Research Article Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane

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1 Abstract and Applied Analysis Volume 2011, Article ID , 10 pages doi: /2011/ Research Article Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane Stevo Stević, 1 Ajay K. Sharma, 2 ands.d.sharma 3 1 Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, Serbia 2 School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, J & K Katra , India 3 Department of Mathematics, University of Jammu, Jammu , India Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs Received 25 January 2011; Accepted 3 May 2011 Academic Editor: Narcisa C. Apreutesei Copyright q 2011 Stevo Stević 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 ψ be a holomorphic mapping on the upper half-plane Π {z C : Jz > 0} and ϕ be a holomorphic self-map of Π. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on ψ, we also characterize the compactness of these operators. 1. Introduction and Preliminaries Let Π {z C : Iz >0} be the upper half-plane, Ω a domain in C or C n,andh Ω the space of all holomorphic functions on Ω. Letψ H Ω, andletϕ be a holomorphic self-map of Ω. Then by W ϕ,ψ ( f ) z ψ ( f ϕ ) z, z Ω, 1.1 is defined a linear operator on H Ω which is called weighted composition operator. If ψ z 1, then W ϕ,ψ becomes composition operator and is denoted by C ϕ,andifϕ z z, then W ϕ,ψ becomes multiplication operator and is denoted by M ψ. During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in C or C n see, e.g., 1 22 and the references therein. There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and

2 2 Abstract and Applied Analysis Bergman spaces are certain weighted composition operators see 23, 24. For some other operators related to weighted composition operators, see and the references therein. While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk D, there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane see, e.g., 2, 3, 5, 7 9, 11, 12, 16 18, 31 and the references therein. For related results in the setting of the complex plane see also papers The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk D. For example, there are holomorphic selfmaps of Π which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map ϕ of D induces a bounded composition operator on the Hardy and weighted Bergman spaces on D. Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators see 3, 5. For 0 <p< and α 1,, letl p Π,dA α denote the collection of all Lebesgue p-integrable functions f : Π C such that Π f z p daα z <, 1.2 where da α z 1 π α 1 2I z α da z, 1.3 da z dxdy,andz x iy. Let A p α Π L p Π,dA α H Π. For 1 p<, A p α Π is a Banach space with the norm defined by ( ) fa p α Π 1/p f z p daα z <. 1.4 Π With this norm A p α Π becomes a Banach space when p 1, while for p 0, 1 it is a Fréchet space with the translation invariant metric d ( f, g ) f g p A p α Π, f,g Ap α Π. 1.5 Recall that for every f A p α Π the following estimate holds: f ( x iy ) p C f p A p α Π y α 2, 1.6 where C is a positive constant independent of f.

3 Abstract and Applied Analysis 3 Let >0. The weighted-type space or growth space on the upper half-plane A Π consists of all f H Π such that fa sup Π Iz f z <. 1.7 It is easy to check that A Π is a Banach space with the norm defined above. For weightedtype spaces on the unit disk, polydisk, or the unit ball see, for example, papers 10, 32, 33 and the references therein. Given two Banach spaces Y and Z, we recall that a linear map T : Y Z is bounded if T E Z is bounded for every bounded subset E of Y. In addition, we say that T is compact if T E Z is relatively compact for every bounded set E Y. In this paper, we consider the boundedness and compactness of weighted composition operators acting from A p α Π to the weighted-type space A Π. Related results on the unit disk and the unit ball can be found, for example, in 6, 13, 15. Throughout this paper, constants are denoted by C; they are positive and may differ from one occurrence to the other. The notation a b means that there is a positive constant C such that a Cb. Moreover, if both a b and b a hold, then one says that a b. 2. Main Results The boundedness and compactness of the weighted composition operator W ϕ,ψ : A p α Π A Π are characterized in this section. Theorem 2.1. Let 1 p<, α> 1, >0, ψ H Π, and let ϕ be a holomorphic self-map of Π.ThenW ϕ,ψ : A p α Π A Π is bounded if and only if Iz M : sup ( ) ψ z α 2 /p <. 2.1 Moreover, if the operator W ϕ,ψ relationship holds: : A p α Π A Π is bounded then the following asymptotic Wϕ,ψ A p α Π A Π M. 2.2 Proof. First suppose that 2.1 holds. Then for any z Π and f A p α Π,by 1.6 we have Iz ( W ϕ,ψ f ) z Iz ψ z f ) Iz ( ) α 2 /p ψ z f A p α Π, 2.3 and so by 2.1, W ϕ,ψ : A p α Π A Π is bounded and moreover Wϕ,ψ A p α Π A Π M. 2.4

4 4 Abstract and Applied Analysis Conversely suppose W ϕ,ψ : A p α Π A Π is bounded. Consider the function Iw α 2 /p f w z z w, w 2α 4 /p Π. 2.5 Then f w Aα Π p and moreover sup w Π f w p A α Π 1 see, e.g., Lemma 1 in 18. Thus the boundedness of W ϕ,ψ : Aα Π p A Π implies that Iz ψ z fw ) Wϕ,ψ f w A Π Wϕ,ψ A p α Π A Π, 2.6 for every z, w Π. In particular, if z Π is fixed then for w ϕ z, weget Iz ( ) α 2 /p ψ z W ϕ,ψ A p α Π A Π. 2.7 Since z Π is arbitrary, 2.1 follows and moreover M W ϕ,ψ A p α Π A Π. 2.8 If W ϕ,ψ : A p α Π A Π is bounded then from 2.4 and 2.8 asymptotic relationship 2.2 follows. Corollary 2.2. Let 1 p<, α> 1, and >0 be such that p α 2 and ψ H Π.Then M ψ : A p α Π A Π is bounded if and only if ψ X, where A α 2 /p Π X H Π if α 2 <p, if α 2 p. 2.9 Example 2.3. Let 1 p<, α> 1and>0 be such that p α 2andw Π.Letψ w be a holomorphic map of Π defined as 1 ψ w z z w α 2 /p Iw z w if α 2 <p, if α 2 p. 2.10

5 Abstract and Applied Analysis 5 For z x iy and w u iv in Π, we have sup Iz α 2 /pψ w z sup z x iy Π sup z x iy Π y α 2 /p ( x u 2 ( y v ) 2 ) p α 2 /2p y α 2 /p ( y v ) α 2 /p Thus ψ w A α 2 /p Π if α 2 <p. Similarly ψ w H Π if α 2 p.bycorollary 2.2, it follows that M ψw : Aα Π p A Π is bounded. Corollary 2.4. Let 1 p<, α> 1, >0, and let ϕ be a holomorphic self-map of Π.Then C ϕ : A p α Π A Π is bounded if and only if Iz sup ( ) α 2 /p < Corollary 2.5. Let ϕ be the linear fractional map ϕ z az b, a,b,c,d R,ad bc > cz d Then necessary and sufficient condition that C ϕ : A p α Π A Π is bounded is that c 0 and α 2 p. Proof. Assume that C ϕ : A p α Π A Π is bounded. Then sup Iz ( ) α 2 /p sup z x iy Π ( cx d 2 c 2 y 2) α 2 /p y ad bc α 2 /p y α 2 /p, 2.14 which is finite only if c 0andα 2 p. Conversely, if c 0andα 2 p, then from 2.13 we get a / 0, and by some calculation Iz sup ( ) α 2 /p ( ) d < a Hence C ϕ : A p α Π A Π is bounded. Corollary 2.6. Let 1 p<,α > 1, and >0 be such that p α 2. Letϕ be a holomorphic self-map of Π and ψ ϕ. Then the weighted composition operator W ϕ,ψ acts boundedly from A p α Π to A Π.

6 6 Abstract and Applied Analysis Proof. By Theorem 2.1, W ϕ,ψ : A p α Π A Π is bounded if and only if Iz sup ( ) ϕ z < By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map ϕ of Π and all z Π ϕ z 1 Iz, 2.17 where the equality holds when ϕ is a Möbius transformation given by From 2.17, condition 2.16 follows and consequently the boundedness of the operator W ϕ,ψ : A p α Π A Π. Corollary 2.6 enables us to show that there exist 1 p <, α> 1, >0, and holomorphic maps ϕ and ψ of the upper half-plane Π such that neither C ϕ : Aα Π p A Π nor M ψ : Aα Π p A Π is bounded, but W ϕ,ψ : Aα Π p A Π is bounded. Example 2.7. Let 1 p<, α> 1, and >0besuch that p α 2. Let ϕ z az b / cz d, a,b,c,d R, ad bc > 0, and c / 0. Then by Corollary 2.5, C ϕ : Aα Π p A Π is not bounded. On the other hand, if ψ z ( ϕ z ) ( ) ad bc, 2.18 cz d 2 then ψ / H Π andsobycorollary 2.2, M ψ : Aα Π p A Π is not bounded. However, by Corollary 2.6, we have that W ϕ, ϕ : Ap α Π A Π is bounded. The next Schwartz-type lemma characterizes compact weighted composition operators W ϕ,ψ : Aα Π p A Π and it follows from standard arguments 4. Lemma 2.8. Let 1 p<, α> 1, >0,ψ H Π, and let ϕ be a holomorphic self-map of Π. Then W ϕ,ψ : Aα Π p A Π is compact if and only if, for any bounded sequence f n n N Aα Π p converging to zero on compacts of Π, one has lim Wϕ,ψ f A n n Π Theorem 2.9. Let 1 p<, α> 1, > 0, ψ H Π and ϕ be a holomorphic self-map of Π. If W ϕ,ψ : A p α Π A Π is compact, then lim sup r 0 <r Iz ( ) ψ z α 2 /p

7 Abstract and Applied Analysis 7 Proof. Suppose W ϕ,ψ : A p α Π A Π is compact and 2.20 does not hold. Then there is a δ>0 and a sequence z n n N Π such that Iϕ z n 0and Iz n ( Iϕ zn ) ψ z α 2 /p n >δ 2.21 for all n N. Letw n ϕ z n, n N, and f n z Iw n α 2 /p, n N α 4 /p z w n Then f n is a norm bounded sequence and f n Lemma 2.8 it follows that 0 on compacts of Π as Iϕ z n 0. By lim Wϕ,ψ f A n n Π On the other hand, W ϕ,ψ f n A Π Iz n ( W ϕ,ψ f n ) zn Iz n ψ z n f n ( ϕ zn ) Iz n 2 2α 4 /p( Iϕ z n ) ψ z α 2 /p n δ > 2, 2α 4 /p 2.24 which is a contradiction. Hence 2.20 must hold, as claimed. Before we formulate and prove a converse of Theorem 2.9, we define, for every a, b 0, such that a<b, the following subset of Π : Γ a,b {z Π : a Iz b} Theorem Let 1 p<, α> 1, >0, ψ H Π, and let ϕ be a holomorphic self-map of Π and W ϕ,ψ : Aα Π p A Π be bounded. Suppose that ψ A Π and Iz ψ z 0 as Rϕ z within Γ a,b for all a and b, 0 <a<b<. ThenW ϕ,ψ : Aα Π p A Π is compact if condition 2.20 holds. Proof. Assume 2.20 holds. Then for each ε>0, there is an M 1 > 0 such that Iz ( ) α 2 /p ψ z <ε, whenever <M

8 8 Abstract and Applied Analysis Let f n n N be a sequence in A p α Π such that sup n N f n A p α Π M and f n 0 uniformly on compact subsets of Π as n.thusforz Π such that <M 1 and each n N, we have Iz ψ z fn ) Iz ( ) α 2 /p ψ z fn A p α Π <εm From estimate 1.6 we have fn z fna p α Π Iz M α 2 /p Iz. α 2 /p 2.28 Thus there is an M 2 >M 1 such that f n ) <ε, 2.29 whenever >M 2. Hence for z Π such that >M 2 and each n N we have Iz ψ z fn ) <ε ψ A Π If M 1 M 2, then by the assumption there is an M 3 > 0 such that Iz ψ z <ε, whenever Rϕ z >M 3. Therefore, for each n N we have Iz ( ) f A n p ψ z fn ϕ z ε α Π ( ) α 2 /p ε M M α 2 /p 1, 2.31 whenever M 1 M 2 and Rϕ z >M 3. If M 1 M 2 and Rϕ z M 3, then there exists some n 0 N such that f n ϕ z <εfor all n n 0,andso Iz ψ z fn ) <ε ψ A Π Combining , we have that W ϕ,ψ f n A Π <εc, 2.33 for n n 0 and some C>0 independent of n. Since ε is an arbitrary positive number, by Lemma 2.8, it follows that W ϕ,ψ : A p α Π A Π is compact.

9 Abstract and Applied Analysis 9 Example Let 1 p<, α> 1, and >0 be such that α 2 p. Letϕ z z i and ψ z 1/ z i, then Rϕ z x and y 1. It is easy to see that ψ A Π.Beside this, for z Γ a,b, we have Iz ψ z y ( x 2 ( y 1 ) 2 ) /2 b 0 as Rϕ z x x 2 a 2 /2 Also Iz sup ( ) ψ z sup y 1 ( ) z x iy Π y 1 (x 2 ( y 1 ) 2 ) /2 1 <, 2.35 and the set {z : < 1} is empty. Thus ϕ and ψ satisfy all the assumptions of Theorem 2.10, and so W ϕ,ψ : A p α Π A Π is compact. Acknowledgments The work is partially supported by the Serbian Ministry of Science, projects III and III The work of the second author is a part of the research project sponsored by National Board of Higher Mathematics NBHM /DAE, India Grant no. 48/4/2009/R&D-II/426. References 1 C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, V. Matache, Composition operators on H p of the upper half-plane, Analele Universităţii din Timişoara, Seria Ştiinţe Matematice, vol. 27, no. 1, pp , V. Matache, Composition operators on Hardy spaces of a half-plane, Proceedings of the American Mathematical Society, vol. 127, no. 5, pp , H. J. Schwartz, Composition operators on H p, Ph.D. thesis, University of Toledo, Ohio, USA, J. H. Shapiro and W. Smith, Hardy spaces that support no compact composition operators, Journal of Functional Analysis, vol. 205, no. 1, pp , A. K. Sharma and S. D. Sharma, Weighted composition operators between Bergman-type spaces, Communications of the Korean Mathematical Society, vol. 21, no. 3, pp , S. D. Sharma, A. K. Sharma, and S. Ahmed, Carleson measures in a vector-valued Bergman space, Analysis and Applications, vol. 4, no. 1, pp , S. D. Sharma, A. K. Sharma, and S. Ahmed, Composition operators between Hardy and Bloch-type spaces of the upper half-plane, Bulletin of the Korean Mathematical Society, vol. 44, no. 3, pp , S. D. Sharma, A. K. Sharma, and Z. Abbas, Weighted composition operators on weighted vectorvalued Bergman spaces, Applied Mathematical Sciences, vol. 4, no , pp , A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Transactions of the American Mathematical Society, vol. 162, pp , R. K. Singh and S. D. Sharma, Composition operators on a functional Hilbert space, Bulletin of the Australian Mathematical Society, vol. 20, no. 3, pp , R. K. Singh and S. D. Sharma, Noncompact composition operators, Bulletin of the Australian Mathematical Society, vol. 21, no. 1, pp , 1980.

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