Generalized Composition Operators on Zygmund-Orlicz Type Spaces and Bloch-Orlicz Type Spaces

B spaces appear in the literature of a natural way when one studies the properties of some operators of holomorphic functions in a certain space. For instance, Attele in [3] proves that the Hankel operator induced by a function f in the Bergman space is bounded if and only iff ∈B1 (where μ1 = w(z) log(z/w(z)), with z ∈ D).The spaceB1 is known as the log-Bloch space or the weighted Bloch space.The logarithmic Bloch-type space is introduced by Krantz and Stević [4, 5], where some properties of this space are studied. In the last decade, there was a big interest in the investigation of Blochtype spaces and various concrete linear operatorsL : X → Y, where at least one of the spaces X and Y is Bloch space. For some other recent results in the areas, see, for example, [4, 6– 10] and a lot of references therein. Recently, Fernándz in [8] uses Young’s functions to define the Bloch-Orlicz space as a generalization of Bloch space. More precisely, let φ : [0,∞) → [0,∞) be an N-function; that is, φ is a strictly increasing convex function such that φ(0) = 0 and limt→∞(t/φ(t)) = limt→0(φ(t)/t) = 0. The Bloch-Orlicz space associated with the function φ, denoted byB, is the class of all analytic functions f ∈ D such that


Introduction
Let D be a unit disk in complex plane C, and let (D) be the space of all holomorphic functions on D with the topology of uniform convergence on compact subsets of D. The Bloch space, B, consists of all functions  ∈ (D) for which          := sup ∈D (1 − || 2 )        ()      < ∞.
Let  > 0, and the -Bloch space, denoted by B  , consists of all holomorphic functions  on D such that          := sup -Bloch space is introduced and studied by numerous authors.The general theory of -Bloch function spaces is referred to in [2].Recently, many authors studied different class of Bloch-type spaces, where the typical weight function, () = 1 − || B  spaces appear in the literature of a natural way when one studies the properties of some operators of holomorphic functions in a certain space.For instance, Attele in [3] proves that the Hankel operator induced by a function  in the Bergman space is bounded if and only if  ∈ B  1 (where  1 = () log(/()), with  ∈ D).The space B  1 is known as the log-Bloch space or the weighted Bloch space.The logarithmic Bloch-type space is introduced by Krantz and Stević [4,5], where some properties of this space are studied.In the last decade, there was a big interest in the investigation of Blochtype spaces and various concrete linear operators  :  → , where at least one of the spaces  and  is Bloch space.For some other recent results in the areas, see, for example, [4,[6][7][8][9][10] and a lot of references therein.Recently, Fernándz in [8] uses Young's functions to define the Bloch-Orlicz space as a generalization of Bloch space.More precisely, let  : [0, ∞) → [0, ∞) be an N-function; that is,  is a strictly increasing convex function such that (0) = 0 and lim  → ∞ (/()) = lim  → 0 (()/) = 0.The Bloch-Orlicz space associated with the function , denoted by B  , is the class of all analytic functions  ∈ D such that sup for some  > 0 depending on .Without loss of generality, we can suppose that  −1 is continuous and differentiable.In fact, if  −1 is not differentiable everywhere, we set the function then  is differentiable, whence  −1 is differentiable everywhere on [0, ∞).Furthermore,  is a strictly increasing and convex function satisfying (0) = 0; then the function ()/,  > 0, is increasing and for all  ≥ 0. Hence B  = B  .By the convexity of , it is not difficult to see that the Minkowski's functional defines a seminorm for B  , which in this case is known as Luxemburg's seminorm, where In fact, it can be shown that B  is a Banach space with the norm We observe that, for any  ∈ B  \ {0}, the following relation holds.
The inequality above allows us to obtain for all  ∈ B  and for all  ∈ D. This last inequality implies that the evaluation function is defined as   () := (), where  ∈ D is fixed and  ∈ B  is continuous on B  .In fact, let  ∈ D be fixed and any  ∈ B  ; we have From the definition of Luxemburg seminorm and the expression (11), we have for any  ∈ B  .
As an easy consequence of (11), we have that the Bloch-Orlicz space is isometrically equal to -Bloch space when with  ∈ D. Thus, for any  ∈ B  , we have Denote by Z the class of all  ∈ (D) ∩ (D) such that where the supremum is taken over all   ∈ D and ℎ > 0.
From the theorem of Zygmund (see [11,  From (18) it is easy to obtain For some other information and operators on this space, see, for example, [6,12,13].
Inspired by the way Bloch-Orlicz spaces were defined (see [8,14]), we define the Zygmund-Orlicz space, which is denoted by Z  , as the class of all analytic functions  in D such that sup for some  > 0 depending on .
The following useful lemmas are easily obtained.
Lemma 1 (see [15]).Proof.In the same way as the case of Bloch-Orlicz space, so the details are omitted here.
Lemma 2 allows us to obtain that for all  ∈ Z  and all  ∈ D.
From the definition of Luxemburg seminorm and expression (25), we have Also, as an easy consequence of (25), we have that Zygmund-Orlicz space is isometrically equal to -Zygmund space, where For more information about Z  , see [7,10,16].Specially, if  is an N-function such that is bounded for all  ∈ D, then we get Z  ⊂  ∞ , the space of all bounded analytic functions on D. However, there exists N-functions for which   () is not a bounded function; for instance, consider () =  log(1 + ) with  ≥ 0.
Let  be an analytic self-map of D; then the composition operator on (D) is given by Composition operators acting on various spaces of analytic functions have been the object for recent years.In particular, the problems of relating operator-theoretic properties of   to function-theoretic properties of  are interesting and have been widely discussed.See the book of Cowen and MacCluer [17] and Shapiro [9] for discussions of composition operators classical spaces of analytic functions.Assume that  : D → C is a holomorphic map of the disk D, for  ∈ (D); we define a linear operator as follows: The operator    is called the generalized composition operator, when  =   .We see that this operator is essentially composition operator, since the difference    −  is constant.Therefore,    is a generalization of the composition operator, which was introduced in [6].
Recall that if  and  are Banach spaces and  :  →  is a linear operator, then  is said to be compact if, for every bounded sequence {  } in , the sequence ({  }) has a convergent subsequence.The book [17] contains plenty of information on this topic.By the standard arguments (see, e.g., Proposition 3.11 in [17]), the following lemma follows.Some characterization of the boundedness and compactness of the composition operator, as well as Volterra type operator, on Bloch-Orlicz-type space and Zygmund space can be found in [2,[18][19][20][21][22].In [6], the boundedness and compactness of the generalized composition operator on Zygmund space and Bloch-type spaces are characterized by Li and Stević.In this paper, we are devoted to investigating the boundedness and compactness of generalized composition operators between Zygmound-Orlicz type spaces and Bloch-Orlicz type spaces.The paper is organized as follows.In Section 2 we give the necessary and sufficient conditions for the boundedness and compactness of the operator    : Z → B  .In Section 3 we obtain the necessary and sufficient conditions for the boundedness and compactness of the operator    on Zygmund type spaces.Throughout this paper, we use the letter  to denote a generic positive constant that can change its value at each occurrence.The notation  ≤  means that there is a positive constant  such that  ≤ .If  ≤  and  ≤  hold, then one says that  ≍ .

The Boundedness and Compactness of 𝐶
where we use Lemma Let and put for any  ∈ D such that 1/ √ 2 < || < 1.Then we have It follows that That is, (42) So we have for all  ∈ D. In particular, for  = (), we have This concludes the proof of the theorem. where We have Set for  ∈ D, such that || > 1/2.Then, Similar to the case of   , we have ℎ  ∈ Z and From this and by the facts that Since  is an arbitrary positive number it follows that the last limit is equal to zero.Employing Lemma 3 the implication follows.

Theorem 4 .
:Z → B  Now, we are ready to state and prove the main results in this section.Let  : [0, ∞) → [0, ∞) be an N-function,  ∈ (D), and  an analytic self-map of D. Then    : Z → B  is bounded if and only if Theroem 5.3]) and the closed graph theorem, we see that  ∈ Z if and only if sup ∈D (1 − || 2 )|  ()| < ∞.It is easy to see that Z is a Banach space under the norm ‖ ⋅ ‖ Z , where for any  ∈ D. But, on the other hand,