The important purpose of this current work is to study a new class of operators, the so-called Toeplitz-superposition operators as an expansion of the weighted known composition operators, induced by such continuous entire functions mapping on bounded specific sets. Minutely, we have deeply discussed the conditions for boundedness of this new type of operators between certain types of some holomorphic Bloch classes with some specific values of the weighted functions.

1. Introduction

Fundamentals of the needed analytic function spaces as well as the types of concerned operators are briefly introduced. The paper focuses first on the concerned setting of certain classes of function spaces and the new defined operator, which in turn is motivated essentially by some certain classical concepts of known operators such as superposition operators as well as Toeplitz operator. There is an emphasis in the concerned paper on intensive tying together the needed type of analytic function spaces and the concerned operators, to illustrate the roles of the obtained results.

All of the needed information to justify the target of this research is collected in this concerned section. Moreover, here, basic concerned concepts, the Bloch space of analytic-type, certain needed concerned lemmas, and superposition and Toeplitz operators are presented.

Let be the open unit disk in , and let denote the class of all analytic functions in . Let denote the concerned Lebesgue measures on

Numerous intensive studies on analytic Bloch-type spaces are researched in literature (see [15] and others).

Let and , the -Bloch space is defined by

The space is called the Bloch space and denoted by (see [3]).

The following interesting needed lemma has been proved in [6].

Lemma 1. For a given, let the function. Then, we have

The following useful integral estimate is well known and can be found in [7].

Lemma 2. Letand. Then

For and , the weighted Bergman spaces is the space of all functions for which

When , we simply write for , and when , is a Hilbert space. It is well known that the Bergman kernel of the Hilbert space is given by where . The Bergman projection is the orthogonal projection from onto Hilbert space which given as:

For and , the Toeplitz-type operator with symbol is defined by

This paper is organized as follows: during Section 2, we have defined the Toeplitz-superposition operators on the normed (metric) subspaces. Throughout Section 3, we establish the conditions for the Toeplitz-superposition operators to be bounded from -Bloch space into -Bloch space , in the case and or . Section 4 is devoted to a study the boundedness of Toeplitz-superposition operators between weighted Bloch spaces in the case or .

Remark 3. It is concerned remarkable to say that two concerned quantitiesand, where both depending on the concerned function, the expressioncan be satisfied when we have a concerned positive constantfor which. When, the expressioncan be written to say that there is an equivalence relation between the concerned quantitiesandFurthermore, whenwe deduce that

2. Toeplitz-Superposition Operators

Let denote the set of all entire functions on the complex plane . For a function , the superposition operator is defined by . Moreover, if and , the weighted superposition operator is defined by , for all and . Note that, if , then , for any .

For any normed subspace , we will consider the set , defined by

Now, we define the Toeplitz-superposition operators acting on .

Definition 4. Let two functionsand. Then, the Toeplitz-superposition operatorson the normed (metric) subspaceare given by

Let be the scalers if is a fixed entire function and . Then, from the definition of Toeplitz-superposition operators, we have which holds for all , and hence, the Toeplitz-Superposition operators are linear on the normed subspace .

It can be seen that whenever , then, the operator becomes the operator . So, Toeplitz-superposition operators can be taken as an extension of weighted superposition operators. The present paper is interested in answering the following interesting questions. (i)Can we transform one holomorphic function space into another by what kinds of entire functions?(ii)What are the holomorphic spaces that can be transformed one into another by certain weighted classes of entire functions such as specific analytic polynomials of a certain degree and certain entire-type functions of given type and order?(iii)When does the holomorphic function induces a Toeplitz-superposition operators to form one holomorphic function space into another?

As a concerned result, the obtained results will introduce answers of the above mentioned questions by using the class of Toeplitz-superposition operators that are acting between different classes of Bloch functions.

Also, the answers for some of these concerned questions have been introduced by several authors; the following citations can be stated for interesting and intensive studies [820].

3. Boundedness in the case and or

Several important discussions on boundedness property of the new operator acting on the analytic Bloch spaces are presented in this concerned section. Furthermore, some essential equivalent characterizations for its boundedness are established too.

Now, we will introduce the main results of boundedness.

Theorem 5. Forand. Suppose thatand let, with. Then, the Toeplitz-superposition operatoris bounded.

Proof. First, assume that . Let , since we have

Now, let the constant where such that , by Lemma 1, we have . Set , then . Since , we have the fact that , and since , we have that Thus, where depended only on , and . This shows that is bounded.

Theorem 6. For, letbe harmonic and let. Then, the Toeplitz-superposition operatoris bounded if and only ifandis a constant entire function.

Proof. It is trivial that if and is constant, then is bounded. If is constant, not identically , and maps into then it is clear that . Assume now that and is not constant, and set maps into . Let be the constant function defined by , for all , such that . Since , it follows that . This implies that since . Finally, since is not constant, then there is a disk and , on which . Set the test function . Then, for all , we have

But, along with the positive radius, we get , as . This shows that is not bounded.

4. Boundedness in the case or

Theorem 7. For, letbe harmonic and let. Then,is bounded onif and only ifandis an affine function (linear function plus a translation).

Proof. First, suppose that and is an affine function. It is easy to explain is bounded from into itself.

On the other hand, assume that and does not linear function. Then, by using the Cauchy estimates for , we can find a sequence , for each such that as and . Also, since the weight is typical, we can find a sequence of points such that, with and such that , for all . Now consider the sequence of functions contained in satisfies and. Furthermore, we can suppose that . Hence,

Because as . This shows that cannot be bounded if is not a linear function.

Theorem 8. For, letbe harmonic and letbe an increasing and continuous function. Then,is bounded if and only if, and for each, there is a positive constantwhenever, such that

Proof. First, suppose that and (15) is true. Now, consider and let satisfy and select such that . Then, there is , such that , whenever . Thus, since is a compact set and is a continuous function, we can assume that , for all . Hence,

Using that the function is increasing and the fact that , we have

This shows that is bounded.

On the other hand, assume that and does not satisfy (15). Then, we can find and a sequence such that as and , for all . Since the weight is typical, we can find a sequence of points such that as . Thus, we can consider a sequence of functions contained in satisfies and . Now, let and set the function for all . Then, we have and . For large enough , we obtain

Then, we conclude that cannot be bounded.

Theorem 9. For, letbe harmonic and let. Then, the following are equivalent:(i) maps into (ii) and is a polynomial of degree at most (iii) is bounded

Proof. First, suppose that (i) holds, let be a polynomial with , then, we have that .

Now, suppose that the entire function is a polynomial of degree . Then, for an integer and a positive constant , there is a sequence such that . We may assume without losing generality that and , for an integer . Now, we let , then, we show that . The point such that and , and satisfies that , for an integer . Thus, we have

Since , then, we obtain

This implies that . Based on the above it is clear that (i) (ii).

Second, assume that and are a polynomial of degree . For all and , by Lemma 1 and 2, where is bounded, we have

This shows that (ii) (iii). Thus, the proof has been completed.

Theorem 10. For, letbe harmonic and let, with orderand type. Then, the following are equivalent:(i) maps into (ii) and with or (with and );(iii) is bounded

Proof. First, assume that maps into . Now, we assume on the antithesis that (ii) does not hold. Then, the function with or (with and ). Thus, there is a positive constant and a sequence of complex numbers such that and

Hence, as in the proof of Theorem 8, we can consider the sequence and satisfies and . Now, let and set the function for all . Then, we have and . For large enough, , since , we obtain

Then, we conclude that cannot be bounded. Based on the above results, it is clear that (i) (ii).

Second, set , where , the order of is

If , then the type of is

For given , the condition (ii) implies that (see for example [18])

Moreover, for a function , with , we know that


Thus, we have

This shows that is bounded. So, (ii) (iii).

5. Conclusion and Future Study

This manuscript deals with a radical study of a concerned class of Toeplitz superposition operators acting between some certain classes of analytic function spaces of Bloch-type. Global discussions of the boundedness property of the new class of operators are presented class of the univalent Bloch functions. All concerned entire functions which transform a class of holomorphic Bloch-type spaces into another using the so-called Toeplitz superposition operators in terms of their order and type or the degree of polynomials are characterized in this paper. Moreover, all the defined Toeplitz-superposition operators induced by concerned entire functions are cleared to be bounded actually. We have cleared that for two spaces of normed-type which belonging to where and we can find certain concerned functions and , with and , for which the newly Toeplitz-superposition operators can map into for some specific values of and . Furthermore, the operator is shown to be actually bounded.

Data Availability

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

Conflicts of Interest

The authors declare that they have no competing interest.


The authors would like to thank Taif University Researchers supporting Project number (TURSP-2020/159), Taif University—Saudi Arabia. The authors would like to thank the anonymous concerned reviewers for their valuable remarks on this study.