Julio C. Ramos Fernández

María T. Malavé Ramírez

Malavé Ramírez, María T.

Ramos Fernández, Julio C.

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 23–26Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical AnalysisOn a criterion for continuity and compactness of composition operatorsacting on α-Bloch spacesSur un critère de continuité et compacité des opérateurs de composition agissant sur lesespaces α-BlochMaría T. Malavé Ramírez, Julio C. Ramos FernándezDepartamento de Matemática, Universidad de Oriente, 6101 Cumaná, Edo. Sucre, Venezuelaa r t i c l e i n f o a b s t r a c tArticle history:Received 8 November 2012Accepted after revision 28 November 2012Available online 3 January 2013Presented by Jean-Pierre KahaneIn this Note we give a new criterion for the continuity and compactness of compositionoperators acting on α-Bloch spaces.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette Note, nous donnons un nouveau critère pour la continuité et la compacité desopérateurs de composition agissant sur les espaces α-Bloch.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionGiven a parameter α > 0, the α-Bloch space on the unit disk D of the complex plane C, denoted as Bα = Bα(D), consistsof all holomorphic functions f on D such that‖ f ‖α = supz∈D(1 − |z|2)α∣∣ f ′(z)∣∣ < ∞.The above relation defines a seminorm, and the α-Bloch functions form a complex Banach space Bα with the norm ‖ f ‖Bα =| f (0)| + ‖ f ‖α . When α = 1 we get back the classical Bloch space B.A holomorphic function φ from the unit disk D into itself induces a linear operator Cφ , defined by Cφ( f ) = f ◦ φ, wheref ∈ H(D), the space of all holomorphic functions on D, which is equipped with the topology of uniform convergence oncompact subsets of D. Cφ is called the composition operator with symbol φ. Composition operators continue to be widelystudied on many subspaces of H(D) and particularly in Bloch-type spaces.The study of the properties of composition operators on Bloch-type spaces began with the celebrated work of Madiganand Matheson in [2], where they characterized the continuity and compactness of composition operators acting on the Blochspace B. Many extensions of the Madigan and Matheson’s results have appeared (see for instance [3] and a lot of referencestherein). In particular, Xiao in [8] showed the following result:E-mail addresses: mtmalave@udo.edu.ve (M.T. Malavé Ramírez), jcramos@udo.edu.ve (J.C. Ramos Fernández).1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2012.11.01324 M.T. Malavé Ramírez, J.C. Ramos Fernández / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 23–26Theorem 1.1. (See [8].) Given α,β > 0, the composition operator Cφ : Bα → Bβ is compact if and only if φ ∈ Bβ andlim|φ(z)|→1(1 − |z|2)β(1 − |φ(z)|2)α∣∣φ′(z)∣∣ = 0.Recently, many authors have found new criteria for the continuity and compactness of composition operators acting onBloch-type spaces en terms of the n-th power of the symbol φ and the norm of the n-th power of the identity functionon D. The first result of this kind appears in 2009 and it is due to Wulan, Zheng, and Zhu [7], in turn, their result wasextended to α-Bloch spaces by Zhao in [9]. Another criterion for the continuity and compactness of composition operatorson Bloch space is due to Tjani in [5] (see also [6] or more recently [7]), she showed the following result:Theorem 1.2. (See [5].) The composition operator Cφ is compact on B if and only if φ ∈ B andlim|a|→1−‖ϕa ◦ φ‖B = 0,where ϕa is a Möbius transformation from the unit disk onto itself ; that is, ϕa(z) = (a − z)/(1 − az), with z ∈ D.The aim of this Note is to extend the above result to α-Bloch spaces. With this in mind, in the next section we defineand study the properties of certain functions σa which will be the substitutes for automorphisms of the disk ϕa , thus inSection 3, we give a new criterion for the continuity of composition operators Cφ acting on α-Bloch spaces involving thecomposition of the symbol ϕ with the functions σa . Finally, in Section 4, we extend Theorem 1.2 to α-Bloch spaces.2. Certain special functions for α-Bloch spacesThe key to our results lies in considering the following family of functions. For a ∈D fixed, we defineσa(z) =(1 − |a|)((1 − az)−α − 1) (z ∈D).Clearly, for each a ∈ D, the function σa has bounded derivative and for this reason we have that σa ∈ Bα . In fact, it is easyto see that supa∈D ‖σa‖Bα  α2α . Furthermore, it is clear that if 12 < |a| < 1, then∣∣σ ′a(a)∣∣  α4(1 − |a|2)α . (1)Also, we can see that σa goes to zero uniformly on compact subsets of D as |a| → 1− .3. ContinuityNow we can enunciate and prove our first result, this give a new criterion for the continuity of the composition operatorsacting on α-Bloch spaces.Theorem 3.1. The composition operator Cφ is continuous on Bα if and only ifsupa∈D‖σa ◦ φ‖α < ∞. (2)Proof. Let us suppose first that Cφ : Bα → Bα is continuous, then there exists a constant M > 0 such that ‖Cφ( f )‖Bα M‖ f ‖Bα for all f ∈ Bα . In particular, for each a ∈ D we have‖σa ◦ φ‖α  ‖σa ◦ φ‖Bα  M‖σa‖Bα  Mα2α.This shows the relation (2).If condition (2) holds, then there exists a constant L > 0 such that (1 − |z|2)α |σ ′a(φ(z))||φ′(z)|  L for all a, z ∈ D. Takinga = φ(z) we have (1 − |z|2)α |σ ′φ(z)(φ(z))||φ′(z)|  L for all z ∈ D. In particular, using the relation (1), we obtain(1 − |z|2)α(1 − |φ(z)|2)α∣∣φ′(z)∣∣  L (3)whenever 12 < |φ(z)| < 1. Furthermore, since α4 ( 12 )α+1‖φ‖α  ‖σ1/2 ◦ φ‖α  L, then we can see that φ ∈ Bα . Hence, thefunction on the left in the above inequality is bounded when |φ(z)|  12 , then we can assume that the expression (3) isvalid for all z ∈D. This shows that the composition operator Cφ is continuous on Bα . M.T. Malavé Ramírez, J.C. Ramos Fernández / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 23–26 254. CompactnessNow we will to extend the Tjani’s result in [5] to α-Bloch spaces. The following lemma is well known and it is conse-quence of a more general result due to Tjani in [4]:Lemma 4.1. The composition operator Cφ is compact on Bα if and only if given a bounded sequence { fn} in Bα such that fn → 0uniformly on compact subsets of D, then ‖Cφ( fn)‖α → 0 as n → ∞.As a consequence of this lemma, we can show the following result.Theorem 4.2. The composition operator Cφ is compact on Bα if and only if φ ∈ Bα andlim|a|→1−‖σa ◦ φ‖α = 0. (4)Proof. Suppose first that the composition operator Cφ is compact on Bα , let {an} be any sequence in D such that |an| → 1as n → ∞. Then since the sequence {σan } is bounded in Bα and σan → 0 uniformly on compact subsets of D, then as aconsequence of Lemma 4.1 we obtain the expression (4).Conversely, if the relation (4) holds then given ε > 0 there exists r1 ∈ ( 12 ,1) such that ‖σa ◦ φ‖α < ε whenever r1 <|a| < 1. Hence, if z ∈ D satisfies |φ(z)| > r1, then we have‖σφ(z) ◦ φ‖α = supw∈D(1 − |z|2)α∣∣σ ′φ(z)(φ(w))∣∣∣∣φ′(w)∣∣ < ε.In particular, if we put w = z in the above expression, we haveα(1 − |z|2)α4(1 − |φ(z)|2)α∣∣φ′(z)∣∣ (1 − |z|2)α∣∣σ ′φ(z)(φ(z))∣∣∣∣φ′(z)∣∣ < εwhenever |φ(z)| > r1, where we have used the relation (1) in the first inequality. The result follows now by Theorem 1.1. Remark 1. Due to the characterizations for the compactness of composition operators on α-Bloch spaces given by us inTheorem 4.2, Xiao in [8] and by Zhao in [9], we can conclude that for φ ∈ Bα , the limit in (4) and the following limits areequivalents:(i) lim|φ(z)|→1− ( 1−|z|21−|φ(z)|2 )α |φ′(z)| = 0,(ii) limn→∞ nα−1‖φn‖α = 0.A similar result was found by Giménez, Malavé and Ramos-Fernández in [1], but for the composition operator Cφ : B → Bμ ,where the weight μ can be extended to non-vanishing, complex valued holomorphic function that satisfy a reasonablegeometric condition on the Euclidean disk D(1,1). A function f belongs to Bμ if ‖ f ‖μ = supz∈D μ(z)| f ′(z)| < ∞ and μ isa continuous, positive and bounded function defined on D. We include a short and direct proof of the implication (ii) ⇒ (4)(without compactness of composition operator) for the benefit of the reader.Suppose that the condition (ii) holds and let ε > 0, there exists N ∈ N such that nα−1‖φn‖α < ε for all n  N . Hence,we can write‖σa ◦ φ‖α (1 − |a|)N∑n=0bn|a|n∥∥φn∥∥α+ ε(1 − |a|)+∞∑n=N+1n1−αbn|a|n,where bn is the n-th coefficient of the Maclaurin serie of the function H(w) = (1 − w)−α , with |w| < 1. Observe that thefirst sum on the right side of the above inequality tends to zero as |a| → 1− . Furthermore, by the Stirling’s formula we havethat cn = n1−αbn → 1Γ (α) > 0 as n → ∞, hence we can ensure that there exists a constant Kα > 0, depending only on α,such that cn  Kα for all n ∈N. Thus we can conclude that lim|a|→1− ‖σa ◦ φ‖α  εKα which shows (4).References[1] J. Giménez, R. Malavé, J. Ramos-Fernández, Composition operators on μ-Bloch type spaces, Rend. Circ. Mat. Palermo 59 (2010) 107–119.[2] K. Madigan, A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995) 2679–2687.[3] J.C. Ramos-Fernández, Composition operators between μ-Bloch spaces, Extracta Math. 26 (1) (2011) 75–88.[4] M. Tjani, Compact composition operators on some Möbius invariant Banach space, PhD dissertation, Michigan State University, 1996.[5] M. Tjani, Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355 (11) (2003) 4683–4698.26 M.T. Malavé Ramírez, J.C. Ramos Fernández / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 23–26[6] H. Wulan, Compactness of composition operators on BMOA and VMOA, Sci. China Ser. A 50 (7) (2007) 997–1004.[7] H. Wulan, D. Zheng, K. Zhu, Compact composition operators on BMO and the Bloch space, Proc. Amer. Math. Soc. 137 (2009) 3861–3868.[8] J. Xiao, Composition operators associated with Bloch-type spaces, Complex Var. Theory Appl. 46 (2001) 109–121.[9] R. Zhao, Essential norms of composition operators between Bloch type spaces, Proc. Amer. Math. Soc. 138 (2010) 2537–2546.

On a criterion for continuity and compactness of composition operators acting on <i>α</i>-Bloch spaces

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On a criterion for continuity and compactness of composition operators acting on <i>α</i>-Bloch spaces

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