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Product of differentiation and composition operators on the logarithmic Bloch space
Journal of Inequalities and Applications volume 2014, Article number: 453 (2014)
Abstract
We obtain a criterion for the boundedness and compactness of the products of differentiation and composition operators on the logarithmic Bloch space in terms of the sequence . An estimate for the essential norm of is given.
MSC:47B38, 30H30.
1 Introduction
Denote by the space of all analytic functions on the unit disk in the complex plane. Let denote the space of bounded analytic functions on . An is said to belong to the Bloch space ℬ if
The logarithmic-Bloch space, denoted by , consists of all satisfying
is a Banach space with the norm . It is well known that is the space of multipliers of the Bloch space ℬ (see [1, 2]). For some results on logarithmic-type spaces and operators on them, see, for example, [3–10].
Let φ be an analytic self-map of . The composition operator is defined by
The differentiation operator D is defined by , . For a nonnegative integer , we define
The product of differentiation and composition operators is defined as follows:
A basic problem concerning concrete operators on various Banach spaces is to relate the operator theoretic properties of the operators to the function theoretic properties of their symbols, which attracted a lot of attention recently, the reader can refer to [4–37].
It is a well-known consequence of the Schwarz-Pick lemma that the composition operator is bounded on ℬ. See [21–24, 27, 33–35, 37] for the study of composition operators and weighted composition operators on the Bloch space. The product-type operators on or into Bloch type spaces have been studied in many papers recently; see [12–20, 26, 28–32, 34, 36] for example.
Let X and Y be two Banach spaces. Recall that a linear operator is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. The essential norm of an operator T between X and Y is the distance to the compact operators K, that is, , where is the operator norm. It is easy to see that if and only if T is compact. For some results in the topic, see, for example, [11, 20, 22, 24, 26, 28, 37].
In [34], Wu and Wulan obtained a characterization for the compactness of the product of differentiation and composition operators acting on the Bloch space as follows:
Theorem A Let φ be an analytic self-map of , . Then is compact if and only if
The purpose of the paper is to extend Theorem A to the case of . We will characterize the boundedness and compactness of in terms of the sequence . Moreover, an estimate for the essential norm of will be given. The main results are given in Sections 3 and 4.
In the paper, we say that a real sequence is asymptotic to another real sequence of and write ‘’ if and only if
In addition, we say that if there exists a constant C such that . The symbol means that .
2 Auxiliary lemmas
In this section, we state and prove some auxiliary results which will be used to prove the main results in this paper.
Lemma 2.1 For , define the function by
Then the following statements hold:
-
(i)
For and , there is a unique such that is the absolute maximum of .
-
(ii)
(2.2)
and
-
(iii)
(2.4)
Proof Directly computing we have
Define
It is easy to see that is continuous on [0,1) and , . Furthermore,
Then is decreasing on . When , we get . When , the intermediate value theorem of continuous function gives the result that there exists a unique such that . So we have
(i) has been proved. By (2.5), we have . Thus
It follows from and that (2.2) holds. Also, gives the result that
So we have
This gives the result (2.3). The proof of (ii) is complete.
Note that
This and (2.2) give
By (2.3) and (2.6) we obtain
which shows that (iii) hold. The proof is complete. □
Lemma 2.2 Let and . Let . Then is increasing on and
Consequently,
Proof Since , we have
By Lemma 2.1, we have , where is given as in Lemma 2.1. Since for , we see that is increasing on . Thus
Applying the important limit we obtain the result that (2.7) holds.
By Lemma 2.1 we have
where is given in Lemma 2.1. By Lemma 2.1 we have
This gives (2.8). The proof is complete. □
Lemma 2.3 [3]
For . Then if and only if
Moreover,
3 The boundedness of on
In this section, we will state the boundedness criterion for the operator on . Since the boundedness of on gives , we may always assume that . The main result of this section is stated as follows.
Theorem 3.1 Let and φ be an analytic self-map of such that . Then is bounded on if and only if
Proof ⇒) Assume that is bounded on , that is, . Since the sequence is bounded in the logarithmic Bloch space , we have
for any , from which the implication follows.
⇐) We now assume that the condition (3.1) holds. On the one hand, for the case , there is an such that . By (3.1), for any given , we have
The last estimate shows that the operator is bounded on .
On the other hand, for the case . Let N be the smallest positive integer such that is not empty, where
and is given in Lemma 2.2. Note that , when , , by (2.8) we obtain
For any given , by Lemma 2.3 we have
The proof is complete. □
4 The essential norm of on
Denote for . Then is a compact operator on the space . It is easy to see that . We denote by I the identity operator.
In order to give the lower and upper estimate for the essential norm of on , we need the following result.
Lemma 4.1 There is a sequence , with tending to 1, such that the compact operator
on satisfies:
-
(i)
for any , ,
(iia) ,
(iib) , for any ,
-
(iii)
.
Proof (i) follows from (iib) by Cauchy’s formula. The proof of (iii) is similar to the proof of Proposition 8 in [25]. Hence we omit it. Next we prove (iia) and (iib). The argument is much like that given in the proof of Proposition 2.1 of [25] or Lemmas 1 and 2 in [22]. For any , we choose an increasing sequence tending to 1 such that . For any given and , , there exists an such that
For any with , we have
Thus
This shows that (iia) holds.
If , by the equality (4.1), we have
The above estimate gives (iib). The proof is complete. □
The following lemma can be proved in a standard way; see, for example Proposition 3.11 in [11].
Lemma 4.2 Let and φ be an analytic self-map of . Then is compact on if and only if is bounded on and for any bounded sequence in which converges to zero uniformly on compact subsets of , then as .
Theorem 4.3 Let and φ be an analytic self-map of . Suppose that is bounded on . Then the estimate for the essential norm of on is
Proof We first give the lower estimate for the essential norm. Without loss of generality, we assume that . Choose the sequence of function , . Then , and converges to zero weakly on as . Thus we have
for any given compact operator K on . The basic inequality gives
Thus we obtain
So we have
Now we give the upper estimate for the essential norm. For the case of , there is a number such that . In this case, the operator is compact on . In fact, choose a bounded sequence in which converges to zero uniformly on compact subset of . From Cauchy’s integral formula, converges to zero on compact subsets of as . It follows that
Then the operator is compact on by Lemma 4.2. This gives
On the other hand, by Lemma 2.1 and (2.9) we obtain
which implies that
Combining the last inequality with (4.3), we get the desired result.
Next, we assume that . Let be the sequence of operators given in Lemma 4.1. Since is compact on and is bounded on , then is also compact on . Hence
where
and
It follows from Lemma 4.1(iib) and Cauchy’s integral formula that .
For each positive integer , we define
where is given in Lemma 2.1. Let k be the smallest positive integer such that . Since , is not empty for every integer and , we have
where
and
Here N is a positive integer determined as follows.
By (2.8),
Hence, for any given , there exists an N such that
when . For such N it follows that
Thus
By (ii) of Lemma 4.1 and Cauchy’s integral formula, we have
which together with (4.4) implies that
From (4.5) we obtain
By the arbitrariness of ε, we get
The proof is complete. □
From Theorem 4.3, we obtain the following result.
Corollary 4.4 Let and φ be an analytic self-map of such that is bounded on . Then is compact on if and only if
Especially, when , from the proof of Theorem 4.3, we get the exact formula for essential norm of composition operator on .
Corollary 4.5 Let φ be an analytic self-map of . Suppose that is bounded on ; then
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Acknowledgements
The project is supported by Department of education of Anhui Province of China (No. KJ2013A101), the Natural Science Foundation of Guangdong (No. S2013010011978) and NSF of China (No. 11471143).
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Zhou, J., Zhu, X. Product of differentiation and composition operators on the logarithmic Bloch space. J Inequal Appl 2014, 453 (2014). https://doi.org/10.1186/1029-242X-2014-453
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DOI: https://doi.org/10.1186/1029-242X-2014-453