The Product-Type Operators from Hardy Spaces into th Weighted-Type Spaces

Abstract

The main goal of this paper is to investigate the boundedness and essential norm of a class of product-type operators from Hardy spaces into th weighted-type spaces. As a corollary, we obtain some equivalent conditions for compactness of such operators.

1. Introduction

Let denote the open unit disc of the complex plane and denotes the space of all holomorphic functions on . The space of bounded holomorphic functions on is denoted by ; it is a Banach space with the equipped norm

Let . A Hardy space consists of all such that

When , is a Banach space with the norm . If , is a nonlocally convex topological vector space and it is a complete metric space (see [1]).

Let and be a weight, continuous, and positive function on . The th weighted-type space , consists of all such that

It is a Banach space with the following norm

The little th weighted-type space is a closed subspace of such that for any

For more information about th weighted-type spaces, see [2–4]. Let . Then, (growth space), (Bloch-type space), and (Zygmund-type space). Also (weighted-type space), (weighted Bloch space), (weighted Zygmund space), and coincide with the logarithmic Bloch space .

Let such that ; the partial Bell polynomials are triangular where the sum is taken over all nonnegative integers such that

More information about Bell polynomials can be found in ([5], p 134).

Let , and be the set of all holomorphic self-map of . In [6], Stevi, Sharma and Krishan defined a new product-type operator as follows:

When , we obtain the Stevi-Sharma-type operator, and for , we get the generalized weighted composition operators . Product-type operators on some spaces of analytic functions on the unit disc have become a subject of increasing interest in the recent years. We refer the reader to [6–10] and the references therein.

Liu and Yu have considered boundedness and compactness of operator from Hardy spaces and into the logarithmic Bloch space in [11, 12]. Also, Zhang and Liu have found some characterizations for boundedness and compactness of operator from Hardy spaces into the weighted Zygmund space in [10]. Recently, the boundedness, compactness, and norm of operator are considered in [13].

Motivated by previous works, the results found in them will be generalized for operator . For this purpose in the second section of this paper, we give some characterizations for boundedness of operator where and . In the third section, some new estimates are obtained for the essential norm of such operators. As a corollary, some equivalent conditions are acquired for compactness of such operators.

Throughout this paper, if there exists a constant such that , we use the notation . The symbol means that .

2. Boundedness

In this section, some equivalent conditions are found for the boundedness of operator from into th weighted-type spaces. Firstly, we state some lemmas.

Lemma 1 (see [14], Propositions 8). Let . Then, for any and ,

Lemma 2 (see [15], Lemma 2.1). Let and . The sequence is bounded in and

From Lemma 1, Proposition 5.1.2 [16] and [1], the next lemma is obtained.

Lemma 3. Let , and . Then,

Lemma 4 (see [4]). Let and . Then, for any In this paper, we set

By using the functions , we obtain the following lemma. Since the proof of it resembles the proof of Lemma 1 [2], therefore, it is omitted.

Lemma 5. Let be Kronecker delta. For any , , and , there exists a function such that

In this case, , where are independent of choice .

Theorem 6. Let , , , be a weight and . Then, the following statements are equivalent (a)The operator is bounded(b)The operator is bounded(c)The operator is bounded(d)(e)For each and (f)For each ,

Proof. Since , we get .
It follows from Lemma 2.
For each and , So, Hence, . It is remained to show that for each , Applying the operator to , by using Lemma 4, we have Now, assume that we have the following inequalities for , where . By applying the operator for and using Lemma 4, we get Since , so from the triangle inequality, we have For any and , by using Lemmas 4 and 5, we obtain Therefore from the last inequality, On the other hand, from we have For any , by using Lemmas 1 and 4, we obtain Also for each , we have So, the operator is bounded.
From Lemma 3, , so we obtain .
It is clear that and . Hence, for each , The proof of the second part of is similar to the proof , so it is omitted. The proof is complete.☐

3. Essential Norm

In this section, we find some approximations for the essential norm of operator from Hardy spaces into nth weighted type-spaces. As a corollary, we give some equivalent conditions for compactness of such operators.

Let and be Banach spaces and be the continuous linear operator. The essential norm of is the distance from to the compact operators, that is,

It is clear that is compact if and only if .

Theorem 7. Let , , , and be a weight such that be bounded. Then where

Proof. For each , and uniformly on compact subsets of as . Applying Lemma 2.10 from [17], for any compact operator from into , we have Hence, for any So, Now, we prove that Let be a sequence in such that . Since is bounded, by using Lemmas 4 and 5 for any compact operator and , we obtain So, from the definition of the essential norm, we get (34).
For , we define . It is apparent that is a compact operator on . Let be a sequence such that as . Since uniformly on compact subsets of as , then, for any positive integer , the operator is compact. Based on the definition of the essential norm, we obtain So, it is sufficient to show that Let such that and for all , , therefore, For any and compact subset of , uniformly, hence, from Theorem 6, we obtain On the other hand Now, estimate for is obtained. Employing Lemmas 1 and 5, Taking the limit when , we get Likewise, we have Thus, by using (38), (39), (40), (42) and (43) we obtain Hence, from (36), Consequently, The proof is complete.☐

Theorem 8. Let , , , , be a weight. If be bounded then

Proof. It is evident that On the other hand, since , for any compact operator , from Lemma 2.10 in [17], for any , we get So, from the last inequality and Theorem 7The proof is complete.☐

Theorem 9. Let , , , be a weight and such that be bounded. Then,

Proof. Let be any positive integer and . It is clear that , , and converge to uniformly on compact subsets of . By using Lemma 2.10 in [17], for any compact operator from into , we get Hence, So, Now, we prove that From Theorem 6, for any fixed positive integer and , we have where Letting , we obtain Applying Theorem 7, we get It is clear that ; so, from the last inequalities, we have The proof is complete.☐

4. Some Applications

For , by using Lemma 3, we have . Also, for , and are a Banach space with the norm . In this case, we get the following corollary.

Corollary 10. Let , , and be a weight and . The operator is bounded if and only if the operator be bounded.

Corollary 11. Let , , and be a weight. If be bounded, then,

Proof. It is clear that and . So, for any compact operator , from Lemma 2.10 in [17], for any , we obtain Hence, from the last inequality and Theorem 7, The proof is complete.☐

From Theorems 7, 8 and 9 and Corollary 11, the next corollaries are obtained.

Corollary 12. Let , , , and be a weight such that be bounded. Then, the following statements are equivalent. (a)The operator is compact(b)The operator is compact(c)The operator is compact(d)(e)For each , (f)For each

Corollary 13. Let , , and be a weight such that be bounded. Then, the following statements are equivalent. (a)The operator is compact(b)The operator is compact(c)The operator is compact(d)The operator is compact(e)(f)For each , (g)For each

Remark 14. By putting in Theorems 6, 7, 8, and 9 and Corollaries 12 and 13, some characterizations are acquired for boundedness, essential norm, and compactness of the generalized weighted composition operator from Hardy spaces into th weighted-type spaces.
Since we obtain the next remark.

Remark 15. Let . Setting in Theorems 6, 7, 8, and 9 and Corollaries12 and 13 and using (64) we get similar results for operator (see [11, 12]).

Remark 16. Putting in Theorems 6, 7, 8, and 9 and Corollaries 12 and 13 and applying (65), similar results are achieved for operator (generalizing Theorems 7 and 9 [10]).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.