On Stević-Sharma Operators from General Class of Analytic Function Spaces into Zygmund-Type Spaces

Bakhit, M. A.; Kamal, A.

doi:https://doi.org/10.1155/2022/6467750

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Volume 2022 | Article ID 6467750 | https://doi.org/10.1155/2022/6467750

On Stević-Sharma Operators from General Class of Analytic Function Spaces into Zygmund-Type Spaces

M. A. Bakhit¹and A. Kamal^2,3

Academic Editor: Jingshi Xu

Received07 Sept 2021

Accepted19 Apr 2022

Published01 Jul 2022

Abstract

A Stevi-Sharma operator denoted by is a generalization product of multiplication, differentiation, and composition operators. In this paper, we characterize the bounded and compact Stevi-Sharma operator from a general class of Banach function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. Using several restrictive terms, we show that all bounded operators from into the little Zygmund-type spaces are compact. As an application, we show that our results hold up for some other domain spaces of , such as the Hardy space and the weighted Bergman space.

1. Introduction

Meanwhile, in this work, is the open unit disk of the complex plane . We denote by the family of analytic functions in ; also, we denote by the family of analytic self-maps of and the set of conformal automorphisms of the disk . As is usual, the Banach space is the family of bounded functions , defined by the norm .

For any weighted function , the weighted Banach space is the space of functions such that

Moreover, the little weighted Banach space is the space of functions such that

The Bloch-type space is the Banach space of functions such that , with the norm . The little Bloch-type space is the closed subspace of consisting of functions such that .

The space of all functions is said to be a weighted Zygmund-type space , if all functions are such that . The space becomes a Banach space with the norm

The little weighted Zygmund-type space is the closed subspace of and consists of functions , such that .

Exactly as in the aforementioned spaces, when , we get the Zygmund-type space . In addition, the space is a Banach space with the norm , where . The Zygmund spaces also satisfy whenever . When , we get the classical Zygmund space denoted by . In fact, the motivation for the name and for studying the Zygmund-type spaces comes from the Zygmund class; see, for example, Chapter 5 of Duren’s book [1]. The Zygmund space and its subspace play an important role in connection to the theory of the Hardy spaces . Indeed, in [2], it was shown that can be viewed as the dual of the Hardy space . More generally, the spaces defined by replacing the function in the definition of with its th derivative, under the assumption for , can be viewed as duals of the Hardy spaces . A recent study of several operators on Zygmund-type spaces has attracted significant research attention; see, for example, [3–5].

For any and , we have the following linear operators: (1): the differentiation operator(2): the multiplication operator(3): the composition operator(4): the weighted composition operator

For a unified manner of treatment of these operators, many researchers seek to present the various products of multiplication, composition, and differentiation operators; see, for example, [3, 4, 6–8]. Stevi et al. was the first to introduce the operator in [7]. So, the operator is called the Stevi-Sharma operator, which is defined as

Over the past 10 years, the boundedness and compactness of this operator have been studied extensively in the most well-known spaces of analytic functions; for example, see [5, 9].

This paper proceeds as follows. In Section 2, we present basic facts and prerequisites required for the Banach spaces. During Section 3, we characterize the boundedness of the operators from a large class of Banach spaces into the Zygmund-type space , also we give their norm estimates. Throughout Section 4, we study the compactness of from a large class of Banach spaces into the Zygmund-type space , under specific conditions on class . The boundedness and compactness of from to are investigated in Section 5. Finally, in Section 6 we apply our results to the Hardy spaces , and the weighted Bergman space , ( and ).

1.1. Suppositions

In this paper, always, we will suppose the following: (i) and , unless otherwise stated(ii)For any , we will do the following differentiation over and over again:

So, to simplify the formulation of our results, we will define the following quantities: (iii)For any two quantities and that are jointly dependent on , we stipulate that , meaning that there is a positive constant that fulfills . Thus, when , we hold that and the quantities and are said to be equivalent. If , then if and only if

2. Some of the Prerequisites Required for Banach Spaces

Following popular terminology in functional analysis, we stipulate a Banach space whose elements are functions , with norm and whose functionals of point-evaluation are bounded. Always we will suppose through this paper that is a Banach space of functions .

Let indicate the functional of point-evaluation at . So, for any , we can define

Thus, for each and , we obtain

Now, under appropriate modifications, we will list the specific conditions given in [10], including the conditions used to formulate the results of the present work. (I)For all and , we get (II)With respect to the uniform convergence topology on compact subsets of , the unit ball of a Banach space is comparatively compact(III)By a positive constant in compact subsets of , the norm is bounded below and (IV)For , let where . Then, the linear map is a compact map on , also (V)For all and , we have and , where (VI)For all , let ; then , where

The following proposition also is proved in [10].

Proposition 1. For , the map is bounded on compact subsets of . Further, if a Banach space is reflexive, then is compact and shows relatively of uniform convergence, with respect to the topology on a compact subset of .

The Lemma 2 helps to distinguish the properties of the operator concerned with studying in this article.

Lemma 2. Let there be a Banach space satisfying the above conditions (V) and (VI). Suppose that , then for any and , such that , there is a set of functions defined in , for , such that . Moreover, the sequences converge to uniformly on compact subsets of , when .

Proof. Let , which is the automorphism of that changes zero and , that is, , for any . Note that Now, fix , such that , and consider the functions , for all and . Let in condition (V); then for all , we have and Also, it is not difficult to prove that Finally, the uniform convergence to of the sequences is self-evident, when .

3. Boundedness of

In this section, we characterize the boundedness of the operators from a large class of Banach spaces into the Zygmund-type space ; also we give their norm estimates.

First of all, to simplify the formulation of the next results, for as in (4), we will define the following:

Now, we introduce the main results of this section.

Theorem 3. Let there be a Banach space satisfying the above conditions (V) and (VI). Then, is a bounded operator if and only if all the quantities are finite, for . Moreover, in which case

Proof. First, suppose that is bounded. Then, Set in Lemma 2, we observe that Also, we have and .
Then, we obtain For any with , by (5), we have Next, we prove that is finite. So, set in Lemma 2; we observe that , but Also, we have and .
Then, we obtain Using the triangle inequality, we get Condition (VI) for shows that Thus, by (16), we have Using (15), we obtain For the third time, we go back to Lemma 2 with ; we know , also , and .
Then, We repeat using the triangle inequality; we get Repeat condition (VI) for and since , we obtain Using (15) and (21), we obtain Finally, we go to prove that is finite. For fix , such that . Then, we have Since is bounded, by condition (VI) for , we observe that Now, using (15), (21), and (26), we have Because of this, all the quantities are finite, where , and On the other hand, suppose all the quantities are finite, where . For fix we let , such that . For , by conditions (VI) and (4), we obtain In addition to that, We can choose a sufficiently large , the highest estimate Moreover, from (1) and (33), we have That ends the proof of Theorem 3.

In the case of and in Theorem 3, we define the following quantities:

Now we deduce the following result as a corollary of Theorem 3.

Corollary 4 ([4], Theorem 3.1). Let a Banach space be satisfying the above conditions (V) and (VI). Then, is a bounded operator if and only if all the quantities are finite, for . Moreover, in which case , where

4. Compactness of

Here, we study the compactness of from a large class of Banach spaces into the Zygmund-type space , under specific conditions on class . Before that, we mention the Lemma 5, which can be proved in a standard way; for example, see [6, 10].

Lemma 5. Let a Banach space be satisfying condition (II) and whose the functionals of point-evaluation are continuous. Then, the bounded operator from into Zygmund-type space is compact if and only if for any bounded sequence in which converges uniformly to as on compact subsets of , .
Moreover, to simplify the formulation of the main results in this section, for as in (4) and , we set

Now, we introduce the main results of this section.

Theorem 6. Let a Banach space be satisfy the above conditions (II), (V), and (VI). Suppose that is a bounded operator, and let Then, is a compact operator if and only if all the quantities , for .

Proof. It is clear that condition (38) holds if the constants contain in , that is . First, assume that is a compact operator. Obviously, the result is intuitive if . So from now, we suppose . Also from now to the end of the proof, we let a sequence in , be such that , for , Since for each and , then, for each and fix , there exists such that and Clearly, is a norm-bounded sequence; also is a bounded map on compact sets in . Using the fact , then is a uniformly bounded sequence on compact sets. For all and , set the functions Then, we see that Clearly, also converges uniformly to on compact subsets of . By the condition (V), we obtain and .
Since , by the functions defined in Lemma 2 for , we can consider the functions By Lemma 2, we observe that Also, we have and , for all .
Using the hypothesis that condition (II) holds, by applying Lemma 5, for all we obtain In the case of in (45) and (47), we get Thence, From (40) and (43), we have Thus, by (38), we have We know from Theorem 3 and the boundedness of the operator , that all the quantities are finite, where . Therefore, we obtain from (50) and (51): Since is an arbitrary constant, we conclude that Next, we will show that . In the case of in (45) and (47), we observe that Also Then, there is an integer that corresponds with the number fixed above, whenever such that Since and , by condition (VI) for , using the triangle inequality, for every , we obtain By (55), we conclude that Again from (40) and (43), for every sufficiently large and , we have Once secondly by (38), we obtain Therefore, from (59) and (61), we get Since is an arbitrary constant, we conclude that .
We go to what is needed to prove that . In the case of in (45) and (47), we observe that Also, Then, there is an integer that corresponds with the number fixed above, whenever such that Repeat condition (VI), for , and using (55), since , we get By (40) and (43), for every sufficiently large and , we have By (38), we obtain Therefore, by (60) and (61), we have Since is an arbitrary constant, we conclude that .
Finally, we go to prove that . For each and fix , let and be instituted as in (41) in terms of the sequence and . Using the hypothesis that condition (II) holds, with apply Lemma 5, we guarantee that Corresponding to the number there is an integer , whenever , such that Since , by condition (VI), we observe that Thus, for every sufficiently large, By (38), we deduce Hence, for every sufficiently large, we have which means .
Conversely, assume that the quantities , for all . By applying Lemma 5, to prove that is a compact operator, it is enough to find a sequence in converges uniformly to , on compact subsets of and such that .
Now, fix , for each , suppose is a sequence in . Let , whenever , such that Since a sequence converges uniformly to on , it does so all sequences , for . Thus, there is an integer , for all , such that , for and all . Hence, as in Theorem 3 if and , using the quantities , for , then Therefore, one last time by (38) for , we obtain Let , and let be large sufficient. Then, by (38), condition (VI) for , and the hypothesis (76), we have The last step, since the sequences and are converged uniformly to on compact subsets of , then, they converge pointwise. Thus, By this with the last two inequalities, we see that . According to Lemma 5, we already got the compactness of .
Now, in the case of and in Theorem 6, by using the quantities in (35), we deduce the following result as a corollary, see Theorem 3.2 in [4].

Corollary 7. Let a Banach space be satisfying the above conditions (II), (V), and (VI). Suppose that is a bounded operator and suppose that (38) holds. Then, is a compact operator if and only if all the quantities , for , where

5. Boundedness and Compactness of

In this part, we will focus on the operators mapping a large class of a Banach spaces into the little Zygmund-type space . In particular, especially, we develop the compactness and the boundedness conditions of .

As is the case in the previous two sections, to simplify the formulation of the results, we define the following limits for as in (4).

Now, we go to the main results of this section.

Theorem 8. Let a Banach space be satisfying the above conditions (II), (V), and (VI) and suppose that (38) holds. Then, the equivalence of the following conditions is certain. (i) is a compact operator(ii) is a bounded operator(iii)The quantities , for

Proof. is evident.
To prove (ii)(iii), assume that is a bounded operator. Then,

Firstly, we let a sequence in , be such that . Fix with , by the functions defined in Lemma 2 for , we can consider the functions

By Lemma 2, we inferred that

Also by using condition (V), we have that all the sequences are bounded in . Now, set in (88), we observe that

Since , then, we obtain

Over all, with , by taking the supremum and by (5), we have

Similarly, we prove that . So, set in (88); we observe that

Since , then, we obtain

Then, by condition (VI) for with (92), we have

Again, over all with , by taking the supremum and by (5) we have

Third likewise, we prove that . Set in (88), we observe that

Also, since , by condition (VI) for with (92) and (96), we obtain

Again, over all with , by taking the supremum and by (5), we have

Finally, we will prove that . For each as above. Using the fact that , we get

Since , by condition (VI), we observe that

Thus, over all with , by taking the supremum and by (5) we have which means .

Conversely, we prove (iii)(i), assume that all the quantities , for all . By applying Theorem 6, we have the compactness of the operator . Thus, to prove that is a compact operator, it is enough to show for all that . So, let , then by condition (VI) for , and (5), we observe that

Thus, as desirable, .

In the case and in Theorem 8, by using the quantities in (35), we deduce the following result as a corollary; see Theorem 3.2 in [4].

Corollary 9. Let a Banach space be satisfying the above conditions (II), (V), and (VI) and suppose that (38) holds. Then, the equivalence of the following conditions is certain: (i) is a compact operator(ii) is a bounded operator(iii)The quantities , for , where

6. Implementation of the Results on Other Domains

In this section, we will show that our estimations hold up for any domain space choice provided that the domain is reflexive. Note that all spaces handled below contain the constant functions, so the following condition holds for all:

6.1. The Banach Space

Observe that the map is a constant mapping. Clearly for the space, all the conditions (II)-(V) hold. By Theorem 5.4 in [11], for any integer , we know that the space and the Bloch space are connected by the condition

Then, the space satisfies also the condition (VI). So, we deduce that all the results in Section 3–5 are applicable. We set the following quantities:

The theory below summarizes the results are applicable for the domain space .

Theorem 10. The operator is a bounded operator if and only if all the quantities are finite, for , in which case .
Moreover, assume that is a bounded operator; then, it is a compact operator if and only if , for all .

Theorem 11. The following conditions are equivalent: (i) is a compact operator(ii) is a bounded operator(iii)The quantities , for and

6.2. The Weighted Bergman Space

For all and , the weighted Bergman space is the set of all functions , such that where is the weighted Lebesgue measure. When , the positive measure was normalized to be a probability measure. The weighted Bergman-type space is a Banach space when ; also, it is a complete metric space when . Several properties of spaces are discussed in [12, 13].

Lemma 12 was proved in [12] and Lemma 5.3 in [10].

Lemma 12. For a positive integer and , let and and let . Then, Set the following quantities: Now, we summarize the results are applicable for the domain space .

Theorem 13. The operator is a bounded operator if and only if all the quantities are finite, for , in which case .
Moreover, assume that is a bounded operator; then, it is a compact operator if and only if , for all .

Proof. From Lemma 12, we have that the conditions (V) and (VI) hold for all . Then, the norm estimate of the operator is an outcome instant from Theorem 3.
The results for are an outcome instant from Theorem 8.

Theorem 14. The following conditions are equivalent: (i) is a compact operator(ii) is a bounded operator(iii)The quantities , for

Proof. From Lemma 12, we have that the conditions (V) and (VI) hold for all . Then, the results are an outcome instant from Theorem 8.

6.3. The Hardy Space

For and , the Hardy space is the Banach space with the norm

We know that the explicit formula for Hardy space is given by

Moreover, for a positive integer and if , then,

From (112) and (113), also, we have that the conditions (V) and (VI) hold for all on Hardy space . Then, the norm estimate of the operator is an outcome instant from Theorem 3.

The results for are an instant outcome from Theorem 8. So, we can summarize the results as follows.

Theorem 15. The operator is a bounded operator if and only if all the quantities are finite, for , in which case .
Assume that is a bounded operator, then it is a compact operator if and only if , for all , where

Theorem 16. The following conditions are equivalent: (i) is a compact operator(ii) is a bounded operator(iii)The quantities , for

Data Availability

The data is not applicable to this concerned article as no concerned datasets were created or used through this concerned study.

Conflicts of Interest

The authors declare that they have no conflicts interest.

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Copyright © 2022 M. A. Bakhit and A. Kamal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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