Research Article | Open Access
Zhong-Shan Fang, "Compact Composition Operators between Bloch Type Spaces in the Polydisk", Abstract and Applied Analysis, vol. 2012, Article ID 732709, 7 pages, 2012. https://doi.org/10.1155/2012/732709
Compact Composition Operators between Bloch Type Spaces in the Polydisk
Wulan et al. (2009), Wulan et al. characterized the compactness of composition operators on the Bloch space in the unit disk by the th power of the induced analytic function. This paper will generalize the result to the Bloch type space in the polydisk.
Let be the polydisk of with boundary . The class of all holomorphic functions on the domain will be denoted by . Let be a holomorphic self-map of . The composition operator is defined as follows: for any and .
The study of the composition operator dates back to the late 60s. From then on, the boundedness and compactness of composition operators between several spaces of holomorphic functions have been studied extensively. We refer the interested readers to the books in [1–3]. Recently, there has been lot of work for Bloch type spaces. For example, see [4–9], as well as the related references therein. There are still many unsolved problems of interest to numerous mathematicians.
For , the Bloch type space consists of those such that with this norm, it becomes a Banach space. When , it is the classical Bloch space.
In , Wulan et al. obtained a new result about the compactness of the composition operators on the Bloch space in the unit disk. We state it as follows.
Theorem 1.1. Let be an analytic self-map of the unit disk . Then is compact on the Bloch space if and only if where here means the th power of .
Along with the further research, it is natural to consider the higher-dimensional case. The goal of this paper is to extend the above result in the unit disk to the polydisk.
Throughout this paper, let be the set of the positive integers, and .
2. Some Lemmas
In this section, we present some lemmas which will be used in the proofs of our main results in the next section.
Lemma 2.1. Let , and . And let . Then for , is decreasing on , and
Lemma 2.2 (see Corollary 3, ). Let and . Suppose that is a holomorphic self-map of . Then is compact if and only if(1) ;(2) .
Lemma 2.3 (see Corollary 4, ). Let and . Suppose that is a holomorphic self-map of . Then is compact if and only if(3) is bounded;(4) .
The following lemma is the crucial criterion for the compactness of , whose proof is an easy modification of the proof of Proposition 3.11 in .
Lemma 2.4. Assume that is a holomorphic self-map of . Then is compact if and only if is bounded and for any bounded sequence in which converges to zero uniformly on compact subsets of , we have as .
3. Main Theorems
In [6, 8], the authors characterized the boundedness and compactness of composition operators between different Bloch type spaces in the polydisk. In this section, we will give some new results about the old problems.
Theorem 3.1. Let and . Suppose that is a holomorphic self-map of . Then is compact if and only if(5) ; (6) ; (7) .
Proof. For any , set
For the proof of the sufficiency, by Lemma 2.2, we only have to show that for every . There are two cases to consider.
Case 1. If with , we have Therefore, for any , is equivalent to .
Case 2. If with , then for each , let where .
For each fixed and every , there exists a with such that whenever .
Since as , we may choose sufficiently large such that . If , then thus There exists with such that Let . Then So we have Letting and by Lemma 2.1, we have From which and (6), we know . Combining the two cases, thus (2) holds. Note that conditions (5) and (1) are the same, it follows from Lemma 2.2 that is compact.
Now we turn to prove the necessity. The result (5) follows by Lemma 2.2. Using Lemma 2.1, we see that for any .
For any , we consider the test functions . It is clear that and uniformly on compact subsets of as . If is compact, then This shows that (6) holds.
For any , from the discussion of Case 1 in the proof of the sufficiency, it follows that . And this fact implies that condition (7) holds. Now the proof of the theorem is completed.
Remark 3.2. For , if , it is well known that is always bounded on the Bloch space, which implies that (5) is true. Note that in this case, conditions (6) and (7) are the same. Thus, we immediately obtain the result on the Bloch space in  by Theorem 3.1.
Theorem 3.3. Let and . Suppose that is a holomorphic self-map of . Then is compact if and only if(8) is bounded;(9) .
Proof. Suppose first that is compact. It is clear that is bounded. Taking the test functions , and using the same arguments as in the proofs of Theorem 3.1, we obtain
for any .
This proves the necessity.
Conversely, by Lemma 2.4, it suffices to show that (4) holds. In fact, for any with , there exists some positive number close enough to , such that the set is empty. Without loss of generality, we may assume that For any , and , let where .
Therefore, we have This along with condition (9) yields that for any .
Combining the results of the two cases for and , we get (4). By Lemma 2.3, we know that is compact. This completes the proof of the theorem.
The authors would like to thank the referees for the useful comments and suggestions which improved the presentation of this paper.
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