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

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


Introduction
Let D denote the open unit disc of the complex plane ℂ and HðDÞ denotes the space of all holomorphic functions on D. The space of bounded holomorphic functions on D is denoted by H ∞ ; it is a Banach space with the equipped norm g k k H ∞ = sup z∈D g z ð Þ j j: ð1Þ Let 0 < p < ∞. A Hardy space H p consists of all g ∈ HðDÞ such that g k k H p = sup 0<r<1 1 2π When 1 ≤ p < ∞, H p is a Banach space with the norm k·k H p . If 0 < p < 1, H p is a nonlocally convex topological vector space and it is a complete metric space (see [1]).
where the sum is taken over all nonnegative integers j 1 , ⋯, j n−k+1 such that More information about Bell polynomials can be found in ( [5], p 134).
Let m ∈ ℕ 0 , u, v ∈ HðDÞ and φ ∈ SðDÞ be the set of all holomorphic self-map of D. In [6], Stevic′, Sharma and Krishan defined a new product-type operator T m u,v,φ as follows: When m = 0, we obtain the Stevic′-Sharma-type operator, and for v ≡ 0, we get the generalized weighted composition operators D m u,φ . 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][7][8][9][10] and the references therein.
Liu and Yu have considered boundedness and compactness of operator T 0 u,v,φ from Hardy spaces and H ∞ into the logarithmic Bloch space in [11,12]. Also, Zhang and Liu have found some characterizations for boundedness and compactness of operator T 0 u,v,φ from Hardy spaces into the weighted Zygmund space in [10]. Recently, the boundedness, compactness, and norm of operator T 0 u,v,φ : H p ⟶ W n μ are considered in [13].
Motivated by previous works, the results found in them will be generalized for operator T m u,v,φ . For this purpose in the second section of this paper, we give some characterizations for boundedness of operator T m u,v,φ : H p ⟶ W ðnÞ μ where m, n ∈ ℕ and 0 < p ≤ ∞. 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 c such that a ≥ cb, we use the notation a ⪰ b. The symbol a ≈ b means that a ⪰ b ⪰ a.

Boundedness
In this section, some equivalent conditions are found for the boundedness of operator T m u,v,φ ðm ∈ ℕÞ from H p ð0 < p≤∞Þ into nth weighted-type spaces. Firstly, we state some lemmas.

Lemma 5.
Let δ ik be Kronecker delta. For any 0 ≠ a ∈ D, m ∈ ℕ 0 , and i ∈ f0, ⋯, n + 1g, there exists a function g i,a ∈ H p such that In this case, g i,a ðzÞ = ∑ n+2 j=1 c i j f j,a ðzÞ, where c i j are independent of choice a. Theorem 6. Let m, n ∈ ℕ, 0 < p ≤ ∞, u, v ∈ HðDÞ, μ be a weight and φ ∈ SðDÞ. Then, the following statements are equivalent Abstract and Applied Analysis Proof. ðbÞ ⟹ ðcÞ Since B 1+ð1/pÞ 0 ⊂ B 1+ð1/pÞ , we get ðcÞ. ðcÞ ⟹ ðdÞ It follows from Lemma 2. ðdÞ ⟹ ðeÞ For each i ∈ f0, ⋯, n + 1g and a ∈ D, So, ð17Þ Now, assume that we have the following inequalities for where j ≤ n + 1. By applying the operator T m u,v,φ for p j+m ðzÞ and using Lemma 4, we get Since kφk ≤ 1, so from the triangle inequality, we have ðeÞ ⇒ ð f Þ For any φðaÞ ≠ 0 and i ∈ f0, ⋯, n + 1g, by using Lemmas 4 and 5, we obtain Therefore from the last inequality, On the other hand, from ðeÞ, we have ð f Þ ⟹ ðbÞ For any f ∈ B 1+1/p , by using Lemmas 1 and 4, we obtain Also for each 0 ≤ k < n, we have The proof of the second part of ðeÞ is similar to the proof ðdÞ ⟹ ðeÞ, so it is omitted. The proof is complete. ☐

Essential Norm
In this section, we find some approximations for the essential norm of operator T m u,v,φ from Hardy spaces into nth weighted type-spaces. As a corollary, we give some equivalent conditions for compactness of such operators.
Let X and Y be Banach spaces and T : X ⟶ Y be the continuous linear operator. The essential norm of T is the distance from T to the compact operators, that is, It is clear that T is compact if and only if kTk e,X⟶Y = 0.
Theorem 7. Let m, n ∈ ℕ, 0 < p ≤ ∞, u, v ∈ HðDÞ, φ ∈ SðDÞ, and μ be a weight such that T m u,v,φ : B 1+ð1/pÞ ⟶ W ðnÞ μ be bounded. Then where Proof. For each i ∈ f0, ⋯, n + 1g, sup a∈D k f i+1,a k B 1+ð1/pÞ < ∞ and f i+1,a ⟶ 0 uniformly on compact subsets of D as jaj ⟶ 1. Applying Lemma 2.10 from [17], for any compact operator K from B 1+ð1/pÞ into W ðnÞ μ , we have Hence, for any i ∈ f0, ⋯, n + 1g, So, Now, we prove that Let fz j g j∈ℕ be a sequence in D such that lim j⟶∞ jφðz j Þj ⟶ 1. Since T m u,v,φ : B 1+ð1/pÞ ⟶ W ðnÞ μ is bounded, by using Lemmas 4 and 5 for any compact operator K : B 1+ð1/pÞ ⟶ W ðnÞ μ and i ∈ f0, ⋯, n + 1g, we obtain So, from the definition of the essential norm, we get (34). For r ∈ ½0,1Þ, we define K r f ðzÞ = f r ðzÞ = f ðrzÞ. It is apparent that K r is a compact operator on B 1+ð1/pÞ . Let fr j g ⊂ ð0, 1Þ be a sequence such that r j ⟶ 1 as j ⟶ ∞. Since f r ⟶ f uniformly on compact subsets of D as r ⟶ 1, then, for any positive integer j, the operator T m u,v,φ K r j : B 1+ð1/pÞ ⟶ W ðnÞ μ is compact. Based on the definition of the essential norm, we obtain So, it is sufficient to show that Let f ∈ B 1+ð1/pÞ such that k f k B 1+ð1/pÞ ≤ 1 and for all j ≥ N, r j ≥ ð3/4Þ, therefore, 4 Abstract and Applied Analysis For any s ∈ ℕ 0 and compact subset of D, ð f − f r j Þ ðsÞ ⟶ 0 uniformly, hence, from Theorem 6, we obtain On the other hand Now, estimate for M k is obtained. Employing Lemmas 1 and 5, Taking the limit when N ⟶ ∞, we get Likewise, we have Thus, by using (38), (39), (40), (42) and (43), we obtain Hence, from (36), Consequently, The proof is complete. ☐ So, from the last inequality and Theorem 7 The proof is complete. ☐ From Theorem 6, for any fixed positive integer k ≥ m and where Q = sup j≥m j ð1/pÞ kT m u,v,φ p j k W ðnÞ μ : The proof is complete. ☐

Some Applications
For 0 < p < ∞, by using Lemma 3, we have H p ⊂ B 1+ð1/pÞ 0 . Also, for p = ∞, H ∞ ⊈ B 0 and H ∞ ∩ B 0 are a Banach space with the norm k·k H ∞ . In this case, we get the following corollary.