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On a product operator from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces
Journal of Inequalities and Applications volume 2014, Article number: 404 (2014)
Abstract
Let be the open unit disk, φ an analytic self-map of and ψ an analytic function in . Let be the differentiation operator and the weighted composition operator. The boundedness and compactness of the product operator from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces on are characterized.
MSC:47B38, 47B33, 47B37.
1 Introduction
Let ℂ be the complex plane, the open unit disk, the space of all analytic functions on and the normalized Lebesgue measure on . For , let be the weighted Lebesgue measure on . The weighted Bergman-Nevanlinna space consists of all such that
It is a Fréchet space with the translation invariant metric
For some details of this space, see, e.g., [1–3] and [4].
For , the weighted-type consists of all such that
This space is a non-separable Banach space with the norm defined by
The closure of the set of polynomials in is denoted by , which is a separable Banach space and consists exactly of those functions f in satisfying the condition
For , the weighted Bloch space is defined by
and it is well known (see, e.g., [5]) that, if , then if and only if . Under the norm
it is a Banach space. The closure of the set of polynomials in is called the little weighted Bloch space and denoted by . For a good source for such spaces, we refer to [5].
For , the weighted Zygmund space consists of all such that
It is a Banach space with the norm
The little weighted Zygmund space consists those functions f in satisfying
and it is a closed subspace of the weighted Zygmund space.
Recently, many authors have studied the properties of some concrete operators between various spaces of analytic functions in the unit disk, the upper half plane, the unit ball and the unit polydisk. For some operators on weighted-type spaces, weighted Bloch spaces, and weighted Zygmund spaces on these domains, see, e.g., [4, 6–38] and the references therein.
Let φ be an analytic self-map of and let ψ be an analytic function in . It is well known that the weighted composition operator on is defined by
If , is called the composition operator. If , it is called the multiplication operator. It is of interest to provide function theoretic characterizations when φ and ψ induce a bounded or compact weighted composition operator. Sharma and Abbas have studied the boundedness and compactness of weighted composition operators from weighted Bergman-Nevanlinna spaces to Bloch spaces in [39]. Kumar and Sharma have characterized the boundedness and compactness of weighted composition operators from weighted Bergman-Nevanlinna spaces to Zygmund spaces in [40].
Now we list several operators, which will be considered in this paper. Let be the differentiation operator on defined by
Operator has been studied, for example, in [10, 15, 19, 20, 31, 35, 41–45]. Here we want to mention that Sharma has studied the following two operators from Bergman-Nevanlinna spaces to Bloch-type spaces in [46]. They are defined as follows:
and
for and . These operators on weighted Bergman spaces have been also studied by Stević et al. in [47] and [48]. If we consider the product operator , it is clear that and . Quite recently, the operator from weighted Bergman spaces to weighted Zygmund spaces has been considered in [49]. This paper is devoted to characterizing the boundedness and compactness of the operator from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces. This paper can be regarded as an continuousness of our work.
Since the weighted Bergman-Nevanlinna space is a Fréchet space and not a Banach space, it is necessary to introduce several definitions needed in this paper. Let X and Y be topological vector spaces whose topologies are given by translation invariant metrics and , respectively, and let be a linear operator. It is said that L is metrically bounded if there exists a positive constant K such that
for all . When X and Y are Banach spaces, the metrical boundedness coincides with the usual definition of bounded operators between Banach spaces. Recall that is metrically compact if it maps bounded sets into relatively compact sets. When X and Y are Banach spaces, the metrical compactness coincides with the usual definition of compact operators between Banach spaces. When and Y is a Banach space, we define
and we often write by .
Throughout this paper, an operator is bounded (respectively, compact), if it is metrically bounded (respectively, metrically compact). Constants are denoted by C, they are positive and may differ from one occurrence to the next. The notation means that there exists a positive constant C such that .
2 The operator
Our first lemma characterizes the compactness in terms of sequential convergence. Since the proof is standard, it is omitted (see Proposition 3.11 in [50]).
Lemma 2.1 Let , and . Then the bounded operator is compact if and only if for every bounded sequence in such that uniformly on every compact subset of as , it follows that
The next result can be found, for example, in [4].
Lemma 2.2 Let and . Then, for all and , there exists a positive constant C independent of f such that
Now we consider the boundedness of the operator .
Theorem 2.3 Let , , φ an analytic self-map of and . Then, for all , the following statements are equivalent:
-
(i)
The operator is bounded.
-
(ii)
The operator is compact.
-
(iii)
,
and
Proof (i) ⇒ (iii). Suppose that (i) holds. Take the functions and , respectively. Since the operator is bounded, we have
and
Inequality (2) shows . From (1), (2), and the boundedness of φ, it follows that
Taking the functions and , respectively, we have
and
By (2) and the boundedness of φ, we have
and
From (3), (4), (6), and the boundedness of φ, it follows that
Inequalities (3), (5), (7), (8), and the boundedness of φ give
For , we choose the functions
and
By a direct calculation, it follows that
and
Using and , we define the function . Applying (10) and (11) to , and , we find
It is obvious that
where
From the proof in [46], we see that and . Since the operator is bounded, it follows that
which means that, for all ,
Replacing z by w in (12), we obtain
and then
Taking the limit as in (13), it gives
For , we choose the functions
and
From a calculation, we obtain
Define the function . Then by (14),
and by a direct calculation,
where . Since is bounded, we have
and so
for all . Letting in (15) yields
Thus
Taking the limit as in (16), we have
For , we choose the function
where
and
We also choose the function
For the functions and , we have
and
Consequently, (17) and (18) make the function to satisfy
and
where
By the boundedness of the operator , we find
Thus
For , we choose the functions
and
Then
and . From this and (19), for the function we have
and
where
By the boundedness of ,
and from which we get
This shows that
The proof of the implication is finished.
-
(iii)
⇒ (ii). Let be a sequence in with and uniformly on every compact subset of as . We have, for all , and especially for constant C in Lemma 2.2, for any , the result that there exists a constant such that whenever , it follows that
and
Then by Lemma 2.2, we have
By Cauchy’s estimation, if converges to zero on each compact subset of , then , , and also do as . From this, and since both and are compact subset of , we can choose a natural number N such that, whenever , it follows that
and
where . Consequently, for all it follows that
which shows that the operator is compact.
-
(ii)
⇒ (i). This implication is obvious. The proof is finished. □
Now, we consider the boundedness of operator . We first have the following result.
Lemma 2.4 Let and . Let φ be an analytic self-map of and ψ an analytic function on . Then, for all , the following statements are equivalent:
-
(i)
-
(ii)
and .
Proof Suppose that (i) holds. Since
for all , we have
as . Hence . Since implies , it follows that the first assertion in (ii) holds.
Now suppose that (ii) holds, but (i) is not true. Then there exist constants , and a sequence tending to as such that
Since , it follows from (20) that the sequence has a subsequence with . Therefore, applying to the first assertion in (ii), we arrive at a contradiction to (20), finishing the proof. □
By Lemma 2.4, the following result follows similar to the proof of Theorem 2.3. Hence, we omit the details.
Theorem 2.5 Let and . Let φ be an analytic self-map of and ψ an analytic function on . Then, for all , the following statements are equivalent:
-
(i)
The operator is bounded.
-
(ii)
The operator is compact.
-
(iii)
and
-
(iv)
and
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Acknowledgements
The author would like to thank the referee for his or her helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11201323), the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (Grant No. 2013QZJ01, No. 2013QYY01), the Key Fund Project of Sichuan Provincial Department of Education (Grant No. 12ZB288) and the Introduction of Talent Project of SUSE (Grant No. 2014RC04).
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Jiang, Zj. On a product operator from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces. J Inequal Appl 2014, 404 (2014). https://doi.org/10.1186/1029-242X-2014-404
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DOI: https://doi.org/10.1186/1029-242X-2014-404