Research Article Weighted Composition Operators on the Zygmund Space

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1 Abstract and Applied Analysis Volume 0, Article ID 4648, 8 pages doi:0.55/0/4648 Research Article Weighted Composition Operators on the Zygmund Space Shanli Ye and Qingxiao Hu Department of Mathematics, Fujian Normal University, Fuzhou , China Correspondence should be addressed to Shanli Ye, ye shanli@yahoo.com.cn Received 5 February 0; Accepted 9 April 0 Academic Editor: Ljubisa Kocinac Copyright q 0 S. Ye and Q. Hu. 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 characterize the boundedness and compactness of the weighted composition operator on the Zygmund space Z {f HD : z f z < } and the little Zygmund space Z 0.. Introduction Let D {z : z < } be the open unit disk in the complex plane C, lett {z : z } be its boundary, and let HD denote the set of all analytic functions by D. For f HD,let fz { z ) f z } : z D.. An analytic function f HD is said to belong to the Zygmund space Z if f Z <, and the little Zygmund space Z 0 consists of all f Zsatisfying lim z z f z 0. From a theorem of Zygmund see, vol. I, page 63 or, Theorem 5.3,weseethatf Zif and only if f is continuous in the close unit disk D {z : z } and the boundary function fe iθ such that f e iθh ) f e iθ h) f e iθ) h <,. for all e iθ T and all h>0. It can easily proved that Z is a Banach space under the norm: f f0 f 0 f Z.3

2 Abstract and Applied Analysis and that Z 0 is a closed subspace of Z. It is easily obtained that f z f 0 f z Z for f Z,.4 z f z lim 0 for f Z 0..5 z / z For some other information on this space and some operators on it, see, for example, 3 5. An analytic self-map ϕ : D D induces the composition operator C ϕ on HD, defined by C ϕ f fϕz for f analytic on D. It is a well-known consequence of Littlewood s subordination principle that the composition operator C ϕ is bounded on the classical Hardy, Bergman, and Bloch spaces see, e.g., 6 9. Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. It is interesting to provide a function theoretic characterization of when ϕ induces a bounded or compact composition operator on various spaces. The book 0 contains plenty of information on this topic. Let u be a fixed analytic function on the open unit disk. Define a linear operator uc ϕ on the space of analytic functions on D, called a weighted composition operator, by uc ϕ f u f ϕ, where f is an analytic function on D. We can regard this operator as a generalization of a multiplication operator and a composition operator. In recent years, the weighted composition operator has been received much attention and appears in various settings in the literature. For example, it is known that isometries of many analytic function spaces are weighted composition operators e.g., see. The boundedness and compactness of it has been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type spaces, see, for example, 6. Also, it has been studied from one Banach space of analytic functions to another, one may see in 7 6. The purpose of this paper is to consider the weighted composition operators on the Zygmund space Z and the little Zygmund space Z 0. Our main goal is to characterize boundedness and compactness of the operators uc ϕ on Z in terms of function theoretic properties of the symbols u and ϕ. We also characterize boundedness and compactness of uc ϕ on Z 0. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other.. Auxiliary Results In order to prove the main results of this paper, we need some auxiliary results. Lemma.. If f Z,then i fz f for every z D; ii f z e/ z f for every z D.

3 Abstract and Applied Analysis 3 Proof. Suppose f Z, z D and 0 <t<, then f zt f 0 f zt Z zt,. by.4. It follows that fz f0 z f ztdt 0 z f 0 f 0 Z ) zt dt zt z f 0 f Z z z f 0 z f Z, 0 s s ds. hence fz f0 f 0 f Z f..3 One may easily prove ii by.4. The details are omitted here. Lemma.. Suppose f Z,then f t f, 0 <t<, wheref t z ftz. One may easily obtain it by a calculation. Lemma.3. Suppose uc ϕ : Z 0 operator. Z 0 is a bounded operator. Then uc ϕ : Z Z is a bounded Proof. Suppose uc ϕ is bounded in Z 0. It is clear that for any f Z, we have f t Z 0 for every 0 <t<. According to Lemma., weobtainthat ucϕ ft ) ucϕ ft ucϕ f <..4 Then ) ) ) ucϕ f lim ucϕ ft ucϕ ft ucϕ f <. t 0<t<.5 Hence, uc ϕ : Z Zis a bounded operator. 3. Boundedness of uc ϕ In this section, we characterize bounded weighted composition operators on the Zygmund space Z and the little Zygmund space Z 0.

4 4 Abstract and Applied Analysis Theorem 3.. Let u be an analytic function on the unit disc D and ϕ an analytic self-map of D.Then uc ϕ is bounded on the Zygmund space Z if and only if u Zand the following are satisfied: z ) uz ϕ z ) ϕz < ; 3. z ) ϕ zu z ϕ zuz <. ϕz 3. Proof. Suppose uc ϕ is bounded on the Zygmund space Z. Then we can easily obtain the following results by taking fz andfz z in Z, respectively: u Z; 3.3 z ) ϕ zu z ϕ zuz ϕzu z <. 3.4 By 3.3, 3.4, and the boundedness of the function ϕz, weget K z ) ϕ zu z ϕ zuz <. 3.5 Let fz z in Z again, in the same way we have z ) 4ϕzϕ zu z ϕ zu z uz ϕzϕ z ϕ z ) ) <. 3.6 Using these facts and the boundedness of the function ϕz again, we get K z ) ϕ z ) uz <. 3.7 Fix a D with a > /, we take the test functions: f a z haz a a ) z 0 aω dω, 3.8 for z D, where hz z ) ). 3.9 z

5 Abstract and Applied Analysis 5 Then we have ) f az ) az a az, f az a ) az a az a az, 3.0 and /< a < f a C by 3, where C is not dependent on a. Therefore, for all λ D with ϕλ > /, we have C fa ucϕ f a z ) ) z ucϕ f a z ) ϕ zu z ϕ zuz ) f ) a ϕz 3. f a ϕz ) ϕ z ) uz u zf a ϕz ). Let a ϕλ, it follows that C f a λ ) ϕ λu λ ϕ λuλ ) f ) ϕλ ϕλ f ϕλ ) ϕλ ϕ λ ) uλ u ) λf ϕλ ϕλ λ ) ϕ λ ) ϕλ uλ ) ϕλ u λf ϕλ ϕλ λ ) ϕ λ ) ϕλ uλ ϕλ λ ) u ) λf ϕλ ϕλ. 3. Then, by Lemma. and 3.3, we have λ ) ϕ λ ) ϕλ uλ ϕλ λ ) u ) λf ϕλ ϕλ Cfa 3.3 u Z fa C fa. Hence ϕλ >/ λ ) ϕ λ ) uλ ϕλ ϕλ >/ λ ) ϕ λ ) ϕλ uλ ϕλ 3.4 C fa <.

6 6 Abstract and Applied Analysis For all λ D with ϕλ /, by 3.7, we have λ D λ ) uλ ϕ λ ) ϕλ 4 3 λ ) uλ ϕ λ ) <. 3.5 λ D Hence 3. holds. Next, we will show that 3. holds. Fix a D with a > /, we take another test functions: g a z haz a a ) 3.6 for z D. It is proved that /< a < g a C above, where C is not dependent on a. Therefore, for all λ D with ϕλ > /, we have C ga ucϕ g a z ) ) z ucϕ g a z ) ϕ zu z ϕ zuz ) g ) a ϕz g a ϕz ) ϕ z ) uz u zg a ϕz ) z ) ϕ zu z ϕ zuz ) ) aϕz a ) ϕ a aϕz z ) uz aϕz a ) g a ϕz u z. ) 3.7 Let a ϕλ, it follows that C ga λ ) ϕ λu λ ϕ λuλ ) ) ϕλ ϕλ ϕλ ϕλ ϕ λ ) uλ u ) λg ϕλ ϕλ )

7 Abstract and Applied Analysis 7 λ ) ϕ λu λ ϕ λuλ ) ϕλ λ ) ϕλ ϕ λ ) ϕλ uλ λ ) u ) λg ϕλ ϕλ. ) 3.8 Hence λ ) ϕ λu λ ϕ λuλ λ ) ϕλ ϕλ ϕ λ ) ϕλ uλ λ ) u ) λg ϕλ ϕλ C ga. 3.9 By 3., Lemma., and the boundedness of the function ϕz, weget ϕλ >/ ϕλ >/ λ ) ϕ λu λ ϕ λuλ λ ) ϕ λ ) ϕλ uλ ϕλ ϕλ >/ u Z gϕλ C ga <. 3.0 For all λ D with ϕλ /, by 3.5, we have ϕλ / 4 3 λ ) ϕ λu λ ϕ λuλ ϕλ / ϕλ λ ) ϕ λu λ ϕ λuλ <. 3. Hence 3. holds. Conversely, pose that u Z, 3. and 3. hold. For f Z, by Lemma., we have the following inequality: z ) ucϕ f ) z z ) ϕ zu z ϕ zuz ) f ϕz ) f ϕz ) ϕ z ) uz u zf ϕz )

8 8 Abstract and Applied Analysis z ) ϕ zu z ϕ zuz ) f ϕz ) z ) f ϕz ) ϕ z ) uz z ) u zf ϕz ) z ) ϕ zu z ϕ zuz e ϕz f z ) ϕ z ) uz ) ϕz ϕz f ϕz ) z ) u z f z ) ϕ zu z ϕ zuz e ϕz f z ) ϕ z ) uz ϕz fz u f Z C f. 3. This shows that uc ϕ is bounded. This completes the proof of Theorem 3.. Theorem 3.. Let u be an analytic function on the unit disc D and ϕ an analytic self-map of D.Then uc ϕ is bounded on the little Zygmund space Z 0 if and only if u Z 0, 3. and 3. hold, and the following are satisfied: lim z ) uz ϕ z ) 0; z 3.3 lim z ) ϕ zu z ϕ zuz 0. z 3.4 Proof. Suppose that uc ϕ is bounded on the little Zygmund space Z 0. Then u uc ϕ Z 0. Also uϕ uc ϕ z Z 0,thus z ) ϕ zu z ϕ zuz ϕzu z 0 z ). 3.5 Since ϕ andu Z 0, we have lim z z ϕ zu zϕ zuz 0. Hence 3.4 holds. Similarly, uc ϕ z Z 0, then z ) 4ϕzϕ zu z ϕ zu z uz ϕzϕ z ϕ z ) ) 0 z ). 3.6

9 Abstract and Applied Analysis 9 By 3.4, ϕ andu Z 0, we get that lim z z uzϕ z 0, that is, 3.3 holds. On the other hand, by Lemma.3 and Theorem 3.,weobtainthat3. and 3. hold. Conversely, let M z ) uz ϕ z ) ϕz < ; M z ) ϕ zu z ϕ zuz <. ϕz 3.7 For all f Z 0, we have both z f z 0and f z / / z 0as z by.5. Since u Z 0,giventhatɛ>0, there is a 0 <δ< such that z u z <ɛ/3 f, z f z <ɛ/3m and fz / / z <ɛ/3m for all z with δ< z <. If ϕz >δ, it follows that z ) ucϕ f ) z z ) ϕ zu z ϕ zuz ) f ϕz ) f ϕz ) ϕ z ) uz u zf ϕz ) z ) ϕ zu z ϕ zuz ) f ϕz ) z ) f ϕz ) ϕ z ) uz z ) u zf ϕz ) ) f ϕz M / ϕz )) M ) ϕz f ϕz ) z ) u z f ɛ 3 ɛ 3 ɛ ɛ We know that there exists a constant M 3 such that fz M 3, f z M 3 and f z M 3 for all z δ. If ϕz δ, it follows that z ) ucϕ f ) z z ) ϕ zu z ϕ zuz ) f ϕz ) f ϕz ) ϕ z ) uz u zf ϕz ) M 3 z ) ϕ zu z ϕ zuz M 3 z ) ϕ z ) uz M3 z ) u z. 3.9

10 0 Abstract and Applied Analysis Thus, we conclude that z uc ϕ f z 0as z. Hence uc ϕ f Z 0 for all f Z 0. On the other hand, uc ϕ is bounded on Z by Theorem 3.. Hence uc ϕ is a bounded operator on the little Zygmund space Z 0. The following corollary is just as Theorem. in 7. Corollary 3.3. Let ϕ be an analytic self-map of D.ThenC ϕ is a bounded operator on Z if and only if z ) ϕ z ) ϕz <, 3.30 z ) ϕ z <. ϕz 3.3 Corollary 3.4. Let ϕ be an analytic self-map of D. ThenC ϕ is a bounded operator on Z 0 if and only if ϕ Z 0, 3.30 and 3.3 hold. Proof. By Theorem 3., C ϕ is a bounded operator on Z 0 if and only if ϕ Z 0, lim z z ϕ z 0, 3.30 and 3.3 hold. However, by.5, ϕ Z 0 implies that lim z z ϕ z 0. Then, C ϕ is a bounded operator on Z 0 if and only if ϕ Z 0, 3.30 and 3.3 hold. 4. Compactness of uc ϕ In order to prove the compactness of uc ϕ on the Zygmund space Z, we require the following lemmas. Lemma 4.. Suppose that uc ϕ be a bounded operator on Z. ThenuC ϕ is compact if and only if for any bounded sequence {f n } in Z which converges to 0 uniformly on compact subsets of D, we have uc ϕ f n 0 as n. The proof is similar to that of Proposition 3. in 0. The details are omitted. Lemma 4.. Let {f n } be a bounded sequence in Z which converges to 0 uniformly on compact subsets of D.Thenlim n f n z 0. Proof. Let K n f n <. Givenanyε>0, there exist 0 <t< such that t / <ε.if t< z <, by Lemma., it follows that ) t f nz f n z z t/ z zf nztdt K e z t/ z zt dt e / K t/ z z zt ) / dt Ke / t / <Ke / ε, 4.

11 Abstract and Applied Analysis where we use the fact that x / e/x e / for all x 0,. Then fn z Ke / ε fn z. t< z < z t 4. Noting that {f n } converges to 0 uniformly on compact subsets of D,weget lim n fn z lim n Ke / ε fn z ) z t Ke / ε. 4.3 Hence, lim n f n z 0. Theorem 4.3. Let u be an analytic function on the unit disc D and ϕ an analytic self-map of D. Suppose that uc ϕ be a bounded operator on Z. ThenuC ϕ is compact if and only if the following are satisfied: i lim ϕz z ) uz ϕ z ) ϕz 0; ii lim z ) ϕ zu z ϕ zuz ϕz ϕz Proof. Suppose that uc ϕ is compact on the Zygmund space Z. Let{z n } be a sequence in D such that ϕz n asn. Without loss of generality, we may pose that ϕz n > / for all n. We take the test functions: f n z ϕz nz ϕz n ) ) ϕz n z ϕzn a n, 4.5 where ϕzn ) ) a n ϕz n ϕzn ϕzn 4.6 such that lim n a n 0. By a direct calculation, we may easily prove that {f n } converges to 0 uniformly on compact subsets of D. From the proof of Theorem 3., weseethat n f n <. Then {f n } is a bounded sequence in Z which converges to 0 uniformly on compact subsets of D. Then lim n uc ϕ f n 0 by Lemma 4.. Notethat f n ϕzn ) 0, f n ϕzn ) ϕzn, n f ϕzn ) ϕz n ϕzn, 4.7

12 Abstract and Applied Analysis it follows that ucϕ f n ucϕ f n Z z n ) u z n ϕ z n ϕ z n uz n ) f n ϕzn ) uz n f n ϕzn ) ϕ z n ) u z n f n ϕzn ) z n ) u z n ϕ z n ϕ z n uz n ) ϕ z n ) ϕz n uzn ϕzn z n ) u z n ϕ z n ϕ z n uz n ) z n ) ϕzn uz n ϕ z n ) ϕzn. ϕz n ϕzn 4.8 Then lim z n ) u z n ϕ z n ϕ z n uz n ) n ϕzn lim z n ) z n ) ϕzn uz n ϕ z n ) n ϕz n, 4.9 if one of these two limits exits. On the other hand, let ) ) h ϕz n z h n z ϕz n ϕz n z 0 3 ϕz n ω dω ϕz n ), 4.0 so ) ) ) h nz ϕz n z ϕzn 3 ϕz n z ϕzn,

13 Abstract and Applied Analysis 3 h nz ϕz n ϕz n z ϕz n z 3ϕz n ϕz n z ϕz n z ϕzn ϕzn ) ). 4. One may obtain that h n 0 n on compact subsets of D by a direct calculation and n h n C< by the proof of Theorem 3.. Consequently, {h n } is a bounded sequence in Z which converges to 0 uniformly on compact subsets of D. Then lim n uc ϕ h n 0 by Lemma 4..Notethatu Z,h nϕz n 0 and lim n h n z 0byLemma4.,it follows that 0 uc ϕ h n uc ϕ h n Z z n ) uzn h n ϕzn ) ϕ z n ) u z n h n ϕzn ) z n ) uz n ϕ z n ) ϕzn ϕzn z n ) u z n h n z n z n ) uz n ϕ z n ) ϕzn, 4. as n. Then lim n z n uz n ϕ z n / ϕz n 0. The proof of the necessary is completed. Conversely, Suppose that i and ii hold. Let {f n } be a bounded sequence in Z which converges to 0 uniformly on compact subsets of D. LetM n f n <. We only prove lim n uc ϕ f n 0 by Lemma 4.. This amounts to showing that w ) ϕ wu w ϕ wuw ) ) f n ϕw 0, w D w D w ) uw ϕ w ) ) f n ϕw 0, w ) u ) wf n ϕw 0. w D 4.3 By Lemma 4. and uc ϕ bounded on Z, which implies that u Z, then If ϕw r<, by 3.5, then w ) u ) wf n ϕw u Z fn z 0. w D w ) ϕ wu w ϕ wuw ) ) f n ϕw K max f n z. z r

14 4 Abstract and Applied Analysis If ϕw >r, by Lemma., then w ) ϕ wu w ϕ wuw ) ) f n ϕw M w ) ϕ wu w ϕ wuw ) e. 4.6 ϕw Thus, w ) ϕ wu w ϕ wuw ) ) f n ϕw w D K maxf nw M w r ϕw >r w ) ϕ wu w ϕ wuw ) e ϕw. 4.7 First, letting n tend to infinity and subsequently r increase to, one obtains that w ) ϕ wu w ϕ wuw ) ) f n ϕw 0, 4.8 w D as n. The third statement is proved similarly. If ϕw r<, by 3.7, then w ) uw ϕ w ) ) f n ϕw K max f nz. z r 4.9 If ϕw >r, then w ) uw ϕ w ) ) w f ) uw ϕ w ) n ϕw M. 4.0 ϕw Thus, w ) uw ϕ w ) ) f n ϕw K max f nz w D z r w ) uw ϕ w ) M ϕw, ϕw >r 4. which also implies that w ) uw ϕ w ) ) f n ϕw 0, 4. w D as n. This completes the proof of Theorem 4.3.

15 Abstract and Applied Analysis 5 In order to prove the compactness of uc ϕ on the little Zygmund space Z 0,werequire the following lemma. Lemma 4.4. Let U Z 0.ThenU is compact if and only if it is closed, bounded, and satisfies lim z ) f z z f U The proof is similar to that of Lemma in 6, we omit it. Theorem 4.5. Let u be an analytic function on the unit disc D and ϕ an analytic self-map of D.Then uc ϕ is compact on the little Zygmund space Z 0 if and only if u Z 0 and the following are satisfied: i lim z z ) uz ϕ z ) ϕz 0; ii lim z ) ϕ zu z ϕ zuz z 0. ϕz 4.4 Proof. Assume that i and ii hold, and u Z 0. By Theorem 3., we know that uc ϕ is bounded on the little Zygmund space Z 0.Fromii, we can show that lim z ) ϕ zu z ϕ zuz z Suppose that f Z 0 with f. We obtain that z ) ucϕ f ) z z ) ϕ zu z ϕ zuz ) f ϕz ) z ) f ϕz ) ϕ z ) uz z ) u zf ϕz ) z ) ϕ zu z ϕ zuz e ϕz f z ) ϕ z ) uz ) ϕz ϕz f ϕz ) z ) u z f z ) ϕ zu z ϕ zuz ) ϕz z ) ϕ z ) uz ϕz z ) u z, 4.6

16 6 Abstract and Applied Analysis thus, { z ) ucϕ f ) z : f Z0, f } z ) ϕ zu z ϕ zuz z ) ϕ z ) uz ϕz z ) u z, ϕz ) 4.7 and it follows that { lim z ) ucϕ f ) z : f Z0, } f 0, z 4.8 hence, uc ϕ is compact on Z 0 by Lemma 4.. Conversely, pose that uc ϕ is compact on Z 0. First, it is obvious uc ϕ is bounded on Z 0, then by Theorem 3., we have u Z 0 and that 3.4 holds. On the other hand, by Lemma 4. we have { lim z ) ucϕ f ) z : f Z0, } f M 0, 4.9 z for some M>0. Next, note that the proof of Theorem 3. and the fact that the functions given in 3.8 are in Z 0 and have norms bounded independently of a, weobtainthat lim z z ) uz ϕ z ) ϕz Similarly, note that the functions given in 3.6 are in Z 0 and have norms bounded independently of a, weobtainthat lim z ) ϕ zu z ϕ zuz z ϕz C lim z ) ) z ucϕ g a lim z ) u z ga z z lim z ) ϕz z uz ϕ z ) ϕz, 4.3 for ϕz > /. So by 4.30 and u Z 0, it follows that lim z z ) ϕ zu z ϕ zuz ϕz 0, 4.3

17 Abstract and Applied Analysis 7 for ϕz > /. However, if ϕz /, by 3.4, we easily have lim z ) ϕ zu z ϕ zuz z ϕz 4 3 lim z ) ϕ zu z ϕ zuz 0. z 4.33 This completes the proof of Theorem 4.5. Corollary 4.6. Let ϕ be an analytic self-map of D.ThenC ϕ is a compact operator on Z 0 if and only if lim z z ) ϕ z ) ϕz lim z ) ϕ z z 0, ϕz In the formulation of corollary, we use the notation M u on HD defined by M u f uf for f HD. Corollary 4.7. Let u be an analytic function on the unit disc D. Then the pointwise multiplier M u : Zresp. Z 0 Zresp. Z 0 is a compact operator if and only if u 0. Acknowledgment The research was ported by the Natural Science Foundation of Fujian Province, China Grant no. 009J0004. References A. Zygmund, Trigonometric Series, Cambridge, 959. P. L. Duren, Theory of H p Spaces, Academic Press, New York, NY, USA, S. Li and S. Stević, Generalized composition operators on Zygmund spaces and Bloch type spaces, Journal of Mathematical Analysis and Applications, vol. 338, no., pp. 8 95, S. Li and S. Stević, Products of Volterra type operator and composition operator from H and Bloch spaces to Zygmund spaces, Journal of Mathematical Analysis and Applications, vol. 345, no., pp. 40 5, S. Li and S. Stević, Weighted composition operators from Zygmund spaces into Bloch spaces, Applied Mathematics and Computation, vol. 06, no., pp , K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Transactions of the American Mathematical Society, vol. 347, no. 7, pp , K. M. Madigan, Composition operators on analytic Lipschitz spaces, Proceedings of the American Mathematical Society, vol. 9, no., pp , W. Smith, Composition operators between Bergman and Hardy spaces, Transactions of the American Mathematical Society, vol. 348, no. 6, pp , R. Yoneda, The composition operators on weighted Bloch space, Archiv der Mathematik, vol. 78, no. 4, pp , 00.

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