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

Abstract

The boundedness and compactness of generalized composition operators on Zygmund-Orlicz type spaces and Bloch-Orlicz type spaces are established in this paper.

1. Introduction

Let be a unit disk in complex plane , and let be the space of all holomorphic functions on with the topology of uniform convergence on compact subsets of . The Bloch space, , consists of all functions for which becomes a Banach space when it is equipped with the norm (see, e.g., [1]).

Let , and the -Bloch space, denoted by , consists of all holomorphic functions on such that -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, , is replaced by a bounded continuous positive function defined on . More precisely, a function is called a -Bloch function, denoted by , if Clearly, if with , is just the -Bloch space . It is readily seen that is a Banach space with the norm 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 (where , with ). The space 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 Bloch-type 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, 6–10] 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 be an -function; that is, is a strictly increasing convex function such that and . The Bloch-Orlicz space associated with the function , denoted by , is the class of all analytic functions such that for some depending on . Without loss of generality, we can suppose that is continuous and differentiable. In fact, if is not differentiable everywhere, we set the function then is differentiable, whence is differentiable everywhere on . Furthermore, is a strictly increasing and convex function satisfying ; then the function , , is increasing and for all . Hence . By the convexity of , it is not difficult to see that the Minkowski’s functional defines a seminorm for , which in this case is known as Luxemburg’s seminorm, where In fact, it can be shown that is a Banach space with the norm We observe that, for any , the following relation holds.

The inequality above allows us to obtain for all and for all . This last inequality implies that the evaluation function is defined as , where is fixed and is continuous on . In fact, let be fixed and any ; we have From the definition of Luxemburg seminorm and the expression (11), we have for any .

As an easy consequence of (11), we have that the Bloch-Orlicz space is isometrically equal to -Bloch space when with . Thus, for any , we have

Denote by the class of all such that where the supremum is taken over all and . From the theorem of Zygmund (see [11, Theroem 5.3]) and the closed graph theorem, we see that if and only if . It is easy to see that is a Banach space under the norm , where 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 , as the class of all analytic functions in such that for some depending on . Same as the Bloch-Orlicz space, since is convex, it is not difficult to see that the Minkowski functional defines a seminorm for . Furthermore, it can be shown that is a Banach space with the norm The following useful lemmas are easily obtained.

Lemma 1 (see [15]). If , then

Also we can observe the following property.

Lemma 2. For any , the following relation holds.

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 and all .

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 with . For more information about , see [7, 10, 16].

Specially, if is an -function such that is bounded for all , then we get , the space of all bounded analytic functions on . However, there exists -functions for which is not a bounded function; for instance, consider with .

Let be an analytic self-map of ; then the composition operator on 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 is a holomorphic map of the disk , for ; 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.

Lemma 3. Let be an -function, , and an analytic self-map of . Let and . Then is compact if and only if is bounded and, for any bounded sequence in which converges to zero uniformly on as , one has as .

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, 18–22]. 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 . 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 .

2. The Boundedness and Compactness of

Now, we are ready to state and prove the main results in this section.

Theorem 4. Let be an -function, , and an analytic self-map of  . Then is bounded if and only if

Proof. Suppose that (31) holds. For arbitrary and , by Lemma 1, we have the following estimate: where we use Lemma 1 (relation (ii)) in the last inequality. From here, we can conclude that . Since , it follows that the generalized composition operator is bounded.
Now, suppose that there exists a constant such that Let and put for any such that . Then we have which implies that for and . Therefore we have It follows that That is, for any . But, on the other hand, and then So we have for all . In particular, for , we have This concludes the proof of the theorem.

Theorem 5. Let be an -function, , and an analytic self-map of . Then is compact if and only if is bounded and

Proof. First assume that is bounded and (45) holds. By the boundedness of with , we see that Let be a sequence in such that and uniformly on compact subsets of as . Then, by Lemma 3, it suffices to show that as . By (45), we have that, for every , there is a constant , such that , which implies for any . From here, we have that where .
On the other hand, let ; by the cauchy estimate, if is a sequence converging to zero on compact subsets of , then the sequence also converges to zero on compact subsets of as . In particular, since is compact, it follows that . Using these facts and letting , we obtain Since as , for given , there exists an , such that whenever , which means that is a compact operator.
To prove the converse, suppose that there exists an such that for any . Given a sequence of real number such that as , we can find a sequence such that and where , and, if necessary, we may suppose that as .
Note that , defined by (35), converges to zero uniformly on compact subset of as and Now, we choose functions defined by From the proof of Theorem 4, we see that Moreover, we can see that converges to zero uniformly on compact subsets of and satisfies Therefore, is not a compact operator. This completes the proof of the theorem.

3. The Boundedness and Compactness of

In this section, we characterize the boundedness and compactness of the operator on Zygmund-Orlicz type spaces.

Theorem 6. Let be an -function, , and an analytic self-map of . Then is bounded if and only if

Proof. Suppose first that (57) holds. For each , we have the following estimate: where and are constants, such that . Here we use relations (26) and (24) in the last seconded inequality. Now, we can conclude that and the generalized composition operator is bounded.
Now assume that is bounded; that is, there exists a constant such that for all . Taking the functions and , respectively, we obtain which implies Then So we obtain Using these facts and the boundedness of the function we get Let and put for such that . We have Then So we obtain Set for , such that . Then, Similar to the case of , we have and . From this and by the facts that and , it follows that From (71) we have By (65) we see that From (72) and (73) we obtain the first inequality of (57). Similarly, from (61) and (68) the second inequality in (57) follows as desired.

Theorem 7. Let be an -function, , and an analytic self-map of . Then is compact if and only if is bounded and

Proof. Suppose that is compact. It is clear that is bounded. Let be a sequence in such that as and defined by Then , and uniformly on compact subsets of as . Since is compact, it gives . Note that We have and, consequently, If one of these two limits exists, set then , , and converges to 0 uniformly on compact subsets of as . Since is compact, we have
On the other hand, Hence Therefore This together with (78) implies The implication follows from the last two equalities.
Conversely, assume that is bounded and (74) holds. From the proof of Theorem 6 we have On the other hand, from (74), we have that, for every , there is a , such that when .
Assume that is a sequence in such that and converges to 0 uniformly on compact subsets of as . Let . Then by (85) and (86) it follows that So we obtain By the Cauchy estimate, if is a sequence converging to zero on compact subsets of , then the sequence and also converges to zero on compact subsets of as . In particular, since is compact it follows that and . Using these facts and letting in the last inequality, we obtain that Since is an arbitrary positive number it follows that the last limit is equal to zero. Employing Lemma 3 the implication follows.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Shengjian Wu for helpful conversations. Part of the work was done during the first author stay at Peking University, in the spring of 2014. The authors also thank the referee for some very valuable suggestions and comments that significantly improved the presentation of the paper. The work is supported in part by CNSF (Grant no. 11101099, Grant no. 11161007, and Grant no. 11171080), Guizhou Foundation for Science and Technology (Grant no. (2012) 2273 and no. (2014) 2044), Guizhou technology foundation for selected overseas scholars, and Ph.D. research foundation of Guizhou Normal University.