmund space to the Bloch-type space on the unit ball,” Glasgow Mathematical Journal, vol. 51, no. 2, pp. 275–287, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
28. S. Stević, “On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball,” Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 426–434, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
29. S. Stević, “On an integral operator between Bloch-type spaces on the unit ball,” Bulletin des Sciences Mathématiques, vol. 134, no. 4, pp. 329–339, 2010. View at: Google Scholar
30. 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. View at: Google Scholar | Zentralblatt MATH
31. K. L. Avetisyan, “Integral representations in general weighted Bergman spaces,” Complex Variables, vol. 50, no. 15, pp. 1151–1161, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
32. K. L. Avetisyan, “Hardy-Bloch type spaces and lacunary series on the polydisk,” Glasgow Mathematical Journal, vol. 49, no. 2, pp. 345–356, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
33. K. D. Bierstedt and W. H. Summers, “Biduals of weighted Banach spaces of analytic functions,” Australian Mathematical Society Journal Series A, vol. 54, no. 1, pp. 70–79, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH

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Copyright © 2011 Stevo Stević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.