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New characterizations for the products of differentiation and composition operators between Bloch-type spaces
Journal of Inequalities and Applications volume 2014, Article number: 502 (2014)
Abstract
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.
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 [20], 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 [10].
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 .
The Bloch space of ν-type
is a Banach space endowed with the norm , where the weight is a continuous, strictly positive and bounded function.
For the standard weights for , we denote and
Similarly, is a Banach space under the norm . When , we get the classical Bloch space ℬ. We refer the readers to the book [21] for more information as regards the above spaces.
The weighted Banach space of analytic functions
is a Banach space endowed with the norm . The weight ν is called radial, if for all . For a weight ν, the associated weight is defined by
For the standard weights (), we have . We refer the interested readers to [[22], p.39]. In this case, we denote and
Then is a Banach space under the norm .
For , , the weighted composition operator is defined by
As for , the weighted composition operator is the usual composition operator . When φ is the identity mapping I, the operator is the multiplication operator .
The differentiation operator D is defined by
The products of differentiation and composition operators and are defined, respectively, as follows:
The essential norm of a continuous linear operator T between two normed linear spaces X and Y is its distance from the compact operators. That is,
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 [[9], Theorem 1]
Let , m be a nonnegative integer and φ be a holomorphic self-map of the unit disk . Then is bounded if and only if
Theorem B [[9], Theorem 2]
Let , m be a nonnegative integer and φ be a holomorphic self-map of the unit disk . Suppose that is bounded. Then the estimate for the essential norm of is
where , , .
Theorem C [[10], Theorem 2.3]
Let , and . Then is bounded if and only if
Theorem D [[10], Theorem 3.5]
Let and . Suppose that is bounded. Then the estimate for the essential norm of is
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.
2 Lemmas
In this section we quote some lemmas for our further application. The first lemma is a well-known characterization for ().
Lemma 2.1 For , and . Then
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 [[23], Proposition 3.1]
Let ν and w be weights. Then the weighted composition operator is bounded if and only if
Moreover, the following holds:
Lemma 2.3 [[23], Theorem 4.4]
Let ν and w be radial, non-increasing weights tending to zero at the boundary of . Suppose is bounded. Then
Lemma 2.4 [[24], Theorem 2.4]
Let ν and w be radial, non-increasing weights tending to zero at the boundary of . Then
-
(a)
is bounded if and only if
with the norm comparable to the above supremum.
-
(b)
.
Lemma 2.5 [[22], Lemma 2.1]
For , we have .
The following criterion for compactness follows from an easy modification of [[25], 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 .
Theorem 3.1 Let , , and . Then the following statements are equivalent:
-
(a)
is bounded.
-
(b)
(3.1)
-
(c)
(3.2)
Proof (a) ⇒ (b). Suppose that is bounded. Choose and
It is easy to verify that and for . By for , we obtain
and
Then it follows that
and
That is, (b) holds.
(b) ⇔ (c). From Lemma 2.2, the condition (b) is a necessary and sufficient condition for the boundedness of weighted composition operator . Further by Lemma 2.4(a) and Lemma 2.5, the boundedness of the weighted composition operator is equivalent to the following:
(b) ⇒ (a). Suppose (b) holds. For every , then it follows from Lemma 2.1 that
Moreover, . Thus , and hence (a) holds. □
Remark 3.2
-
(1)
The relation (a) ⇔ (b) was essentially proved in a very general result in [18]. For convenience of the reader, we sketch the proof in [18].
-
(2)
One can easily see that
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).
Example 3.3 Let for and . Then we study the boundedness of . Firstly, by (3.1), it is clear that
Secondly, by (3.2) we obtain
From each of these conditions, one sees that is bounded.
Next we estimate the essential norm of the operator for all . Denote . Let be defined by . Then we have for . Since when , and further by the equality for all , it follows that
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 .
Lemma 3.4 If and is a bounded operator from to , then
Proof Although the proof is similar to [[20], Lemma 3.1], we will give all the details for convenience of the reader. It is obvious that
Conversely, let be given. Choose an increasing sequence in converging to 1. We denote by the closed subspace of consisting of all constant functions. Then we have
Hence
Since is bounded, it follows that
Thus we obtain . The proof is finished. □
Thus by Lemma 3.4 and (3.3) it follows that
Theorem 3.5 Let , , and . Suppose that is bounded. Then
Proof If , then by [[26], 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.
If , since is bounded, then the boundedness of follows from the proof in Theorem 3.1. Thus by (3.4), Lemma 2.4(b), and Lemma 2.5,
Since , we may choose a sequence such that as . Define
It is easy to show that and converges to zero uniformly on the compact subsets of as . Moreover,
Then for every compact operator , by Lemma 2.6, it follows that . Thus
Consequently,
Since the operator is bounded, then applying Lemma 2.3, Lemma 2.4(b), and Lemma 2.5, we get
Thus
Hence
This completes the proof. □
Remark 3.6
-
(1)
The relation can be proved similarly to [[26], 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.
Corollary 3.7 Let , , and . Then the following statements are equivalent:
-
(a)
is compact.
-
(b)
is bounded and
-
(c)
is bounded and
4 Boundedness and essential norm of
In this section, the corresponding problems for the operator are considered. Let , then for every , define
Then it follows that
By an easy calculation, one can get
and
In 2007, S Li and S Stević gave the following characterizations for the boundedness and compactness of the operator .
Lemma 4.1 Let and . Then the following statements hold:
-
(a)
[[4], Theorem 1] is bounded if and only if
(4.3) -
(b)
[[4], Theorem 2] is compact if and only if is bounded,
First, we will give a brief proof of Theorem C as regards the bounded operator for all .
Theorem 4.2 Let and . Then is bounded if and only if
Proof Lemma 4.1 shows that maps boundedly into if and only if (4.3) holds. On the other hand, Lemma 2.2 shows that (4.3) holds if and only if the weighted composition operators maps boundedly into and maps boundedly into , and hence it follows from Lemma 2.4(a) that (4.3) is equivalent to
Using Lemma 2.5, (4.1) and (4.2), then the boundedness of is equivalent to
and
This completes the proof. □
Now, we give a new proof of Theorem D about the essential norm of for . We denote . Let and be the first-order derivative operator and the second-order derivative operator, respectively. That is,
By Lemma 2.1 we have
For , by Lemma 2.1, , and . Then by the equation , it follows that
Moreover, every compact operator can be extended to a compact operator . Then similar to Lemma 3.4, one can easily get
Thus combining the above equation with (4.4), we obtain
According to (4.5), we only need to estimate the right two essential norms for the upper bound of the essential norm of .
Theorem 4.3 Let and . Suppose that is bounded. Then
Proof By Lemma 4.1(a) and Lemma 2.2, the boundedness of is equivalent to and are bounded weighted composition operators.
The upper estimate. From Lemma 2.4(b) and Lemma 2.5, we obtain
Then it follows from (4.5) that
The lower estimate. Let be a sequence in such that as . Define
We can easily show both and belong to and converge to zero uniformly on the compact subsets of as . Moreover,
Then for every compact operator , by Lemma 2.6 we obtain
Since the weighted composition operators and are bounded. Then applying Lemma 2.3, Lemma 2.4(b), and Lemma 2.5, it follows that
Hence
This completes the proof. □
The following result is an immediate consequence of Theorem 4.3 and Lemma 4.1(b).
Corollary 4.4 Let and . Then the following statements are equivalent:
-
(a)
is compact.
-
(b)
is bounded,
References
Hibschweiler RA, Portnoy N: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mt. J. Math. 2005,35(3):843–855. 10.1216/rmjm/1181069709
Hyvärinen O, Nieminen I: Weighted composition followed by differentiation between Bloch-type spaces. Rev. Mat. Complut. 2014. 10.1007/s13163-013-0138-y
Krantz SG, Stević S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. TMA 2009, 71: 1772–1795. 10.1016/j.na.2009.01.013
Li S, Stević S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 2007, 9: 195–206.
Li S, Stević S: Composition followed by differentiation from mixed norm spaces to α -Bloch spaces. Sb. Math. 2008,199(12):1847–1857. 10.1070/SM2008v199n12ABEH003983
Li S, Stević S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 2008, 338: 1282–1295. 10.1016/j.jmaa.2007.06.013
Li S, Stević S: Composition followed by differentiation between and α -Bloch spaces. Houst. J. Math. 2009,35(1):327–340.
Li S, Stević S: Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput. 2010, 217: 3144–3154. 10.1016/j.amc.2010.08.047
Liang YX, Zhou ZH: Essential norm of product of differentiation and composition operators between Bloch-type spaces. Arch. Math. 2013,100(4):347–360. 10.1007/s00013-013-0499-y
Liang YX, Zhou ZH: New estimate of essential norm of composition followed by differentiation between Bloch-type spaces. Banach J. Math. Anal. 2014, 8: 118–137.
Ohno S: Products of composition and differentiation on Bloch spaces. Bull. Korean Math. Soc. 2009,46(6):1135–1140. 10.4134/BKMS.2009.46.6.1135
Stević S: Characterizations of composition followed by differentiation between Bloch-type spaces. Appl. Math. Comput. 2011, 218: 4312–4316. 10.1016/j.amc.2011.10.004
Stević S: On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. Appl. Math. Comput. 2008, 206: 313–320. 10.1016/j.amc.2008.09.002
Stević S: On a new integral-type operator from the Bloch spaces to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 2009, 354: 426–434. 10.1016/j.jmaa.2008.12.059
Stević S: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 2009, 16: 623–635.
Stević S: Norm and essential norm of composition followed by differentiation from α -Bloch spaces to . Appl. Math. Comput. 2009, 207: 225–229. 10.1016/j.amc.2008.10.032
Stević S: Composition followed by differentiation from and the Bloch space to n -th weighted-type spaces on the unit disk. Appl. Math. Comput. 2010, 216: 3450–3458. 10.1016/j.amc.2010.03.117
Stević S: Weighted differentiation composition operators from and Bloch spaces to n -th weighted-type spaces on the unit disk. Appl. Math. Comput. 2010, 216: 3634–3641. 10.1016/j.amc.2010.05.014
Wu Y, Wulan H: Products of differentiation and composition operators on the Bloch space. Collect. Math. 2012, 63: 93–107. 10.1007/s13348-010-0030-8
Esmaeili K, Lindström M: Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equ. Oper. Theory 2013,75(4):473–490. 10.1007/s00020-013-2038-4
Zhu KH: Operator Theory in Function Spaces. Dekker, New York; 1990.
Hyvärinen O, Lindström M: Estimates of essential norms of weighted composition operators between Bloch-type spaces. J. Math. Anal. Appl. 2012, 393: 38–44. 10.1016/j.jmaa.2012.03.059
Contreras MD, Hernández-Díaz AG: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. A 2000, 69: 41–60. 10.1017/S144678870000183X
Hyvärinen O, Kemppainen M, Lindström M, Rautio A, Saukko E: The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equ. Oper. Theory 2012, 72: 151–157. 10.1007/s00020-011-1919-7
Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton; 1995.
Stević S: Essential norms of weighted composition operators from the α -Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal. 2008., 2008: Article ID 279691
Acknowledgements
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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Liang, YX., Dong, XT. New characterizations for the products of differentiation and composition operators between Bloch-type spaces. J Inequal Appl 2014, 502 (2014). https://doi.org/10.1186/1029-242X-2014-502
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DOI: https://doi.org/10.1186/1029-242X-2014-502