Research Article | Open Access
Stevo SteviΔ, Ajay K. Sharma, S. D. Sharma, "Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane", Abstract and Applied Analysis, vol. 2011, Article ID 989625, 10 pages, 2011. https://doi.org/10.1155/2011/989625
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., [1β22] 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 [25β30] 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, 7β9, 11, 12, 16β18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19β21].
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 FrΓ©chet 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 MΓΆbius 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 ofββ. Then is compact if and only if, for any bounded sequence converging to zero on compacts ofββ, one has
Theorem 2.9. Let and be a holomorphic self-map ofββ . 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)β(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
- 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.
- V. Matache, βComposition operators on Hp of the upper half-plane,β Analele Universităţii din Timişoara, Seria Ştiinţe Matematice, vol. 27, no. 1, pp. 63β66, 1989. View at: Google Scholar
- V. Matache, βComposition operators on Hardy spaces of a half-plane,β Proceedings of the American Mathematical Society, vol. 127, no. 5, pp. 1483β1491, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- H. J. Schwartz, Composition operators on Hp, Ph.D. thesis, University of Toledo, Ohio, USA, 1969.
- J. H. Shapiro and W. Smith, βHardy spaces that support no compact composition operators,β Journal of Functional Analysis, vol. 205, no. 1, pp. 62β89, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- A. K. Sharma and S. D. Sharma, βWeighted composition operators between Bergman-type spaces,β Communications of the Korean Mathematical Society, vol. 21, no. 3, pp. 465β474, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. D. Sharma, A. K. Sharma, and S. Ahmed, βCarleson measures in a vector-valued Bergman space,β Journal of Analysis and Applications, vol. 4, no. 1, pp. 65β76, 2006. View at: Google Scholar | Zentralblatt MATH
- S. D. Sharma, A. K. Sharma, and S. Ahmed, βComposition operators between Hardy and Bloch-type spaces of the upper half-plane,β Bulletin of the Korean Mathematical Society, vol. 44, no. 3, pp. 475β482, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. D. Sharma, A. K. Sharma, and Z. Abbas, βWeighted composition operators on weighted vector-valued Bergman spaces,β Applied Mathematical Sciences, vol. 4, no. 41–44, pp. 2049β2063, 2010. View at: Google Scholar
- A. L. Shields and D. L. Williams, βBonded projections, duality, and multipliers in spaces of analytic functions,β Transactions of the American Mathematical Society, vol. 162, pp. 287β302, 1971. View at: Google Scholar
- R. K. Singh and S. D. Sharma, βComposition operators on a functional Hilbert space,β Bulletin of the Australian Mathematical Society, vol. 20, no. 3, pp. 377β384, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- R. K. Singh and S. D. Sharma, βNoncompact composition operators,β Bulletin of the Australian Mathematical Society, vol. 21, no. 1, pp. 125β130, 1980. View at: Publisher Site | Google Scholar
- S. Stević, βWeighted composition operators between mixed norm spaces and spaces in the unit ball,β Journal of Inequalities and Applications, Article ID 28629, 9 pages, 2007. View at: Google Scholar
- S. Stević, βNorm of weighted composition operators from Bloch space to on the unit ball,β Ars Combinatoria, vol. 88, pp. 125β127, 2008. View at: Google Scholar
- S. Stević, βWeighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball,β Applied Mathematics and Computation, vol. 212, no. 2, pp. 499β504, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Stević, βComposition operators from the Hardy space to Zygmund-type spaces on the upper half-plane and the unit disc,β Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 305β312, 2010. View at: Google Scholar
- S. Stević, βComposition operators from the Hardy space to the th weighted-type space on the unit disk and the half-plane,β Applied Mathematics and Computation, vol. 215, no. 11, pp. 3950β3955, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Stević, βComposition operators from the weighted Bergman space to the th weighted-type space on the upper half-plane,β Applied Mathematics and Computation, vol. 217, no. 7, pp. 3379β3384, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S.-I. Ueki, βHilbert-Schmidt weighted composition operator on the Fock space,β International Journal of Mathematical Analysis, vol. 1, no. 13–16, pp. 769β774, 2007. View at: Google Scholar | Zentralblatt MATH
- S.-I. Ueki, βWeighted composition operators on the Bargmann-Fock space,β International Journal of Modern Mathematics, vol. 3, no. 3, pp. 231β243, 2008. View at: Google Scholar | Zentralblatt MATH
- S.-I. Ueki, βWeighted composition operators on some function spaces of entire functions,β Bulletin of the Belgian Mathematical Society Simon Stevin, vol. 17, no. 2, pp. 343β353, 2010. View at: Google Scholar | Zentralblatt MATH
- X. Zhu, βWeighted composition operators from area Nevanlinna spaces into Bloch spaces,β Applied Mathematics and Computation, vol. 215, no. 12, pp. 4340β4346, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- F. Forelli, βThe isometries of Hpspaces,β Canadian Journal of Mathematics, vol. 16, pp. 721β728, 1964. View at: Publisher Site | Google Scholar
- C. J. Kolaski, βIsometries of weighted Bergman spaces,β Canadian Journal of Mathematics, vol. 34, no. 4, pp. 910β915, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Li and S. Stević, βGeneralized composition operators on Zygmund spaces and Bloch type spaces,β Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282β1295, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Stević, βOn a new operator from to the Bloch-type space on the unit ball,β Utilitas Mathematica, vol. 77, pp. 257β263, 2008. View at: Google Scholar
- S. Stević, βOn an integral operator from the Zygmund space to the Bloch-type space on the unit ball,β Glasgow Mathematical Journal, vol. 51, no. 2, pp. 275β287, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Stević, βOn a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball,β Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 426β434, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Stević, βOn an integral operator between Bloch-type spaces on the unit ball,β Bulletin des Sciences Mathématiques, vol. 134, no. 4, pp. 329β339, 2010. View at: Google Scholar
- X. Zhu, βGeneralized weighted composition operators from Bloch type spaces to weighted Bergman spaces,β Indian Journal of Mathematics, vol. 49, no. 2, pp. 139β150, 2007. View at: Google Scholar | Zentralblatt MATH
- K. L. Avetisyan, βIntegral representations in general weighted Bergman spaces,β Complex Variables, vol. 50, no. 15, pp. 1151β1161, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- K. L. Avetisyan, βHardy-Bloch type spaces and lacunary series on the polydisk,β Glasgow Mathematical Journal, vol. 49, no. 2, pp. 345β356, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- K. D. Bierstedt and W. H. Summers, βBiduals of weighted Banach spaces of analytic functions,β Australian Mathematical Society Journal Series A, vol. 54, no. 1, pp. 70β79, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH
Copyright
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.