Composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions

Transcription

1 Irish Math. Soc. Bulletin Number 79, Summer 07, ISSN Composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions ELKE WOLF Abstract. An analytic self-map φ of the open unit disk in the complex plane induces the so-called composition operator C φ : H) H), f f φ, where H) denotes the set of all analytic functions on. Motivated by [5] we analyze under which conditions on the weight v all composition operators C φ acting between the weighted Bergman space and the weighted Banach space of holomorphic functions both generated by v are bounded.. Introduction Let denote the open unit disk in the complex plane C and H) the space of all analytic functions on endowed with the compactopen topology co. Moreover, let φ be an analytic self-map of. Such a map induces through composition the classical composition operator C φ : H) H), f f φ. Composition operators acting on various spaces of analytic functions have been studied by many authors, since this kind of operator appears naturally in a variety of problems, such as e.g. in the study of commutants of multiplication operators or the study of dynamical systems, see the excellent monographs [8] and [7]. For a deep insight in the recent research on weighted) composition operators we refer the reader to the following papers as well as the references therein: [4], [5], [6], [7], [], [3], [4], [5], [6]. Let us now explain the setting in which we are interested. We say that a function v : 0, ) is a weight if it is bounded and continuous. For a weight v we consider the following spaces: 00 Mathematics Subject Classification. 47B33, 47B38. Key words and phrases. weighted Banach space of holomorphic functions, weighted Bergman space, composition operator. Received on ; revised c 07 Irish Mathematical Society

2 76 WOLF ) The weighted Banach spaces of holomorphic functions defined by H v := {f H); f v := fz) < }. Endowed with the weighted -norm. v this is a Banach space. These spaces arise naturally in several problems related to e.g. complex analysis, spectral theory, Fourier analysis, partial differential and convolution equations. Concrete examples may be found in [3]. Weighted Banach spaces of holomorphic functions have been studied deeply in [] and also in []. ) The weighted Bergman spaces given by { ) } A v := f H); f v, := fz) daz) <, where daz) is the normalized area measure such that area of is. Endowed with norm. v, this is a Hilbert space. An introduction to Bergman spaces is given in [0] and [9]. In [9] we characterized the boundedness of composition operators acting beween weighted Bergman spaces and weighted Banach spaces of holomorphic functions in terms of the involved weights as well as the symbols. In this article we are interested in the question, for which weights v all composition operators C φ : A v Hv are bounded.. Background and basics.. Theory of weights. In this part of the article we want to give some background information on the involved weights. A very important role play the so-called radial weights, i.e. weights which satisfy = v z ) for every z. If additionally lim z = 0 holds, we refer to them as typical weights. Examples include all the famous and popular weights, such as a) the standard weights = z ) α, α, b) the logarithmic weights = log z )) β, β > 0, c) the exponential weights = e z ) α, α.

3 Composition operators 77 In [] Lusky studied typical weights satisfying the following two conditions v n ) L) inf > 0 n N v n ) and v n j ) L) lim < for some j N. n v n ) In fact, weights having L) and L) are normal weights in the sense of Shields and Williams, see [8]. The standard weights are normal weights, the logarithmic weights satisfy L), but not L) and the exponential weights satisfy neither L) nor L). In our context L) is not of interest, while L) will play a secondary role. The formulation of results on weighted spaces often requires the socalled associated weights. For a weight v its associated weight is given by ṽz) := { fz) ; f H), f v }, z. See e.g. [] and the references therein. Associated weights are continuous, ṽ v > 0 and for every z there is f z H) with f z v such that f z z) = ṽz). Since it is quite difficult to really calculate the associated weight we are interested in simple conditions on the weight that ensure that v and ṽ are equivalent weights, i.e. there is a constant C > 0 such that ṽz) C for every z. If v and ṽ are equivalent, we say that v is an essential weight. By [5] condition L) implies the essentiality of v... Setting. This section is devoted to the description of the setting we are working in. In the sequel we will consider weighted Bergman spaces generated by the following class of weights. Let ν be a holomorphic function on that does not vanish and is decreasing as well as strictly positive on [0, ). Moreover, we assume that lim r νr) = 0. Now, we define the weight as follows: := ν z ) for every z. ) Obviously such weights are bounded, i.e. for every weight v of this type we can find a constant C > 0 such that C. Moreover, we assume additionally that νz) ν z ) for every

4 78 WOLF z. Now, we can write the weight v in the following way = min{ gλz), λ = }, where g is a holomorphic function on. Since ν is a holomorphic function, we obviously can choose g = ν. Then we arrive at min{ νλz), λ = } = min{ νλre iθ ), λ = } νe iθ re iθ ) = νr) = ν z ) = for every z. Conversely, by hypothesis, for every λ we obtain for every z νλz) ν λz) ) ν z ) =. Thus, the claim follows. The standard, logarithmic and exponential weights can all be defined like that..3. Composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions. In the setting of weighted Banach spaces of holomorphic functions the classical composition operator has been studied by Bonet, omański, Lindström and Taskinen in [4] and [5]. Among other things they proved that in case that v and w are arbitrary weights the boundedness of the operator C φ : Hv Hw is equivalent to wz) ṽφz)) <. Moreover, they showed that v satisfies condition L) if and only if every composition operator C φ : H v H v is bounded. This was the motivation to study the boundedness composition operators acting between weighted Bergman spaces and weighted Banach spaces of holomorphic functions. oing this we obtain the following results which we need in the sequel. For the sake of understanding and completeness we give the proof here. For p let α p z) := p z pz, z, be the Möbius transformation that interchanges p and 0.

5 Composition operators 79 Lemma. [0], Lemma ). Let = ν z ) for every z with ν H) be a weight as defined in Section.. Then there is a constant M > 0 such that for every f A v. f v, fz) M z ) Proof. As we have seen in Section. a weight as defined above may be written as := min { gλz) ; λ = } for every z, where g is a holomorphic function on. In the sequel we will write g λ z) := gλz) for every z. Now, fix λ C with λ =. Moreover, let p be arbitrary. Then, we consider the map T p,λ : A v A v, T p,λ fz) = fα p z))α pz)g λ α p z)). Let f A v. Then a change of variables yields T p,λ f v, = fα p z)) α pz) g λ α p z)) daz) fα p z)) α pz) vα p z)) daz) fα p z)) α pz) vα p z)) daz) C vt) ft) dat) = C f v,. Next, put h p,λ z) := T p,λ fz) for every z. By the Mean Value Property we obtain v0) h p,λ 0) h p,λ z) daz) h p,λ v, C f v,. Since λ was arbitrary, we obtain Finally, v0) fp) p )vp) C f v,. f v, fp) M p )vp) <.

6 80 WOLF The following result is obtained by using the previous lemma and following exactly the proof of [9] Theorem.. Again, for a better understanding we give the proof. Theorem.. Let = ν z ) for every z with ν H) be a weight as defined in Section.. Then the operator C φ : A v H v is bounded if and only if φz) )vφz)) Proof. First, we assume that ) holds. every f A v we have f v, fz) C z ) for every z. Thus, for every f A v: C φ f v = fφz)) C <. ) Applying Lemma. for f v, φz) )vφz)) <. Hence the operator must be bounded. Conversely, let p be fixed. Then there is fp Hv, fp v with fp p) = ṽp). Now, put g p z) := f p z)α pz) for every z. Changing variables we obtain g p v, = g p z) daz) = f p z) f p z) α pz) daz) α pz) daz) = Next, we assume to the contrary that there is a sequence z n ) n such that φz n ) and Now, we consider vz n ) φz n ) )vφz n )) n for every n N. g n z) := g φzn )z) for every z and every n N

7 Composition operators 8 as defined above. Then g n ) n is contained in the closed unit ball of A v and we can find a constant c > 0 such that c vz n ) g n φz n )) = vz n ) φz n ) )vφz n )) n for every n N. Since we know that under the given assumptions we have v = ṽ this is a contradiction. Having now characterized the boundedness of the composition operator acting between A v and Hv we take the second result of Bonet, omański, Lindström and Taskinen as a motivation to ask the question: For which weights v are all operators C φ : A v Hv bounded? 3. Results Lemma 3.. Let = ν z ) for every z with ν H) be a weight as defined in Section.. Moreover, let z < and C αp : A v Hv be bounded for every p. Then all composition operators C φ : A v Hv are bounded. Proof. Let φ : be an arbitrary analytic function. We have to show that C φ : A v Hv is bounded. Now, φ = α p ψ where p = φ0), ψ = α p φ and ψ0) = 0. Since ψ0) = 0, by the Schwarz Lemma we obtain that ψz) z. Hence we get ψz) )vψz)) z ) = z ) <. Thus, C ψ is bounded. Finally, we can conclude that C φ is bounded since it is a composition of bounded operators. The proof of the following theorem is inspired by [5]. Theorem 3.. Let = ν z ) for every z with ν H) be a weight as defined in Section.. Moreover, let z <. Then the composition operator C φ : A v Hv is bounded for every analytic self-map φ of if and only if n n )v n ) inf n N v n ) > 0. 3) Proof. By Lemma 3. we have to show that condition 3) holds if and only if C αp : A v H v is bounded for every p.

8 8 WOLF First, let each C αp : A v H v be bounded. Then we have that for every p there is M p > 0 such that M p vα p z)) αp z) ) for every z. Since z =r α p z) = p +r ) M p v p +r + p r + p r it follows that ) ) for all z = r. Let lr) = p +r + p r v r) r) ) and s = r. Now, since p + s s p ) + p p s, for s < l s p ) l + p we obtain p + s ) l + p s) s p p + p s) = Next, choose p = 5 and find M > 0 and s 0 > 0 such that s v s) Ml = Mv ) s ) s ) ) for all s ]0, s 0 [. Hence the claim follows. Conversely we assume that 3) holds. Then l as defined above has the property that there are M > 0 and t 0 ]0, [ with ) t v t) Ml for all t t 0. Hence, for any c < we find n N such that c < n and thus lt) M n l ) t c. We take c = + p p. By the first inequality in 4) for all p there is M p > 0 such that v t) M p l p + t ) + p t) for all t > t 0. Clearly this implies that for all p there exists M p > 0 such that for every z = r we have that < M p vα p z)) α p z) ). Example 3.3. a) Let = z ) n, n. Then all composition operators C φ : A v Hv are bounded. To prove this we have to show that the weight v satisfies the following conditions ) z <, ) 4)

9 ) inf k N k k )v k ) v k ) > 0. Indeed, z = Composition operators 83 z ) n z ) + z ) = z ) n + z ) z ) n < since n. Moreover, k k )v k ) inf k N v k ) = inf k N k k ) kn n) = inf k N n k n ) k + n ) ) > 0 for every n. b) The weight = z satisfies neither ) nor ). We obtain z = z ) + z ) Furthermore easy calculations show k k )v k ) inf k N v k ) z ) =. = inf k N k k ) k ) = inf k N k 3 k ) = 0. Hence, in this case there exists a composition operator C φ : A v Hv that is not bounded. For example, the operator C φ generated by the map φz) = z for every z is not bounded, since φz) )vφz)) = + z ) z ) =. c) The exponential weights = e z ) n, n > 0, satisfy condition ), but not condition ). First, we get z = e z ) n z It follows, that ) is fulfilled. Now, <. k k )v k ) inf k N v k ) = inf k N k k ) e kn e k n = 0.

10 84 WOLF Thus, ) is not satisfied. There must be a composition operator C φ : A v Hv that is not bounded. d) The logarithmic weights = log z )), n > 0, neither n satisfy ) nor ). Indeed, and z = k k )v k ) inf k N v k ) z ) log z )) n = = inf k N k k ) log k ) n log k ) n = 0. With the criteria above we cannot decide, whether all composition operators C φ : A v Hv are bounded or not. But, again selecting φz) = z we see that φz) )vφz)) = z ) log z )) =. Hence the corresponding composition operator is not bounded. References [] K.. Bierstedt, J. Bonet, A. Galbis: Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40, no., 993), [] K.. Bierstedt, J. Bonet, J. Taskinen: Associated weights and spaces of holomorphic functions, Studia Math. 7, no., 998), [3] K.. Bierstedt, R. Meise, W.H. Summers: A projective description of weighted inductive limits, Trans. Am. Math. Soc. 7, no., 98), [4] J. Bonet, P. omański, M. Lindström: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 4, no., 999), [5] J. Bonet, P. omański, M. Lindström: J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Series A) ), 0-8. [6] J. Bonet, M. Lindström, E. Wolf: ifferences of composition operators between weighted Banach spaces of holomorphic functions, J. Aust. Math. Soc ), no., 9-0. [7] M.. Contreras, A.G. Hernández-íaz: Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Series A) ), [8] C. Cowen, B. MacCluer: Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 995. [9] P. uren, A. Schuster: Bergman spaces, Mathematical Surveys and Monographs 00, American Mathematical Society, Providence, RI, 004.

11 Composition operators 85 [0] H. Hedenmalm, B. Korenblum, K. Zhu: Theory of Bergman spaces, Graduate Texts in Mathematics 99, Springer-Verlag, New York, 000. [] T. Kriete, B. MacCluer: Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J. 4 99), no. 3, [] W. Lusky: On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. ) 5 995), no., [3] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal ), no., [4] B. MacCluer, S. Ohno, R. Zhao: Topological structure of the space of composition operators on H, Integral equations Operator Theory 40 00), no. 4, [5] P. Nieminen: Compact differences of composition operators on Bloch and Lipschitz spaces, Comput. Methods. Funct. Theory 7 007), no., [6] N. Palmberg: Weighted composition operators with closed range, Bull. Austral. Math. Soc ), no. 3, [7] J.H. Shapiro: Composition Operators and Classical Function Theory, Springer, 993. [8] A.L. Shields,.L. Williams: Bounded projetions, duality and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 99/ ), [9] E. Wolf: Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Rev. Mat. Complut. 008), no., [0] E. Wolf: Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Bull. Acad. Stiinte Repub. Mold. Mat. 04, no., Elke Wolf obtained her Ph in 004 at Paderborn and her Habilitation in 00 at Trier. Currently she works at the University of Paderborn. In her private life she enjoys running, cycling and reading. Institut für Mathematik, Universität Paderborn, Warburger Str. 00, Paderborn, Germany address: lichte@math.uni-paderborn.de