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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 817278, 5 pages
A New Essential Norm Estimate of Composition Operators from Weighted Bloch Space into -Bloch Spaces
1Departamento de Matemáticas, Universidad Nacional de Colombia, AP 360354, Bogotá, Colombia
2Departamento de Matemáticas, Universidad de Oriente, Cumaná 6101, Estado de Sucre, Venezuela
3Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia
Received 15 May 2013; Accepted 27 August 2013
Academic Editor: Alfonso Montes-Rodriguez
Copyright © 2013 René E. Castillo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be any weight function defined on the unit disk and let be an analytic self-map of . In the present paper, we show that the essential norm of composition operator mapping from the weighted Bloch space to -Bloch space is comparable to where for , is a certain special function in the weighted Bloch space. As a consequence of our estimate, we extend the results about the compactness of composition operators due to Tjani (2003).
The weighted Bloch space appears in the literature when we study properties of certain operators acting on certain spaces of analytic functions on the unit disk of the complex plane . For instance, in 1991, Brown and Shields  showed that an analytic function is a multiplier on the Bloch space if and only if Also, in 1992, Attele  showed that Hankel operator induced by a function in the Bergman space (for the definition of Bergman space and the Hankel operator, see ) is bounded if and only if , where is the space of all holomorphic functions on with the topology of uniform convergence on compact subsets of . The set of all functions such that (1) holds is denoted by and it is known that is a Banach space with the norm .
In the last decade, many authors have studied different classes of Bloch-type spaces, where the weight function, , (), is replaced by a bounded, continuous, and positive function defined on . More precisely, a function is called a -Bloch function, denoted as , if If , we get the classical Bloch space (see ), while if with , is just the -Bloch space (see ). It is readily seen that is a Banach space with the norm .
A holomorphic function from the unit disk into itself induces a linear operator , defined by , where . is the so-called composition operator with symbol Composition operators continue to be widely studied on many subspaces of and particularly in Bloch-type spaces.
The study of the properties of composition operators on Bloch-type spaces began with the celebrated work of Madigan and Matheson in , where they characterized the continuity and compactness of composition operators acting on the Bloch space . Many extensions of Madigan and Matheson’s results have appeared (see, e.g.,  and a lot of references therein). An important reference on this subject is the work of Montes-Rodríguez in  (see also ); here the author supposes that weight is radial ( for all ), typical , and decreasing function of defined on . In particular, Yoneda in  has extended the results by Madigan and Matheson in  to composition operators acting between weighted Bloch spaces. The weight considered by Yoneda in  was which is not decreasing on .
Recently, many authors have found new criteria for the continuity and compactness of composition operators acting on Bloch-type spaces in terms of the th power of the symbol and the norm of the th power of the identity function on . The first result of this kind appears in 2009 and it is due to Wulan et al. ; in turn, their result was extended to -Bloch spaces by Zhao in . Another criterion for the continuity and compactness of composition operators on Bloch space is due to Tjani in  (see also  or more recently ); she showed the following result.
Theorem 1 (see ). The composition operator is compact on if and only if and where is a Möbius transformation from the unit disk onto itself; that is, , with .
This last result has been recently extended to -Bloch spaces by Malavé-Ramírez and Ramos-Fernández in .
The essential norm of a continuous linear operator between normed linear spaces and is its distance from the compact operators; that is, , where denotes the operator norm. Notice that if and only if is compact, so that estimates on lead to conditions for to be compact. The essential norm of a composition operator on was calculated by Montes-Rodríguez in . He obtained similar results for essential norms of weighted composition operators between weighted Banach spaces of analytic functions in . Recently, many extensions of the above results have appeared in the literature; for instance, Zhao in  gave a formula for the essential norm of in terms of an expression involving norms of powers of . More precisely, he showed that It follows from the discussion at the beginning of this paragraph that is compact if and only if Zhao’s results in  have been extended recently to the weighted Bloch spaces by Castillo et al. in . Also, Hyvärinen et al. in  obtained necessary and sufficient conditions for boundedness and an expression characterizing the essential norm of a weighted composition operator between general weighted Bloch spaces , under the technical requirements that is radial, and that it is nonincreasing and tends to zero toward the boundary of .
The goal of the present paper is to give a new estimate of the essential norm of composition mapping from to . More precisely, in the next section we will show the following result.
Theorem 2. Let be an analytic self-map of the unit disk . Then for the essential norm of the composition operator , one has
The relation (6) means that there is a positive constant such that and the functions with will be defined at the beginning of the next section. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in  (see Theorem 1) and Malavé-Ramírez and Ramos-Fernández in .
2. A Class of Special Functions for
For convenience, throughout this paper we consider the following weight: in place of the one used by Brown and Shields in (1). It is not hard to see that with this new weight we have the same space , and the seminorm is comparable to defined in (1). Moreover, this weight satisfies the following property which will be of main relevance in the proof of our result.
Lemma 3. For all and all , one has
Proof. The result follows from the fact that the function with is concave and satisfies as .
The key to our results lies in considering the following family of functions: for fixed, we define where for , the function is defined by We have the following result.
Lemma 4. The family satisfies the following properties. (1) converges to zero uniformly on compact subsets of as .(2)There exists a constant such that for all such that .(3)There exists a constant such that
Proof. The property (1) is clear since the function is bounded on compact subsets of .
(2) If , then and the property follows since
(3) Indeed, for , we have since, for each , we have . Note that the last expression is uniformly bounded if , and thus it will be enough to consider the case when . Furthermore, since the function with is increasing, concave and satisfy as , then we can use the fact that and conclude that Hence Also, it is clear that Finally, since then we can deduce that there exists a constant such Now the proof is complete.
3. A Criterion for the Compactness of Operators Acting on Spaces of Analytic Functions
We will need to prove three auxiliary facts. The first of which is the following lemma that appears in  and also in . In both of these places, we point out a typographical error (“point evaluation functionals on " there, as one can see from Relation (16) in , should be “point evaluations on ").
Lemma 5 (Tjani). Let , be two Banach spaces of analytic functions on . Suppose that (1)point evaluation functionals on are bounded; (2)the closed unit ball of is a compact subset of in the topology of uniform convergence on compact sets; (3) is continuous when and are equipped with the topology of uniform convergence on compact sets.
Then is a compact operator if and only if given a bounded sequence in such that uniformly on compact sets, in .
The following facts about -Bloch space are well known. (i)For each compact , there is a such that for all and , we have that (ii)Every point evaluation functional on is bounded.(iii)The closed unit ball of is compact in the topology of uniform convergence on compact subsets of .(iv)If is a weight on and is holomorphic, with , then is continuous with respect to the compact-open subspace topologies on and .
Thus, combining the above and Lemma 5, we obtain the following auxiliary result.
Lemma 6. Let and be weights on and suppose that is holomorphic. Then is compact if and only if given a bounded sequence in such that uniformly on compact subsets of , then as .
As a consequence of Lemma 5, we can see that the dilatation operator is compact on . Recall that for , the linear dilation operator is defined by , where , for each , is given by . It is clear that if then uniformly on compact subsets of as .
Lemma 7. The following statements hold: (1)for , the operator is compact on ;(2)for each
Proof. If is a bounded sequence in such that uniformly on compact subsets of , then
and the item (1) follows by Lemma 5.
On the other hand, if then where we have used Lemma 3. This ends the proof of item (2).
4. Proof of Theorem 2
Now we can show Theorem 2. We set Let be any compact operator, fixed and define where is the constant in the item (3) of Lemma 4. Then goes to zero uniformly on compact subsets of as , for all , and Hence, taking and using Lemma 5, we obtain
Now, we go to show that there exists a constant such that
Hence, if we consider a sequence such that as and we consider the dilatation operators , then by Lemma 7, for all , the operator is a compact from into , and by definition of the essential norm, we have Thus, we have to show that there exists a constant such that To see this, consider any such that ; then since and as , it is enough to show that there exists a constant such that Furthermore, since uniformly on compact subsets of as , we have where is large enough such that for all . Hence we only have to show that there exists a constant such that Indeed, we write , where Then we have where we have used the item (2) of Lemma 4 in the first inequality and the fact that in the second one. Taking limit as , we obtain In a similar way, we have where we have used Lemma 3 in the last inequality. Therefore This completes the proof of the theorem.
Corollary 8. The composition operator is compact from into if and only if and
We want to finish this paper with the following question: Is the above corollary true for the composition operator from into , where and are weights defined on ?
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