Abstract
We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund-type spaces. Consequently, we obtain new characterizations for the compactness of such operators.
1. Introduction
Let denote the open unit ball of the complex plane and denote the space of all complex-valued analytic functions on . By a weight, we mean a strictly positive bounded function . The weighted-type space consists of all functions such that
For a weight , the associated weight is defined by
It is known that for the standard weights, , and for the logarithmic weight, the associated weights and weights are the same.
For each , the Bloch-type space consists of all functions for which
The space is a Banach space equipped with the norm for each . The little Bloch-type space is the closed subspace of consisting of those functions satisfying
The classic Zygmund space consists of all functions which are continuous on the closed unit ball and where the supremum is taken over all and . By [1] (Theorem 5.3), an analytic function belongs to if and only if . Motivated by this, for each , the Zygmund-type space is defined to be the space of all functions for which
The space is a Banach space equipped with the norm for each . The little Zygmund-type space is the closed subspace of consisting of those functions satisfying
Recall that for the Banach spaces and , the space of all bounded operators is denoted by , and the operator norm of is denoted by . The closed subspace of containing all compact operators is denoted by . The essential norm of , denoted by , is defined as the distance from to , that is,
Clearly, an operator is compact if and only if . Therefore, essential norm estimates of bounded operators result in necessary and/or sufficient conditions for the compactness of such operators. Essential norm estimates of different types of operators between various classes of Banach spaces have been studied by many authors (see, for example, [2–7], and references therein).
Let and be analytic functions on such that . The weighted composition operator is defined by for all . When , we get the well-known composition operator given by for all . Weighted composition operators appear in the study of dynamical systems, and also, it is known that isometries on many analytic function spaces are of the canonical forms of weighted composition operators. Operator theoretic properties of (weighted) composition operators have been studied by many authors between different classes of analytic function spaces (see, for example, [4, 5, 8, 9], and the references therein).
For each nonnegative integer , the generalized weighted composition operator is defined by for each and . The class of generalized weighted composition operators include weighted composition operators , composition operators followed by differentiation and composition operators proceeded by differentiation [10]. Also, weighted types of operators and are of the form , that is, and [11].
Boundedness and compactness of generalized weighted composition operators have been studied between Bloch-type spaces and Zygmund-type spaces in [2, 12] and between Bloch-type spaces and weighted-type spaces in [13, 14]. Essential norms of generalized weighted composition operators in these cases have been studied in [2, 3, 6, 15]. In [16], different characterizations for the boundedness and compactness of these operators between Bloch-type spaces are given, and also, their essential norms are investigated in [7]. The essential norm of some extensions of the generalized composition operators between th weighted-type spaces on the unit disk has been calculated by Stević in [17]. In this paper, we first study boundedness of generalized weighted composition operators between Zygmund-type spaces and give new characterizations for the boundedness of these operators. Then, we find estimates for the essential norms of such operators in terms of the new characterizations. Consequently, we obtain different types of characterizations for the compactness of such operators.
The following lemma, which will be used in the next chapters, collects some useful estimates for the functions in Zygmund-type spaces (see, for example, [6], and references therein).
Lemma 1. For every , we have (i) and for (ii) and for (iii), for (iv), for (v), for (vi), for
It is known that for each and , we have for all and (see [18]). Therefore, for each and , we have for all and . Note that, by the definition of Zygmund-type spaces, it is clear that (13) also holds in the case of .
In this paper, for real scalars and , the notation means for some positive constant . Also, the notation means and .
2. Boundedness
For each , the following test functions in will be used in our proofs:
Moreover, in order to simplify the notations, we define
In the next theorem, we give three different characterizations for the boundedness of .
Theorem 2. Let , be an analytic self-map of and . Then, the following statements are equivalent for each . (i) is bounded(ii), where for each and (iii)(iv)
Moreover, this is also equivalent to in the special case of and .
Proof. Suppose that and is a bounded operator. Since is a bounded sequence in (see, for example, [19]), one can see that is a bounded sequence in . Therefore, boundedness of implies that
Let . Recalling the definition of , , and , one can see that
for each . Note that
Also, by Stirling’s formula, we know that as (see [2]). Consequently,
and since was arbitrary, we get . Applying a similar argument to the functions and implies .
For each define
Then, one can see that , , and
Moreover,
and by the definition of , we have
Therefore,
On the other hand,
In order to prove that , the similar approach can be applied using the test functions defined as
Also, for the proof of , the argument is similar, using the test functions
The special case of and can be proved by a similar argument and applying Lemma 1.
By applying (13), for every and , we get
Multiplying both sides by and taking supremum over , one can get (i).
In the case of and , we have the following result.
Theorem 3. For each , is bounded if and only if
Proof. The sufficiency part is a consequence of Lemma 1(ii) and definition of the norm in Zygmund-type spaces. Now, suppose that is bounded. Then, by applying , we get . Also, by defining for each , one can see that , , , and . Therefore, and consequently,
On the other hand, implies that
which completes the proof of (iii). In order to prove (ii), for each , define the test functions
Then, one can prove (ii) by a similar approach as in (iii) and using the facts , , , and .
In order to prove (i), consider the test functions for each , where . Then, one can see that , , and
Since the operator is bounded, we get
Therefore,
On the other hand, since , we have which completes the proof.
3. Essential Norms
In this section, we investigate essential norm estimates of in different cases of , , and . In order to simplify the notations, we define
Theorem 4. Let and with . If is a bounded operator, then
Proof. First, note that, as mentioned in the proof of Theorem 2, is a bounded sequence in . Since polynomials are dense in , by [20] (Proposition 2.1), converges weakly to in and hence in . Therefore, for each compact operator , we have , and consequently,
Next, we prove that
One can see that having uniformly bounded norm. Also, these functions converge to uniformly on compact subsets of , as , which implies their weak convergence to in . Therefore, for each compact operator , we have
The same argument is valid for and , that is,
Since was an arbitrary compact operator, we get
For each and , the function is defined by for all . Note that uniformly on compact subsets of when . Also, the operator given by is a compact operator for each . Let be a sequence in converging to as . Then, is a compact operator for each . Hence,
Therefore, it is enough to prove that
Note that for every with , we have
Since , we have
Therefore, for each , we get
respectively. Since the operator is bounded, by Theorem 2, we have
Also, for each , uniformly on compact subsets of , since uniformly on compact subsets of . These facts imply that and hence, we just need to estimate , , and . By applying (13) and Lemma 1 and using the test functions defined in Theorem 2, we have
Therefore, as , we get
Similarly, by applying the test functions and , one can prove and therefore,
By applying (56), we get and similarly, one can see that and . Therefore,
In order to prove the converse of (61), let be a sequence in such that , as , and define the test functions where , , and are defined in the proof of Theorem 2. Then, , and are bounded sequences in which converge to zero on compact subsets of . Also, for each , we have
Thus, for any compact operator , we get
Hence, and similarly, which implies that
To prove the final case, first note that by Theorem 2, we have
Recall that, for each ,
For each , as in the proof of Theorem 2, we have which implies that
Therefore, when , we get
A similar approach implies that and, by applying (59), we conclude that
This, along with (44), completes the proof.
Corollary 5. Let and with . If is a bounded operator, then the following statements are equivalent: (i) is compact(ii)(iii)(iv)
We next prove the result of Theorem 4 in the case of and .
Theorem 6. Let and be a bounded operator. Then,
Proof. Let be a sequence in such that , as . Consider the test functions and defined in the proof of Theorem 3. Then, and are bounded sequences in which converge to zero on compact subsets of . Also,
Thus, for any compact operator , we have
Therefore, and also,
Now, consider the test functions defined in the proof of Theorem 3. Then, is a bounded sequence in which converges to zero on compact subsets of . Moreover,
Consequently, which implies that
The proof of upper estimate is similar to the proof of Theorem 4 using the operators and Lemma 1.
Corollary 7. Let and be a bounded operator. Then, is compact if and only if
Remark 8. Montes-Rodríguez in [8] (Theorem 2.1), and also Hyvrinen et al. in [4] (Theorem 2.4), proved that ifandare radial and nonincreasing weights tending to zero at the boundary of, then(i)the weighted composition operator maps into if and only ifwith norm comparable to the above supremum. (ii)
Also, by [5] (Lemma 2.1), we know that for each , (iii)(iv)
By applying these facts, our results in this paper containing terms of the type can be restated in terms of and (see, for example, [9], and references therein for these types of results).
Data Availability
Data are available on request.
Disclosure
A preprint of the manuscript can be found in https://arxiv.org/abs/1903.12383.
Conflicts of Interest
The authors declare that they have no conflicts of interest.