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 thatfor 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. ThenAgainMultiplying (18) by and then adding it to (19), we getSimilarly, by considering a sequence in such that as and we can prove thatCombining (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 asLet Then Let Then, we have the fact that is compact. Since is bounded, so is compact. Thuswhere is the identity operator on Let be such that . For any , we can writeIf , 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 thatSince is bounded, where or , soBy 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 ThusSimilarly, we can show thatCombining (31), (32), (40), (46), and (47), we have the fact thatCombining (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.