New characterizations for the products of differentiation and composition operators between Bloch-type spaces
© Liang and Dong; licensee Springer. 2014
Received: 29 September 2014
Accepted: 28 November 2014
Published: 12 December 2014
We use a brief way to give various equivalent characterizations for the boundedness and the essential norm of the operator acting on Bloch-type spaces. At the same time, we use this method to easily get a known characterization for the operator on Bloch-type spaces.
MSC:47B38, 26A24, 32H02, 47B33.
Keywordsessential norm differentiation composition operator Bloch-type space
1 Introduction and preliminaries
Recently there has been a considerable interest on various product-type operators (see, e.g. [1–19]), and among them on products of composition and differentiation operators (see, e.g. [1, 2, 4, 5, 7–12, 15–19]). One of the problems of interest is to characterize the boundedness and compactness of the composition operator acting on Bloch-type spaces in terms of the n th power of the analytic self-mapping φ of the unit disk . Very recently, the first author and Zhou have given the characterizations for the boundedness and the essential norm of the products of differentiation and composition operator and acting on Bloch-type spaces in [9, 10], respectively. Inspired by , we present here an easier way to research the corresponding problem. Moreover, by this brief method, we first give new equivalent characterizations for the boundedness and the essential norm of the operator , and then we obtain the same results for the operator as in the paper .
Let denote the unit disk in the complex plane ℂ. Denote the space of all holomorphic functions on and the collection of all holomorphic self-mappings on . The composition operator is defined by for and .
is a Banach space endowed with the norm , where the weight is a continuous, strictly positive and bounded function.
Similarly, is a Banach space under the norm . When , we get the classical Bloch space ℬ. We refer the readers to the book  for more information as regards the above spaces.
Then is a Banach space under the norm .
As for , the weighted composition operator is the usual composition operator . When φ is the identity mapping I, the operator is the multiplication operator .
where denotes the operator norm. Notice that if and only if T is compact, so the estimate on will lead to the condition for the operator T to be compact.
Throughout this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. The notations , , mean that there maybe different positive constants C such that , , .
For convenience of the reader we list the results related with our conclusions in this paper.
Theorem A [, Theorem 1]
Theorem B [, Theorem 2]
where , , .
Theorem C [, Theorem 2.3]
Theorem D [, Theorem 3.5]
where and . The definitions of and can be found in Section 4.
We would like to point out that the first author and Zhou got the above four theorems by using complex calculations and intricate discussions. In this paper, we will use a brief way to give other equivalent characterizations for the boundedness and the essential norm of on the unit disk in Section 3. In addition, using this method we will show new proofs of Theorem C and Theorem D in Section 4.
In this section we quote some lemmas for our further application. The first lemma is a well-known characterization for ().
So for , the above lemma implies that and more general . Therefore, theories of the weighted composition operator play a key role in the proof of our main results. Here we list some lemmas which will be used later.
Lemma 2.2 [, Proposition 3.1]
Lemma 2.3 [, Theorem 4.4]
Lemma 2.4 [, Theorem 2.4]
- (a)is bounded if and only if
Lemma 2.5 [, Lemma 2.1]
For , we have .
The following criterion for compactness follows from an easy modification of [, Proposition 3.11]. Hence we omit the details.
Lemma 2.6 Let and T be a linear operator from to . Then is compact if and only if is bounded and for any bounded sequence in which converges to zero uniformly on compact subsets of , as .
3 Boundedness and essential norm of
In this section, we give other equivalent characterizations for the boundedness and the essential norm of the operator with .
That is, (b) holds.
Moreover, . Thus , and hence (a) holds. □
Therefore, the characterizations for the boundedness of the operator in Theorem 3.1 are equivalent to that in Theorem A.
As an application of Theorem 3.1, we present an example of the bounded operator , according to either (3.1) or (3.2).
From each of these conditions, one sees that is bounded.
Thus we only need to estimate for the upper bound of the essential norm of . It is obvious that every compact operator can be extended to a compact operator . In fact, for every , , and we can define , which is a compact operator from to , due to has convergent subsequence when is a bounded sequence. In the following lemma we will use the compact operator defined on the space by .
Thus we obtain . The proof is finished. □
Proof If , then by [, Lemma 3.1], the operator is compact. The boundedness (compactness) of is equivalent to the boundedness (compactness) of . In this case, all items in (3.5) are zero.
This completes the proof. □
The relation can be proved similarly to [, Theorem 3.2]. Here we give a complete proof for the reader’s convenience.
- (2)Similar to Remark 3.2, one can get
Therefore, the characterizations for the essential norms of the operator in Theorem 3.5 are equivalent to that in Theorem B.
The following corollary is an immediate consequence of Theorem 3.5.
- (b)is bounded and
- (c)is bounded and
4 Boundedness and essential norm of
In 2007, S Li and S Stević gave the following characterizations for the boundedness and compactness of the operator .
First, we will give a brief proof of Theorem C as regards the bounded operator for all .
This completes the proof. □
According to (4.5), we only need to estimate the right two essential norms for the upper bound of the essential norm of .
Proof By Lemma 4.1(a) and Lemma 2.2, the boundedness of is equivalent to and are bounded weighted composition operators.
This completes the proof. □
The following result is an immediate consequence of Theorem 4.3 and Lemma 4.1(b).
- (b)is bounded,
The authors would like to thank the Journal Editorial Office and the referees for the useful comments and suggestions, which improved the presentation of this article. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11201331; 11301373; 11401431).
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