Essential Norms of Stević–Sharma Operators from General Banach Spaces into Zygmund-Type Spaces

Bakhit, M. A.

doi:https://doi.org/10.1155/2022/1230127

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Volume 2022 | Article ID 1230127 | https://doi.org/10.1155/2022/1230127

Essential Norms of Stević–Sharma Operators from General Banach Spaces into Zygmund-Type Spaces

M. A. Bakhit¹

Academic Editor: V. Ravichandran

Received18 Feb 2022

Accepted18 Mar 2022

Published12 May 2022

Abstract

A Stević–Sharma operator denoted by is a generalization product of multiplication, differentiation, and composition operators. Using several restrictive terms, we characterize an approximation of the essential norm of the Stević–Sharma operator from a general class of holomorphic function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. As an application, we show that our results hold up for several other domain spaces of , such as the Hardy space and the weighted Bergman space.

1. Introduction

Let be 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 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, [3–10].

For any and , we have the following linear operators: , the differentiation operator , the multiplication operator , the composition operator , the weighted composition operator

For a unified manner of treatment of these operators, many researchers seek to present various products of multiplication, composition, and differentiation operators; see, for example, [8, 11–18]. Stević et al. was the first to introduce the operator in [19]. So, for any and , the operator is called the Stević–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 holomorphic functions; for example, see [3, 19–27].

This paper proceeds as follows. In Section 2, we present basic facts and prerequisites required for the general class of holomorphic function spaces. In Section 3, we characterize upper and lower bounds for the essential norm of the operators from into the Zygmund-type space (or ). Finally, in Section 4, we show that our estimations withstand any choice space, provided that it is reflexive, such as the Hardy spaces , and the weighted Bergman space ( and ). This work is a continuation of our not yet published article regarding the boundedness and compactness of between the currently considered spaces; see [20].

In this paper, 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. Preliminaries

Following popular terminology in functional analysis, we stipulate a Banach space (or ) whose elements are functions with the norm and whose the functionals of point-evaluation are bounded.

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

For each and , we obtain

The following sets of conditions on Banach space , given in [28], include the conditions used to formulate the results of the present work. For all and , if , we obtain , for all . By a positive constant , the norm is bounded below in compact subsets of and . With respect to the uniform convergence topology on compact subsets of , the unit ball of a Banach space is comparatively compact. For all and , we have and , where . For , let , where . Then, the linear map is a compact map on . For , . For all , let , then , where .

If a Banach space contains polynomials and satisfies the conditions , , and , then is said to be an admissible space (see [28–30]). If the set of polynomials in an admissible space is dense, it is said to be polynomial dense. Examples of a polynomial dense space include the Hardy spaces and the weighted Bergman space for all and (see [31]).

The following proposition is Proposition 1 in [28].

Proposition 1. For , the map is bounded on compact subsets of in a Banach space . 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 following lemma helps to distinguish the properties of the operator .

Lemma 1. Let there be a Banach space satisfying the above conditions and . Suppose that , then for any and , such that , there is a set of functions defined in , for , such that ,and

Moreover, the sequences converge to 0 uniformly on compact subsets of , when .

Proof. Let , which is the automorphism of that changes zero and , that is, , for any . Note thatandNow, fix , such that and consider the functions , for all and . Let in condition , then for all , we have andAlso, it is not difficult to prove thatFinally, the uniform convergence to 0 of the sequences is self-evident when .

This work is a continuation of [20], where we describe the boundedness and the compactness of . The following boundedness result has been proven.

Theorem 1. Let there be a Banach space satisfying the above conditions and . Then, is a bounded operator if and only if all the quantities are finite, for . Moreover, in which casewhere the quantities are defined as follows:

The following lemma is proved in a standard way; see, for example [11, 28].

Lemma 2. Suppose that are two Banach spaces satisfying the following conditions: The point evaluation functionals are continuous on With respect to the uniform convergence topology on compact subsets of , the closed unit ball of a Banach space is comparatively compact The operator is continuous from into when and are given the topology of uniform convergence on compact sets

Then, the bounded operator is compact if and only if for any bounded sequence in converges uniformly to 0 as on compact subsets of , and we have .

We know that the essential norm of an operator is its distance from the compact operators in the operator norm. Specifically, let and be two Banach spaces and let be a bounded linear operator, then the essential norm of between and is referred to as , and its definition is

Lemma 3 (see [28]). Let be a Banach space of functions , such that . Fix any and choose .(a)If the Banach space includes some functions nonvanishing at zero and fulfill either or , then(b)If the Banach space satisfies one of the following sets of conditions:(i)Both and (ii) and is reflexive(iii) then(c)If the Banach space satisfies and one of the conditions in part (b), then

Remark 1. Fix any and choose . By part (c) of Lemma 3, we have converging uniformly to 0 on compact subsets as . Since satisfies , under the same conditions, we obtain

Remark 2. For any , if we suppose thatwe observe that assumption (20) is sufficient to apply Lemma 3, while part (c) of Lemma 3 requires condition .

3. Essential Norm of

Here, we provide an approximation of the essential norm of from a large class of Banach spaces into the Zygmund-type space , under specific conditions on class on the open unit disk. With quantities defined below, the following result complements Theorem 3 in [32]. To simplify the formulation of the main results in this section, for as in (5) and , we set

Theorem 2. Let a Banach space of functions in be reflexive and satisfy the above conditions , , and , together with either or . Suppose that , if is a bounded operator. Then,

Proof. First, we prove that,If we suppose that , then it follows immediately that for all . This is why, hereafter, we suppose , then we prove that for all .
For each and fix , we choose such that ,By assuming that is reflexive, the unit ball of a Banach space is compact under the uniform convergence topology on compact subsets of . Using the fact that , then is a uniformly bounded sequence on compact sets of . For all and , set the functionsThen, observe thatandClearly converges uniformly to 0 on compact subsets of . By the condition , we obtain and .
Since , by the functions defined in Lemma 1, we consider the functionsBy Lemma 1, observe thatFurther, and , for all .
Using the hypothesis that conditions and hold, then all sequences are bounded in a Banach space . Moreover, the sequences converge to 0 uniformly on compact subsets of , when .

Step 1. In the case of , let be a sequence in , such that andLet in (29), then we haveThus,Now, supposing that is a compact operator, then the operator is bounded. Therefore, by Proposition 1 and Lemma 2, we obtainBy (32) and (33), we haveUsing (24) and (27), we obtainThus, by (30), (34), and (35), we have

Step 2. In the case of , we let a sequence in be such that andLetting in (29), we observe thatSo, we haveBy Proposition 1 and Lemma 2, if is a compact operator, then we obtainBy (39) and (40), we haveHence,Using (24) and (36), then (42) givesSince is an arbitrary constant, combining (43) with (36) and (37), we conclude that

Step 3. In the case of , we let a sequence in be such that andLetting in (29), we observe thatThus, we haveBy Proposition 1 and Lemma 2, if is a compact operator, then we obtainBy (47) and (50), we haveHence,Using (24), (36), (44), and then (50) givesSince is an arbitrary constant, combining (51) with (44) and (45), we conclude that

Step 4. In the case of , for each and fix , let and be such that in terms of the sequence in such that andThen,If is a compact operator, then by Lemma 2, we obtainBy (54) and (55), we haveHence, by condition and (56), we haveUsing (6), (14), (21), and (50), then (57) givesSince is an arbitrary constant, combining (58) with (52) and (53), we conclude thatWe thus obtain the lower estimate on the essential norm.
Now, let us prove the upper estimate. By the following,In the statement of the theorem, we assume that , which is the same as assumption (20) in Remark 2. By reading carefully the detailed proof of Lemma 3 in [28], while part (c) of the lemma requires condition , we can conclude that the assumption is sufficient to apply Lemma 3. Thus, we fix and choose as in Lemma 3 and Remark 2.
By condition , since is a bounded operator, the linear map is compact on , and the product is compact. Therefore,whereBy Lemma 3, we obtainNow, combining condition with Lemma 3, Remark 1, Theorem 1, and (20), we obtainUsing (38), (39), (40), and (67), we obtainSince is arbitrary, if , then we have , and is compact. If , letting , we have

Remark 3. Note that the assumption that the Banach space is reflexive is used only in the case of a lower estimate of Theorem 2. This brings about an interesting question. What is the possibility of approximating the essential norm in Theorem 2 without assuming the reflectivity on the Banach space ?
Actually, by condition in Lemma 2, if is a compact operator and the sequence is bounded in the Banach space converging to zero uniformly on compact subsets of , then . Thus, our argument in the proof of Theorem 2 for taken to be leads to a lower estimate on . An example of a space satisfying condition in Lemma 2 is the Hardy spaces .
Thus, the approximation of also holds if the conditions — in Lemma 2 hold.

In the cases and in Theorem 2, we deduce the following result as a corollary; see Theorem 3.3 in [4].

Corollary 1. Let a Banach space of functions in be reflexive and satisfy the above conditions , , and , together with either or . Suppose that , if is a bounded operator. Then,where

4. Applications of Essential Norm on Other Domains

We now 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. Thus, condition (20) applies to all cases. First, set the following quantities:

Second, we discuss the approximation of on other domain spaces.

4.1. The Hardy Spaces

Observe that the map is a constant 1 mapping. Clearly for the space, all conditions – hold. By Theorem 5.4 in [33], for any integer , we know that the space and the Bloch space are connected by the condition as follows:

Thus, the space also satisfies the condition . So, we deduce that the approximation of the essential norm is applicable. The theory below summarizes this approach, and the results are applicable for the domain space .

Theorem 3. If the operator is a bounded operator, then

For and , the Hardy space is the Banach space with the norm as follows:

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

Moreover, for a positive integer and if , then

From (76) and (77), we have that the conditions and hold for all on Hardy space . Thus, the norm estimate of the operator is an outcome instant from Theorem 1.

The results for are an instant outcome from Theorem 2. We can then summarize the results as follows.

Theorem 4. If is a bounded operator, thenwhere .

4.2. The Weighted Bergman Space

For all and , the weighted Bergman space is the set of all functions , such thatwhere is the weighted Lebesgue measure. When , the positive measure is normalized to become a probability measure. The weighted Bergman-type space is a Banach space when , and also it is a complete metric space when . Several properties of spaces are discussed in [31, 34].

The following lemma is proved in [31] and Lemma 5.3 in [4].

Lemma 4. For a positive integer and , let and and let . Then,(i)(ii).

Let us summarize from the domain space .

Theorem 5. If is a bounded operator, thenwhere .

Proof. From Lemma 4, we have that conditions and hold. Thus, the norm estimate of the operator is an outcome instant from Theorem 1.
For the essential norm estimates of , we observe that conditions – and hold for (see [28]). Since is a reflexive Banach space if , in the case of an arbitrary weight , the hypotheses of Theorem 2 hold.

Data Availability

The research conducted in this paper does not make use of separate data.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright © 2022 M. A. Bakhit. 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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