Product Type Operators Involving Radial Derivative Operator Acting between Some Analytic Function Spaces

: Let N denote the set of all positive integers and N 0 = N ∪ { 0 } . For m ∈ N , let B m = { z ∈ C m : | z | < 1 } be the open unit ball in the m − dimensional Euclidean space C m . Let H ( B m ) be the space of all analytic functions on B m . For an analytic self map ξ = ( ξ 1 , ξ 2 , . . . , ξ m ) on B m and φ 1 , φ 2 , φ 3 ∈ H ( B m ) , we have a product type operator T φ 1 , φ 2 , φ 3 , ξ which is basically a combination of three other operators namely composition operator C ξ , multiplication operator M φ and radial derivative operator R . We study the boundedness and compactness of this operator mapping from weighted Bergman–Orlicz space A Ψ σ into weighted type spaces H ∞ ω and H ∞ ω ,0 .

Denote the open unit ball in C m by B m = {z ∈ C m : |z| < 1}. Let H(B m ) be the space of all analytic functions on B m , S m the boundary of B m called as the sphere in C m . Let dV be the Lebesgue measure on B m and dσ the normalized measure on S m . For σ > −1, we write dV σ (z) = C σ (1 − |z| 2 ) σ dV(z), where C σ is such that V σ (B m ) = 1. For the result in settings of unit ball, refer to Ref. [1] and the references therein.
For 0 < p < ∞ and σ > −1, the weighted Bergman space A p σ (B m ) = A p σ , consists of all those functions f ∈ H(B m ) for which we have the following norm A non-zero function Ψ : [0, ∞) → [0, ∞) is said to be a growth function if it is continuous and non-decreasing. Clearly, every growth function fixes origin, that is Ψ(0) = 0. We say that the function Ψ is of positive upper type(respectively, negative lower type) for every s > 0 and t ≥ 1, if there exist C > 0 and q > 0 (respectively, q < 0) such that Ψ(st) ≤ Ct q Ψ(s). The class of all growth functions Ψ of positive upper type q, (for some q ≥ 1) for which the function t → Ψ(t)/t is non-decreasing on (0, ∞) is denoted by U q . Similarly, for every s > 0 and 0 < t ≤ 1, a function Ψ is said to be of positive lower type (respectively, negative upper type) if there are C > 0 and p > 0 (respectively, p < 0) such that Ψ(st) ≤ Ct p Ψ(s). The set of all growth functions Ψ of positive lower type r, (for some 0 < r ≤ 1) such that the function t → Ψ(t)/t is non-increasing on (0, ∞) is denoted by L r .
For a growth function Ψ, the weighted Bergman-Orlicz space A Ψ σ (B m ) = A Ψ σ is the class of all functions f in H(B m ) such that The quasi-norm on A Ψ σ is defined as follows: If Ψ ∈ U q or Ψ ∈ L r , then the quasi-norm on A Ψ σ is finite and called the Luxembourg norm. A quasi-norm on a linear space X is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by the quasi-triangle inequality, that is, x + y ≤ C( x + y ), for some C > 0 and x, y ∈ X. The smallest C for which quasi-triangle inequality holds will be called the quasi-norm constant of (X, · ). The Luxembourg space equipped with the Luxembourg quasi-norm is really a quasi-normed function space with the same quasi-triangle constant as the one of the quasi-norm. If Ψ(t) = t p , for p > 0, then we get the weighted Bergman space A To know more about these spaces one may refer [2,3] and the references therein.
A positive continuous function ω on B m is called as a weight. The weight ω is called to be a standard weight, if for z ∈ B m , we have ω(z) → 0 as |z| → 1. Further, for z ∈ B m , we call a weight ω to be radial, if ω(z) = ω(|z|). For a weight ω the weighted-type space The little weighted-type space H ∞ ω,0 (B m ) = H ∞ ω,0 is the subspace of the space H ∞ ω and contains all those f ∈ H(B m ) for which will be the classical weighted-type spaces H ∞ σ (respectively, classical little weighted-type spaces H ∞ σ,0 ). For ω ≡ 1, the space H ∞ ω get reduced to the the space H ∞ of bounded analytic function on B m . The weighted-type spaces have been studied by various authors see e.g., [4][5][6] and the references therein.
Let ξ = (ξ 1 , ξ 2 , . . . , ξ m ) be a holomorphic self-map of B m and φ ∈ H(B m ). Then, the composition, multiplication, differential and weighted composition operator on H(B m ) are respectively defined as More results on weighted composition operators on class of holomorphic functions can be found in [7,8] and the references therein. The product-type operators W φ,ξ D and DW φ,ξ were respectively, considered in [2] and [3]. To characterize the product-type operators in a unified way, new product-type operator T φ 1 ,φ 2 ,ξ was introduced which can be found in [9,10] and the references therein.
In this paper, we investigate the boundedness as well as the compactness of the operators T φ 1 ,φ 2 ,φ 3 ,ξ . This paper is represented in a systematic manner. Introduction and literature part is kept in Sections 1 and 2 consists of some auxiliary results which are used to derive the main results. In Section 3, we characterize the boundedness of operators T φ 1 ,φ 2 ,φ 3 ,ξ : A Ψ σ → H ∞ ω and T φ 1 ,φ 2 ,φ 3 ,ξ : A Ψ σ → H ∞ ω,0 . Finally, in Section 4, the compactness of operators T φ 1 ,φ 2 ,φ 3 ,ξ : A Ψ σ → H ∞ ω and T φ 1 ,φ 2 ,φ 3 ,ξ : A Ψ σ → H ∞ ω,0 is given. Throughout the paper, for any two positive quantities a and b, the notation a b means that a ≤ Cb, for some constant C > 0 . The constant C may differ at each occurrence. Further, if both a b and b a hold, then we simply write a b.

Auxiliary Results
To obtain the desired results, we have used the following auxiliary results: Lemma 1. Let Ψ ∈ L p ∪ U q and σ > −1. There is a constant C > 1 such that for any f ∈ A Ψ σ , .
For the proof of Lemmas 1 and 2, we refer to [26,27]. Lemma 3. [28] Let Ψ ∈ L p ∪ U q and σ > −1. Then, there are two positive constants C 1 and C 2 such that for any f ∈ A Ψ σ , σ which is bounded and on compact subsets of B m uniformly converges to zero as m → ∞, The compactness of T φ 1 ,φ 2 ,φ 3 ,ξ from a holomorphic space to H ∞ ω,0 can be obtained by using the following lemma which is similar to Lemma 1 in [21]. So the proof is omitted.
, when |z| ≤ r, we get that f ξ(z k ) converges to zero uniformly on compact subsets of B m . Thus, f k converges to zero uniformly on compact subsets of B m . Therefore, by Lemma 4, it follows that the sequence { f k } uniformly converges to zero on any compact subsets of B m as k → ∞ such that In addition, we have Thus, we obtain which implies that (28) holds. In order to prove condition (30), we define another sequence of functions where . Therefore, Similar to the sequence f k , the sequence g ξ(z k ) and hence the sequence g k converges to zero uniformly on compact subsets of B m , g k ∈ A Ψ σ and sup k∈N g k A Ψ σ ≤ C. By Lemma 4, we have In addition, we have Thus, By condition (36) from which condition (30) follows.
Proof. We omit the proof as it is easy to prove.
Proof. We omit the proof as it is easy to prove.

Discussion and Conclusions
In this paper, we have considered the product type operators formed by the combination of composition, multiplication, differentiation and radial derivative operators acting between weighted Bergman-Orlicz spaces and weighted type spaces taken over the unit ball. We analysed these operators for basic properties including boundedness and compactness. The basic aim of this paper is to give the operator-theoretic characterization of these operators in terms of function-theoretic characterization of their including functions.
Author Contributions: M.D., K.R. and M.M. All authors have contributed equally to the conceptualization, design and implementation of this research work. All authors have read and agreed to the published version of the manuscript.