Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 925901, 6 pages

http://dx.doi.org/10.1155/2013/925901

Research Article

A New Characterization of Generalized Weighted Composition Operators from the Bloch Space into the Zygmund Space

Hao Li¹ and Xiaohong Fu²

¹Department of Mathematics, Henan Normal University, Xinxiang 453007, China

²Department of Mathematics, JiaYing University, Meizhou, Guangdong 514015, China

Received 10 April 2013; Accepted 16 June 2013

Academic Editor: Jiecheng Chen

Copyright © 2013 Hao Li and Xiaohong Fu. 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.

A new criterion for the boundedness and the compactness of the generalized weighted composition operators from the Bloch space into the Zygmund space is given in this paper.

1. Introduction

Let be the space of analytic functions on the unit disk . An is said to belong to the Bloch space, denoted by , if

Let be the little Bloch space, which consists of those such that , as . Denote by the set of all such that

where the supremum is taken over all and . In fact, if and only if by Theorem 5.3 of [1] and the Closed Graph Theorem. The class with the norm

is a Banach space. We call the Zygmund space. For more information on the Zygmund space, see, for example, [1, 2].

Let denote the set of analytic self-map of . Associated with is the composition operator , which is defined by for and . We refer the readers to the book [3] for the theory of the composition operator on various function spaces.

Let . The weighted composition operator, denoted by , is defined by

Let be a nonnegative integer. The generalized weighted composition operator, denoted by , is defined as follows (see [4–7]):

When , then becomes the weighted composition operator. If and , then . If and , then . If and , then . The operators and were studied, for example, in [8–15].

Composition operators, weighted composition operators, and related operators on the Bloch space were studied in [10, 12–22], while composition operators and weighted composition operators between the Zygmund space and some other spaces were studied in [2, 12, 23–28].

Recently, many researchers studied the generalized weighted composition operator on various spaces; see, for example, in [4–7, 25, 26, 29–32]. In [26], Stević has studied the operator from the Bloch space to the th weighted-type space, which includes the Zygmund space. Among other things, he obtained the following result (see [26]).

Theorem A. Let , , an analytic self-map of   and a positive integer. Then the following assertions hold.

(a)The operator is bounded if and only if

(b)Suppose that the operator is bounded then is compact if and only if

Here we give a new criterion for the boundedness or compactness of the operator ; namely, we use three families of functions to characterize the operator .

Throughout the paper, denotes a positive constant which may differ from one occurrence to the other. The notation means that there exists a positive constant such that .

2. Main Results

In this section we give our main results and proofs. For , set

Next, we will use these three families of functions to characterize generalized weighted composition operators .

Theorem 1. Let , be an analytic self-map of , and a positive integer. Then the following conditions are equivalent:

(a)the operator is bounded;(b)the operator is bounded;(c), , ,

Proof. (a)⇒(b). This implication is obvious.

(b)⇒(c). Assume that is bounded. Taking the functions , , and and using the boundedness of , we see that , and and are finite. For each , it is easy to check that . Moreover , are bounded by constants independent of . By the boundedness of , we get

as desired.

(c)⇒(a). Suppose that , and and and are finite. To prove this implication, we only need to show that these conditions imply (6). A calculation shows that

For the simplicity, we denote by . From (11), for , we have

Multiplying (12) by and then adding (13), we get

Multiplying (12) by and (14) by 2, respectively, we obtain

Multiplying (15) by , we get

Subtracting (17) from (16), we obtain

which implies that

From (16) and (18), we obtain

which implies that

By (12), (18), and (22), we have

which implies that

Fix . If , then from (21) we obtain

On the other hand, if , we get

From (30) and (31) we see that is finite. Similarly, from (25) and (29) we can obtain that and are finite as well. The proof of this theorem is finished.

To get the characterization of the compactness of , we need the following criterion, which follows from standard arguments similar to those outlined in Proposition 3.11 of [3].

Lemma 2. Let , , an analytic self-map of and be a positive integer. The operator is compact if and only if is bounded, and for any bounded sequence in which converges to zero uniformly on compact subsets of , one has as .

Theorem 3. Let , be an analytic self-map of , and a positive integer. Suppose that the operator is bounded; then the following conditions are equivalent:

(a)the operator is compact;(b)the operator is compact;(c).

Proof. (a)⇒(b). This implication is clear.

(b)⇒(c). Assume that is compact. Let be a sequence in such that (if such a sequence does not exist, then the limits in (c) automatically hold). Since the sequences , , are bounded in and converge to 0 uniformly on compact subsets of , by Lemma 2, we get , as , which means that (c) holds.

(c)⇒(a). Suppose that the limits in are 0. To prove this implication, we only need to show that (7) hold. Using the inequality (28), we get

as .

Using inequality (24), we get

as .

Using inequality (20), we get

as . The desired result follows. The proof of this theorem is complete.

Acknowledgment

The first author is supported by National Natural Science Foundation of China (no. 11126284).

References

P. L. Duren, Theory of HP spaces, Academic Press, New York, NY, USA, 1970.

S. Li and S. Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 825–831, 2008.

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.

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.

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.

X. Zhu, “Generalized weighted composition operators on weighted Bergman spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 881–893, 2009.

X. Zhu, “Generalized weighted composition operators from Bers-type spaces to Bloch spaces,” Mathematical Inequalities & Applications. In press

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.

S. Li and S. Stević, “Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces,” Sbornik, vol. 199, pp. 1847–1857, 2008.

S. Li and S. Stević, “Composition followed by differentiation between and α-Bloch spaces,” Houston Journal of Mathematics, vol. 35, no. 1, pp. 327–340, 2009.

S. Stević, “Products of composition and differentiation operators on the weighted Bergman space,” Bulletin of the Belgian Mathematical Society, vol. 16, no. 4, pp. 623–635, 2009.

S. Stević, “Composition followed by differentiation from and the Bloch space to nth weighted-type spaces on the unit disk,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3450–3458, 2010.

S. Stević, “Characterizations of composition followed by differentiation between Bloch-type spaces,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4312–4316, 2011.

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.

W. Yang, “Products of composition and differentiation operators from (p, q) spaces to Bloch-type spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 741920, 14 pages, 2009.

F. Colonna, “New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space,” Central European Journal of Mathematics, vol. 11, no. 1, pp. 55–73, 2013.

Z. Lou, “Composition operators on Bloch type spaces,” Analysis, vol. 23, no. 1, pp. 81–95, 2003.

K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995.

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.

M. Tjani, Compact composition operators on some Möbius invariant Banach spaces [Ph.D. dissertation], Michigan State University, 1996.

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.

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.

F. Colonna and S. Li, “Weighted composition operators from into the Zygmund spaces,” Complex Analysis and Operator Theory. In press

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.

S. Stević, “Weighted differentiation composition operators from the mixed-norm space to the n-th weighted-type space on the unit disk,” Abstract and Applied Analysis, vol. 2010, Article ID 246287, 15 pages, 2010.

S. Stević, “Weighted differentiation composition operators from and Bloch spaces to nth weighted-type spaces on the unit disk,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3634–3641, 2010.

Y. Yu and Y. Liu, “Weighted differentiation composition operators from to Zygmund spaces,” Integral Transforms and Special Functions. An International Journal, vol. 22, no. 7, pp. 507–520, 2011.

Y. Zhang, “New criteria for generalized weighted composition operators from mixed norm spaces into Bloch-type spaces,” Bulletin of Mathematical Analysis and Applications, vol. 4, no. 4, pp. 29–34, 2012.

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.

W. Yang and W. Yan, “Generalized weighted composition operators from area Nevanlinna spaces to weighted-type spaces,” Bulletin of the Korean Mathematical Society, vol. 48, no. 6, pp. 1195–1205, 2011.

W. Yang and X. Zhu, “Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces,” Taiwanese Journal of Mathematics, vol. 16, no. 3, pp. 869–883, 2012.

X. Zhu, “Generalized weighted composition operators from Bloch spaces to Bers-type spaces,” Filomat, vol. 26, pp. 1163–1169, 2012.