About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 151929, 9 pages
http://dx.doi.org/10.1155/2013/151929
Research Article

Weighted Differentiation Composition Operators to Bloch-Type Spaces

1Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
2School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, India

Received 2 February 2013; Accepted 8 April 2013

Academic Editor: Pedro M. Lima

Copyright © 2013 Junming Liu 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.

Abstract

We characterized the boundedness and compactness of weighted differentiation composition operators from BMOA and the Bloch space to Bloch-type spaces. Moreover, we obtain new characterizations of boundedness and compactness of weighted differentiation composition operators.

1. Introduction

Let be the open unit disk in the complex plane , the space of all functions holomorphic on , the normalized area measure on , and the space of all bounded holomorphic functions with the norm .

Let . The -Bloch space on is the space of all holomorphic functions on such that The little -Bloch space consists of all such that Both spaces and are Banach spaces with the norm and is a closed subspace of . If , they become the classical Bloch space and little Bloch space , respectively. For any , the space consists of functions such that For information of such spaces, see, for example, [14].

For , let be the automorphism of that interchanges and . Let the Green function in with logarithmic singularity at be given by The space consists of all in the Hardy space such that is a Banach space under following norm (see, e.g., [5]):

Let and be holomorphic maps on the open unit disk such that . For a nonnegative integer , we define a linear operator as follows: We call it weighted differentiation composition operators, which was defined in [6, 7]. If and , becomes induced by , defined as , . If and , then is the differentiation operator defined as . If , then we get the weighted composition operator defined as . If and , then reduces to . When , then reduces to differentiation composition operator (also named as product of differentiation and composition operator). If we put , then , the product of multiplication and differentiation operator.

The boundedness and compactness of differentiation composition operator between spaces of holomorphic functions have been studied extensively. For example, Hibschweiler; Portnoy and Ohno studied differentiation composition operator on Hardy and Bergman spaces in [8, 9]; Li; Stević and Ohno studied on Bloch type spaces in [1012]; Wu and Wulan gave a new compactness criterion of on the Bloch space in [13]. Recently, the weighted differentiation composition operator between different function spaces has also been investigated by several authors (see, for example, [1421]).

Boundedness, compactness, and essential norm of weighted composition operator between Bloch-type spaces have been studied in [2224]. Recently, Manhas and Zhao [25] and Hyvärinen and Lindström [26] gave a new characterization of boundedness and compactness of in terms of the norm of (for the compactness of composition operator, see [27, 28]).

Motivated by [13, 25, 26], we study the operator from and Bloch space to Bloch-type spaces.

Throughout this paper, constants are denoted by ; they are positive and not necessarily the same at each occurrence. The notation means that there is a positive constant such that . When and , we write .

2. Some Lemmas

It is well known that . From the definition of the norm, we know Indeed, Girela proved that in Corollary 5.2 of [5]. The following lemma is from Lemma 5 in [29] (see also Lemma 4.12 of [4]).

Lemma 1. If , then

The following lemma may be known, but we fail to find its reference; so we give a proof for the completeness of the paper.

Lemma 2. Let . Then,

Proof. Applying Littlewood-Paley identity and Lemma 1, we have It follows from the definitions of Bloch space and space that

By Theorem 6.2 of [5] and the proof of Theorem 1 of [30], we have the following lemma.

Lemma 3. Let be a fixed positive integer and with . If then .

Lemma 4. Suppose that is a fixed positive integer. Let , , and If , then there are two positive constants and , depending only on , such that

Proof. The proof is similar to that of Lemma 2.2 of [13] and is so omitted.

3. Boundedness of

In this section, we characterize the boundedness of from and the Bloch space to Bloch-type spaces.

Theorem 5. Let , , , and a holomorphic self-map of . Then, the following statements are equivalent: (a) is bounded. (b) and are bounded. (c) is bounded. (d) and are bounded. (e) is bounded. (f) and are bounded. (g)  and .(h) and .

Proof. It is obvious that , , , and . Thus, we will prove the theorem according to the following steps. (I): , . (II): , . (III): , . (IV): .
(I): , . Suppose that or holds. We choose the test function . By Lemma 2, we get So Taking and using the fact that , we have We now consider the function It is easy to check that and . Moreover, Thus, and We obtain Thus, for any , we have Using (21) yields Combining (26) with (27), we get We next consider the function Similarly, we get and Moreover, So and . We have, as above,
Thus, for any , Applying (20), we get Combining (34) with (35) yields (II): and . Suppose that is bounded or is bounded. Set If , then for any positive integer , we can find such that If , then choose the test function . It is clear that . From Lemma 2, we have
So If , consider the function where . Let . Then, and It is easy to see that So, by Theorems 5.4 and 5.13 of [4], we have and . By Lemma 1 of [31] and Lemma 3, we get . We have Since is arbitrary, we get . This contradicts the boundedness of and that of .
Now, suppose that is bounded or is bounded. Set If , then for any positive integer , exists such that If , then set . The process as above gives If , consider the function where . Let . Then, and Applying Theorems 5.4 and 5.13 of [4] again yields and . We get and Since is arbitrary, we have . This contradicts the boundedness of .
(III): , . Note that The desired results follow.
(IV): . Suppose that is true. It follows from Proposition 5.1 of [4] that . So,
Conversely, assume that is true. It is easy to see that If , then Hence, is true. From , we obtain that is also true.
From now on, we assume that . For any integer , let Let with be the smallest positive integer such that . Since is not empty for every integer and . By Lemma 4, for , So, is bounded. Similar argument implies Thus, is bounded. Theorem 5 is proved.

4. Compactness of

The following criterion for the compactness is a useful tool and it follows from standard arguments, for example, Proposition 3.11 of [32] or Lemma 2.10 of [33].

Lemma 6. Let , , and , or . Suppose that and are in such that . Then, is compact if and only if for any sequence in with , which converges to zero locally uniformly on ; we have .

We now give the compactness of from and the Bloch space to Bloch-type spaces.

Theorem 7. Let , , , and a holomorphic self-map of . Then, the following statements are equivalent: (a) is compact. (b) is compact and is compact. (c) is compact. (d) is compact and is compact. (e) is compact. (f) is compact and is compact. (g), , (h) and .

Proof. The proof is a modification of that of Theorem 5; so we give a sketch of the proof. We will prove the theorem according to the following steps. (I): , . (II): , . (III): , . (IV): .
(I): , . Suppose that or holds. Then by Theorem 5, we have That is, , .
Let be a sequence in such that as . Now, we consider the function Simple computation shows that and It is also easy to check that uniformly on compact subsets of as . Moreover, We have By Lemma 6, we get We next consider the function Similarly, we get and It is easy to see that converges to zero uniformly on compact subsets of as and Thus, Applying Lemma 6 again, we have Since is arbitrary, we proved that is true.
(II) , . Suppose that or holds. A similar argument to (I) shows that , . Now, suppose that the equations in are not true. Then, there exists a sequence in and such that as and Choose a subsequence of if necessary and suppose that . Let Then, it is easy to check that , , uniformly on compact subsets of and Thus, Those contradict the compactness of and .
(III) , . Let be a norm bounded sequence in that converges to zero uniformly on compact subsets of . Let . For , then there exists such that for , we have Thus, for , we have where and . Since uniformly on compact subsets of as , we have as . It follows from Lemma 6 that is compact.
Similar as above, we know From uniformly on compact subsets of , we have and as . So, , are compact.
(IV): . Suppose that is true. Note that and uniformly on compact subsets of as ; by Lemma 6, we have Conversely, assume that is true. It is easy to see that If , from , we get that is true. If , as in the proof of Theorem 5, let And let with be the smallest positive integer such that . For given , there exists a large enough integer with such that whenever . Let be a norm bounded sequence in that converges to zero uniformly on compact subsets of as . Denote . We get Then, where Since uniformly on compact subsets of , then as . Thus, by Lemma 6, is compact. Similar as above, we can prove that is compact. The proof is complete.

Acknowledgments

This work was supported by NNSF of China (Grant no. 11171203) and NSF of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511).

References

  1. P. L. Duren, Theory of Hp Spaces, vol. 38 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1970. View at MathSciNet
  2. H. T. Kaptanoğlu and S. Tülü, “Weighted Bloch, Lipschitz, Zygmund, Bers, and growth spaces of the ball: Bergman projections and characterizations,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 101–127, 2011. View at Zentralblatt MATH · View at MathSciNet
  3. K. Zhu, “Bloch type spaces of analytic functions,” The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 1143–1177, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. K. Zhu, Operator Theory in Function Spaces, vol. 138 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2nd edition, 2007. View at MathSciNet
  5. D. Girela, “Analytic functions of bounded mean oscillation,” in Complex Function Spaces (Mekrijärvi, 1999), vol. 4 of University of Joensuu, Department of Mathematics. Report Series, pp. 61–170, University of Joensuu, Joensuu, Finland, 2001. View at Zentralblatt MATH · View at MathSciNet
  6. X. Zhu, “Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces,” Integral Transforms and Special Functions, vol. 18, no. 3-4, pp. 223–231, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Zhu, “Generalized weighted composition operators on weighted Bergman spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 881–893, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. A. Hibschweiler and N. Portnoy, “Composition followed by differentiation between Bergman and Hardy spaces,” The Rocky Mountain Journal of Mathematics, vol. 35, no. 3, pp. 843–855, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Ohno, “Products of composition and differentiation between Hardy spaces,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 235–243, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Li and S. Stević, “Composition followed by differentiation between Bloch type spaces,” Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195–205, 2007. View at Zentralblatt MATH · View at MathSciNet
  11. S. Ohno, “Products of differentiation and composition on Bloch spaces,” Bulletin of the Korean Mathematical Society, vol. 46, no. 6, pp. 1135–1140, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Stević, “Characterizations of composition followed by differentiation between Bloch-type spaces,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4312–4316, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Wu and H. Wulan, “Products of differentiation and composition operators on the Bloch space,” Collectanea Mathematica, vol. 63, no. 1, pp. 93–107, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. K. Sharma, “Generalized composition operators between weighted Bergman spaces,” Acta Scientiarum Mathematicarum, vol. 78, pp. 187–211, 2012.
  15. A. Sharma and A. K. Sharma, “Carleson measures and a class of generalized integration operators on the Bergman space,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 5, pp. 1711–1724, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. K. Sharma, “Products of multiplication, composition and differentiation between weighted Bergman-Nevanlinna and Bloch-type spaces,” Turkish Journal of Mathematics, vol. 35, no. 2, pp. 275–291, 2011. View at Zentralblatt MATH · View at MathSciNet
  17. S. Stević, “Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces,” Applied Mathematics and Computation, vol. 211, no. 1, pp. 222–233, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Stević, “Weighted differentiation composition operators from H and Bloch spaces to nth weighted-type spaces on the unit disk,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3634–3641, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. Stević and A. K. Sharma, “Iterated differentiation followed by composition from Bloch-type spaces to weighted BMOA spaces,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3574–3580, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Stević and A. K. Sharma, “Composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk,” Annales Polonici Mathematici, vol. 105, no. 1, pp. 77–86, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Y. Yu and Y. Liu, “Weighted differentiation composition operators from H to Zygmund spaces,” Integral Transforms and Special Functions, vol. 22, no. 7, pp. 507–520, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. B. D. MacCluer and R. Zhao, “Essential norms of weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437–1458, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872–884, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191–215, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. S. Manhas and R. Zhao, “New estimates of essential norms of weighted composition operators between Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 32–47, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. O. Hyvärinen and M. Lindström, “Estimates of essential norms of weighted composition operators between Bloch-type spaces,” Journal of Mathematical Analysis and Applications, vol. 393, no. 1, pp. 38–44, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. H. Wulan, D. Zheng, and K. Zhu, “Compact composition operators on BMOA and the Bloch space,” Proceedings of the American Mathematical Society, vol. 137, no. 11, pp. 3861–3868, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. Zhao, “Essential norms of composition operators between Bloch type spaces,” Proceedings of the American Mathematical Society, vol. 138, no. 7, pp. 2537–2546, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Liu and C. Xiong, “Norm-attaining integral operators on analytic function spaces,” Journal of Mathematical Analysis and Applications, vol. 399, no. 1, pp. 108–115, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. R. Aulaskari, M. Nowak, and R. Zhao, “The nth derivative characterisation of Möbius invariant Dirichlet space,” Bulletin of the Australian Mathematical Society, vol. 58, no. 1, pp. 43–56, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. Zhao, “Distances from Bloch functions to some Möbius invariant spaces,” Annales Academiae Scientiarum Fennicae, vol. 33, no. 1, pp. 303–313, 2008. View at Zentralblatt MATH · View at MathSciNet
  32. 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. View at MathSciNet
  33. M. Tjani, Compact composition operators on some Möbius invariant Banach spaces [Ph.D. thesis], Michigan State University, 1996.