Journal of Function Spaces
Volume 2018 (2018), Article ID 4670904, 7 pages
https://doi.org/10.1155/2018/4670904

Research Article

Essential Norm of Difference of Composition Operators from Weighted Bergman Spaces to Bloch-Type Spaces

Ram Krishan,¹ Mehak Sharma,¹ and Ajay K. Sharma²

¹School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, India
²Department of Mathematics, Central University of Jammu, Bagla, Rahya-Suchani, Samba 181143, India

Correspondence should be addressed to Ajay K. Sharma; aksju_76@yahoo.com

Received 10 November 2017; Accepted 31 January 2018; Published 13 March 2018

Academic Editor: Ruhan Zhao

Copyright © 2018 Ram Krishan 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 compute upper and lower bounds for essential norm of difference of composition operators acting from weighted Bergman spaces to Bloch-type spaces.

1. Introduction and Preliminaries

Let be the open unit disk in the complex plane , the space of all holomorphic functions on , and the set of all holomorphic self-maps of . For , the composition operator is a linear operator defined by

For a general reference on composition operator we refer to [1]. To understand the topological structures of spaces of composition operators, many people have studied difference of composition operators on different spaces of holomorphic functions; see, for example, [2–26] and references therein. Recently, Zhu and Yang [26] characterized boundedness and compactness of the differences of two composition operators from weighted Bergman spaces to Bloch spaces. Motivated by these results, in this paper we compute essential norm of difference of composition operators from weighted Bergman spaces to Bloch-type spaces.

Throughout this paper, constants are denoted by ; they are positive and not necessary the same in each occurrence. The notation means that is less than or equal to a constant times and means that is greater than or equal to a constant times . When and , then we write .

For and , the weighted Bergman space is the space of all functions such that

where is the normalized area measure on For any , the following point-evaluation estimate holds:

The following equivalent norm in the weighted Bergman spaces is well-known. Let , , and . Then if and only if, for all , , and

Thus, if , then and Therefore, by (3) applied to the function , for every in , we have

Moreover, by Lemma  2 in [26], for any , there is a constant such that

for all For more about weighted Bergman spaces, we refer the reader to [27, 28].

Let be a positive continuous function on , which we call a weight. A weight is called typical if it is radial; that is, , for each and , decreasingly converges to as Throughout this paper by a weight, we shall mean a typical weight. For a weight , the weighted Bloch-type space on is a Banach space of all analytic functions on such that

The space is a Banach space with the norm

In particular, the space is the classical Bloch space and we denote it by

2. Main Results

In this section, we compute upper and lower bounds for the essential norm of difference of composition operators mapping the weighted Bergman space into Bloch-type spaces.

Given Banach spaces and , we recall that the essential norm of a bounded linear map is defined as

that is, the essential norm of an operator is the distance of from the set of all compact operators from to Clearly, if is compact, then

For , let be the conformal automorphism of that interchanges and :

The pseudo-hyperbolic distance between and is given by

For , a typical weight, and , let and be two functions defined as

Theorem 1. Let , , such that and let be a weight function such that is bounded. Then

Proof. Let be a sequence in such that as and

For each , let be defined as

Now consider the following functions:

Then it is easy to see that and are norm bounded sequences in . Moreover, both sequences and converge to uniformly on compact subsets of Let be any compact operator. Then

Again

Multiplying (18) by and then adding it to (19), we get

Similarly, by considering a sequence in such that as and

we can prove that

Combining (17) and (20) and using the fact that and as , we have

Combining (17) and (22) and using the fact that and as , we have

Again, let be a sequence in such that as and

Then from (18) we have

Using (20) and (26), we have

Combining (17) and (27) and using the fact that and as , we have

Combining (23), (24), and (28), we get the lower bound as

Let Then Let

Then, we have the fact that is compact. Since is bounded, so is compact. Thus

where is the identity operator on Let be such that . For any , we can write

If , then we have

Let , and then and . Let be such that Denote the straight line segment from to by . Then the segment , where . Thus by (5), we have

Similarly, we can show that

Since is bounded, where or , so

By taking in (36), we have

Combining (33), (34), and (37), we have

Using (35) with and , we have

Combining (38) and (39), we have

as Finally, we have

For every and , we have

and by Banach-Steinhaus theorem; it converges to zero uniformly on compact subsets of , so we have

Also, for the boundedness of , we have the facts that and Using these facts and (41), we see that for each and the right hand side of (41) is dominated by a constant multiple of

If , then we see that the right hand side of (41) is dominated by a constant multiple of

Thus

Similarly, we can show that

Combining (31), (32), (40), (46), and (47), we have the fact that

Combining (29) and (48), we get the desired result.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author acknowledges DST (India) for Inspire fellowship (DST/Inspire fellowship/2013/281). The third author acknowledges NBHM (DAE) (India) for the Project Grant no. 02011/30/2017/R&D II/12068.

References

C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press Boca Raton, New York, NY, USA, 1995.

R. F. Allen, K. C. Heller, and M. . Pons, “Compact differences of composition operators on weighted Dirichlet spaces,” Central European Journal of Mathematics, vol. 12, no. 7, pp. 1040–1051, 2014.

J. Bonet, M. Lindström, and E. Wolf, “Differences of composition operators between weighted Banach spaces of holomorphic functions,” Journal of the Australian Mathematical Society, vol. 84, no. 1, pp. 9–20, 2008.

B. R. Choe, H. Koo, and I. Park, “Compact Differences of Composition Operators on the Bergman Spaces Over the Ball,” Potential Analysis, vol. 40, no. 1, pp. 81–102, 2014.

T. Hosokawa, K. Izuchi, and S. Ohno, “Topological structure of the space of weighted composition operators on H∞,” Integral Equations and Operator Theory, vol. 53, no. 4, pp. 509–526, 2005.

T. Hosokawa and S. Ohno, “Differences of composition operators on the Bloch spaces,” The Journal of Operator Theory, vol. 57, no. 2, pp. 229–242, 2007.

S. Li, R. Qian, and J. Zhou, “Essential norm and a new characterization of weighted composition operators from weighted Bergman spaces and Hardy spaces into the Bloch space,” Czechoslovak Mathematical Journal, vol. 67(142), no. 3, pp. 629–643, 2017.

S. Li and S. Stevic, “Weighted composition operators from Bergman-type spaces into Bloch spaces,” The Proceedings of the Indian Academy of Sciences – Mathematical Sciences, vol. 117, no. 3, pp. 371–385, 2007.

M. Lindström and E. Saukko, “Essential norm of weighted composition operators and difference of composition operators between standard weighted Bergman spaces,” Complex Analysis and Operator Theory, vol. 9, no. 6, pp. 1411–1432, 2015.

M. Lindström and E. Wolf, “Essential norm of the difference of weighted composition operators,” Monatshefte für Mathematik, vol. 153, no. 2, pp. 133–143, 2008.

B. MacCluer, S. Ohno, and R. Zhao, “Topological structure of the space of composition operators on H∞,” Integral Equations and Operator Theory, vol. 40, no. 4, pp. 481–494, 2001.

J. Moorhouse, “Compact differences of composition operators,” Journal of Functional Analysis, vol. 219, no. 1, pp. 70–92, 2005.

J. Moorhouse and C. Toews, “Differences of composition operators,” in Trends in Banach Spaces and Operator Theory, Contemporary Mathematics, 2001.

P. J. Nieminen and E. Saksman, “On compactness of the difference of composition operators,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 501–522, 2004.

E. Saukko, “Difference of composition operators between standard weighted Bergman spaces,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 789–798, 2011.

E. Saukko, “An application of atomic decomposition in Bergman spaces to the study of differences of composition operators,” Journal of Functional Analysis, vol. 262, no. 9, pp. 3872–3890, 2012.

A. K. Sharma and R. Krishan, “Difference of composition operators from the space of Cauchy integral transforms to the Dirichlet space,” Complex Analysis and Operator Theory, vol. 10, no. 1, pp. 141–152, 2016.

A. K. Sharma, R. Krishan, and E. Subhadarsini, “Difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces,” Integral Transforms and Special Functions, vol. 28, no. 2, pp. 145–155, 2017.

Y. Shi and S. Li, “Differences of composition operators on Bloch type spaces,” Complex Analysis and Operator Theory, vol. 11, no. 1, pp. 227–242, 2017.

Y. Shi and S. Li, “Essential norm of the differences of composition operators on the Bloch space,” Journal of Inequalities and Applications, vol. 20, no. 2, pp. 543–555, 2017.

S. Stevic, “Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1811–1824, 2010.

S. Stevic and Z. J. Jiang, “Compactness of the differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball,” Taiwanese Journal of Mathematics, vol. 15, no. 6, pp. 2647–2665, 2011.

E. Wolf, “Compact differences of composition operators,” Bulletin of the Australian Mathematical Society, vol. 77, no. 1, pp. 161–165, 2008.

K.-B. Yang and Z.-H. Zhou, “Essential norm of the difference of composition operators on Bloch space,” Czechoslovak Mathematical Journal, vol. 60(135), no. 4, pp. 1139–1152, 2010.

X. D. Yang and L. H. Khoi, “Compact differences of composition operators on Bergman spaces in the ball,” Journal of the Australian Mathematical Society, vol. 89, no. 3, pp. 407–418, 2010.

X. Zhu and W. Yang, “Differences of composition operators from weighted Bergman spaces to Bloch spaces,” Filomat, vol. 28, no. 9, pp. 1935–1941, 2014.

H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Springer, New York, NY, USA, 2000.

K. Zhu, in Operator Theory in Function Spaces, vol. 139, Marcel Dekker, New York, NY, USA, 1990.