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Journal of Operators
Volume 2013 (2013), Article ID 154029, 9 pages
Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space
1Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
2Department of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China
Received 25 February 2013; Accepted 19 April 2013
Academic Editor: Zeljko Cuckovic
Copyright © 2013 Flavia Colonna and Songxiao Li. 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.
We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.
Let be the open unit disk in the complex plane and the space of analytic functions on . A function is said to belong to the Bloch space if Under the norm defined by , is a conformally invariant Banach space. The functions in the Bloch space satisfy the following growth condition:
An important class of Möbius invariant spaces is given by the analytic Besov spaces , with , consisting of the functions such that where denotes the normalized area measure on the unit disk. The quantity is a seminorm and the Besov norm is defined by By Lemma 1.1 of , for , Thus, the space is continuously embedded in the Bloch space. Moreover, by Theorem 9 in , the functions in satisfy the growth condition
Let denote the set of all functions such that where the supremum is taken over all and . By Theorem 5.3 of  and the Closed Graph theorem, an analytic function on belongs to if and only if . Furthermore, The quantities in (8) are just seminorms for the space , as they do not distinguish between functions differing by a linear function. The norm
yields a Banach space structure on , which is called the Zygmund space. For more information on the Zygmund space on the unit disk, see, for example, .
Let denote the set of all analytic self-maps of . Each induces the composition operator on defined by
Let . The multiplication operator is defined as
for and . The composition product of and yields a linear operator on called the weighted composition operator with symbols and .
In recent years, considerable interest has emerged in the study of the weighted composition operators due to the important role they play in the study of the isometries on many Banach spaces of analytic functions, such as the Hardy space (for , ) [6, 7], the weighted Bergman space , and the disk algebra .
There is a very extensive literature on the composition operators and the weighted composition operators between the Bloch space and other spaces of analytic functions in one and several complex variables. The study on such operators between the Zygmund space and some other spaces of analytic functions is more recent and not yet well developed. The weighted composition operators and related operators from the Zygmund space to Bloch-type spaces have been investigated by the second author and Stević in [10, 11].
Research on the composition operators between various Banach spaces of analytic functions has shown that the compact composition operators acting on different Banach spaces are often the same. For example, in , it was shown that the composition operator mapping into the Bloch space is compact as an operator acting on if and only if it is compact as an operator acting on the analytic Besov space (with ), on the space , and on the Bloch space itself. Also, an interesting question that arises in operator theory is whether a countable set of functions on the range of the operator exists whose norm approaching 0 is sufficient to characterize the compactness of the operator. In , it was shown that the sequence of powers of the symbol has this property with regards to the composition operator on the Bloch space as well as the space . In fact, for , the operator is compact if and only if as . For the weighted composition operator , this is not the case, although a few cases are known in which the sequence of norms is sufficient to characterize boundedness and compactness (see [14, 15], see also  for analogous questions for the operators mapping into ). In all examples known to us, the space is small compared to the codomain of the operator. These results prompted us to investigate analogous questions when the operator maps into the Zygmund space.
In this work, besides giving characterizations of the bounded and the compact operator from the Besov and the Bloch spaces into in terms of function theoretic conditions on the symbols and , we provide boundedness and the compactness criteria for the operator , in terms of the norms in of functions in the range of the operator belonging to certain one-parameter families. For both boundedness and compactness, one of these characterizations involves the sequence .
Finally, we show that there are no nontrivial bounded multiplication operators from the Bloch space to the Zygmund space, and that a composition operator from or into is bounded (resp., compact) if and only if it is bounded (resp., compact) as an operator acting on the space of bounded analytic functions on under the supremum norm. One of the characterizing conditions for boundedness (resp., compactness, under the boundedness assumption of the operator) of such operators is that (resp., ) as . We found this phenomenon quite surprising due to the size of the spaces and in comparison to that of the Zygmund space.
We begin the section by characterizing the bounded weighted composition operators from to the Zygmund space. We note that since the constant function 1 belongs to all spaces under consideration, the boundedness of requires that . Thus, we will assume throughout that .
For a fixed and for , set
For an integer and , let .
Theorem 1. Let and . Then the following conditions are equivalent.(a)The operator is bounded.(b), and .(c), , and are finite. (d)The quantities , are finite.
Lemma 2. Let and . Then the following conditions are equivalent.(a). (b)The quantities , , , and are finite.(c)The quantities and are finite.
Proof. (a) ⇒ (b) Since the sequence is bounded in with supremum norms no greater than 1, if is bounded, then for each integer , we have
Therefore, the supremum of over all integers is finite. Likewise, . Therefore,
(b) ⇒ (c) follows at once from Lemma 2.
(c) ⇒ (d) Suppose that , , , , and are finite. By Lemma 2 we see that and are finite. For each nonnegative integer and each , a direct calculation shows that
Setting for , from (17) we obtain Subtracting (19) from (20), we get Hence, from (21) and (22), we deduce Therefore, taking the modulus and multiplying by , we obtain
Taking the supremum over all , we see that is finite.
(d) ⇒ (a) Suppose that (d) holds. As a consequence of Theorem of , for each , Thus, using (2) and (25), for an arbitrary in , we have In addition, again by (2), we have
The boundedness of follows from (27) and (26) after taking the supremum over all .
We now analyze the case of the weighted composition operator acting on the analytic Besov spaces for . The case has been established by the second author in . For and , define Then, as shown in , and is bounded by a constant independent of .
Theorem 3. Let , , and . Then the following conditions are equivalent.(a)The operator is bounded.(b), and .(c)The quantities , , , and in Theorem 1 and are finite.(d)The quantities and in Theorem 1 and are finite.
Proof. Note that, unlike the case of the Bloch space, the sequence is unbounded in . Therefore, first condition in (b) is not an immediate consequence of the boundedness of the operator.
(a) ⇒ (d) Suppose is bounded. Since , it follows that Moreover, for , and are in and their Besov norms are bounded by constants independent of . Indeed, noting that , with , and using the Möbius invariance of the Besov seminorm, we have Likewise, since , and observing that , again by the Möbius invariance of , we obtain
Therefore, by the boundedness of , the quantities and are finite.
Next observe that since and are in and is bounded, and are in , so arguing as in the proof of Theorem 1 in , we see that and are finite as well. By Lemma 2, it follows that and are finite. To prove that is finite, let and note that Recalling the notation , a straightforward calculation shows that Thus, by the boundedness of , we have
Taking the supremum over all , it follows that is finite, as desired.
(d) ⇒ (a) Assume (d) holds. As a consequence of Theorem 5.1.5 of  and (5), for each , Thus, using (6) and (36), for any , we have Moreover, again by (6) and (5), we have
Taking the supremum over all and using (38) and (37), we conclude that is a bounded operator.
The equivalence of (c) and (d) follows from (34) and Lemma 2. The equivalence of (b) and (c) is an immediate consequence of Lemma 2.
We begin the section with a useful compactness criterion which will be used to characterize the compact weighted composition operators from the Bloch space and the Besov spaces to the Zygmund space. Its proof is based on standard arguments similar to those outlined in Proposition 3.11 of .
Lemma 4. Let , , and . Then, the operator from (resp., ) into is compact if and only if it (resp., ) is bounded and , as , whenever is a bounded sequence in (resp., ) converging to zero uniformly on compact subsets of .
We will also make use the following result that follows from the characterization of the compact weighted composition operators from to given in Theorem 2 of .
Lemma 5. Let and be such that any of the equivalent conditions in Lemma 2 hold. Then the following conditions are equivalent.(a). (b).(c), and
In the next theorem we focus on the weighted composition operators acting on the Bloch space. We will use the one-parameter family introduced in (12).
Theorem 6. Let , and suppose that the operator is bounded. Then the following conditions are equivalent. (a)The operator is compact. (b). (c). (d) = , and
Proof. (a) ⇒ (b) Suppose is a compact operator. Since the sequence is bounded in and converges to 0 uniformly on compact subsets of , then by Lemma 4,
On the other hand, if is a sequence in such that as , then is a bounded sequence in converging to 0 uniformly on compact subsets of , so again by Lemma 4, it follows that
(b) ⇒ (c) follows from Lemma 4.
(c) ⇒ (d) Suppose that the limits in (c) are 0. Using the inequality (24), we get
as . The desired result now follows from the last formula and Lemma 4.
(d) ⇒ (a) Assume (d) holds. Then, in particular, Thus, recalling the notation , for any , there is a constant , such that
Let be a sequence in with and converging to uniformly on compact subsets of . In light of Lemma 4, it suffices to show that as . Using (45), for , and recalling (2) and (25), we have By Cauchy's estimate, if is a sequence which converges to zero on compact subset of , then so do the sequences and . Hence, for , we have Since and
as , from (46) and (47) we deduce that , as . This completes the proof.
We now turn our attention to the weighted composition operators mapping into the Zygmund space. We will use the one-parameter family defined in (28).
Theorem 7. Let , , and assume the operator is bounded. Then the following conditions are equivalent. (a)The operator is compact; (b); (c); (d) = , and
Proof. Assume is compact. Since is bounded in and convergent to 0 uniformly on compact subsets of , if is a sequence in such that as , by Lemma 4 it follows that as . Similarly, since the sequences and are bounded in and converge to 0 uniformly on compact subsets of , it follows that and converge to 0 as , proving (c). Therefore, by Lemma 5, it follows that as , so (b) holds as well.
On the other hand, by (34) and Lemma 5, for all , we have
which proves that (d) holds as well. Thus, to prove the equivalence of (a)–(d), it suffices to show that (d) implies (a). Since the proof is similar to the proof of (d) implies (a) in Theorem 6, we omit the details.
Observe that conditions (d) in Theorems 1 and 6 coincide with the corresponding conditions (d) in Theorems 3 and 7 by taking . We could indeed have adopted the notation in place of , as the Bloch space is widely viewed as a limit of as , which could have allowed us to unify parts of Theorems 1 and 3 (resp., Theorems 6 and 7) into a single statement taking . However, we did not do so because the family of functions is not suitable to characterize the bounded and the compact weighted composition operators from the Bloch space into the Zygmund space.
4. Multiplication and the Composition Operators
In this section we highlight the results concerning the boundedness and the compactness of the multiplication operator and the composition operator, which to the best of our knowledge have not appeared in the literature.
For the case of the multiplication operator, the finiteness of in part (d) of Theorems 1 and 3, combined with the continuous inclusions of and of into and Theorems 1 and 2 in , yields the following result.
Corollary 8. For and , the following statements are equivalent.(a) is bounded.(b) is compact.(c) is bounded.(d) is compact.(e) is bounded.(f) is compact.(g) is identically 0.
For the case of the composition operator, noting that implies that , from (23), we see that for any , is a linear combination of and . Furthermore, since for , the function maps into the right half-plane, the image of the function is contained in the horizontal strip . Thus, which does not depend on . Thus, the boundedness of and implies the boundedness of . Moreover, from (34), it follows that for , Therefore, if then, arguing as above, we see that is bounded by a constant independent of , so From these remarks, Corollary 2 of , and Theorem 1 of , Theorems 1 and 3 yield the following result.
Corollary 9. For and , the following statements are equivalent.(a) is bounded.(b).(c), and .(d) and .(e) is bounded.(f) is bounded.
Next note that, by the above remarks, as , Thus, the convergence to 0 of and as implies that and converge to as well. From Theorems 6 and 7, and from Corollary 3 of  and Theorem 2 of , we deduce the following result.
Corollary 10. For such that is bounded and for , the following statements are equivalent.(a) is compact.(b).(c).(d).(e) is compact.(f) is compact.
4.1. Concluding Remarks
In this section, we established that the boundedness (resp., compactness) of the multiplication and the composition operator from to is equivalent to the boundedness (resp., compactness) of the multiplication and composition operator from or to . It is evident that the same can be said for the weighted composition operator mapping between these spaces for any choice of composition symbol when the multiplicative symbol is linear, or for an arbitrary multiplication symbol in the Zygmund space if the range of the composition symbol is relatively compact. This leads to the question of whether this equivalence holds also for completely general weighted composition operators. As a consequence of [17, 19], the answer is affirmative for the case of the spaces and . We suspect this is false for the other Besov spaces or but have not been able to construct examples to support this claim.
The second author is supported by the Foundation for Scientific and Technological Innovation in Higher Education of Guangdong (no. 2012KJCX0096), National Natural Science Foundation of China (no. 11001107), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. LYM11117).
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