BELLWETHERS OF COMPOSITION OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS. E.

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1 Acta Universitatis Apulensis ISSN: No. 54/08 pp doi: 0.74/j.aua BELLWETHERS OF COMPOSITION OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS E. Wolf Abstract. Let D denote the open unit disk in the complex plane and HD) the set of all analytic functions on D. Now, let v : D 0, ) be a weight, A v the weighted Bergman space generated by the weight v and Hv the weighted Banach space of holomorphic functions f on D such that sup fz) <. An analytic self-map φ of D induces the linear composition operator C φ : HD) HD), f f φ. We investigate under which conditions on the symbol φ and the weight v the operator C φ : A v Hv is a bellwether for boundedness of composition operators where C φ being a bellwether means that C φ acts boundedly if and only if all composition operators act boundedly. 00 Mathematics Subject Classification: 47B38, 47B33 Keywords: bellwethers, composition operators, weighted Bergman spaces. Introduction Let D be the open unit disk in the complex plane and HD) be the collection of all analytic functions on D. An analytic self-map φ of D induces through composition a linear composition operator C φ : HD) HD), f f φ. For many reasons these operators play an important role. One of them is that in the classical setting of the Hardy space H they link operator theory with complex analysis. Moreover they provide a large class of operators which gives many examples on basic operator theoretical questions. For more information on composition operators we refer the reader to the excellent monographs of Cowen and MacCluer [7] and of Shapiro [9]. Now, let v : D 0, ) be a continuous and bounded function. Such a map is 5

2 called a weight. We will study composition operators that act between the weighted Bergman space { ) } A v := f HD), f v, := fz) daz) < D where daz) denotes the normalized area measure and the weighted Banach spaces of holomorphic functions given by { } Hv := f HD); f v = sup fz) <. Endowed with the weighted sup-norm. v these are Banach spaces. They arise naturally in a number of problems related to functional analysis, Fourier analysis, partial differential equations and convolution equations. Later, they became of interest themselves, see [3] and the references therein for further and deeper information. In [6] Bourdon studied composition operators acting on weighted Banach spaces generated by the weights that satisfy the following conditions. We say that v is radial, if = v z ) for every z D. A radial weight is called typical, if v is radial, non-increasing w.r.t. z and satisfies lim z = 0. Taking Bourdon s notation we say that C φ is a bellwether for boundedness of composition operators if for each typical weight v the boundedness of C φ : A v Hv ensures the boundedness of all composition operators acting between A v and Hv. Bourdon analyzed the relation between φ having an angular derivative less than and inducing a bellwether for boundedness of composition operators on typicalgrowth spaces. More precisely, if φ has an angular derivative less than, then C φ is a bellwether. Conversely, if all angular derivatives of φ exceed, C φ is not a bellwether. It is still an open question, if C φ being a bellwether implies that the inducing symbol φ must have an angular derivative less than at some point ξ D. In [] we generalized Bourdon s results to the setting of the unit polydisk using two different settings. In this article we remain in the one-dimensional setting but consider operators acting between A v and Hv. We give a characterization for C φ to be a bellwether and compare our results to those obtained by Bourdon in [6]... Theory of weights. Background and basics In this part of the article we give some background information on the involved weights. A very important role play the so-called radial weights, i.e. weights which 6

3 satisfy = v z ) for every z D. 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 > 0, b) the logarithmic weights = log z )) β, β < 0, c) the exponential weights = e z ) γ, γ > 0. In [8] Lusky studied typical weights satisfying the following two conditions L) v n ) inf n N v n > 0 ) and L) lim sup n v n j ) v n ) < for some j N. In fact, weights having L) and L) are normal weights in the sense of Shields and Williams, see [0]. 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 so-called associated weights. For a weight v its associated weight is given by ṽz) := sup{ fz) ; f HD), f v }, z D. See e.g. [] and the references therein. Associated weights are continuous, ṽ v > 0 and for every z D there is f z HD) 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 D. If v and ṽ are equivalent, we say that v is an essential weight. By [4] condition L) implies the essentiality of v. 7

4 .. 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 D 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 D. ) Obviously such weights are bounded, i.e. for every weight v of this type we can find a constant C > 0 such that sup C. Moreover, we assume additionally that νz) ν z ) for every z D. Now, we can write the weight v in the following way = min{ gλz), λ = }, where g is a holomorphic function on D. 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 D. Conversely, by hypothesis, for every λ D we obtain for every z D 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, Domański, Lindström and Taskinen in [4]. Among other things they proved that in case that v and w are arbitrary weights the boundedness of the operator C φ : H v H w is equivalent to sup 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. 8

5 This was the motivation to study the boundedness of composition operators acting between weighted Bergman spaces and weighted Banach spaces of holomorphic functions. Doing this we obtain the following results which we need in the sequel. Lemma [3], Lemma ). Let = ν z ) for every z D with ν HD) 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 ) The following result is obtained by using the previous lemma and following exactly the proof of [] Theorem.. We gave the full proof in [4] Theorem.. Theorem [4], Theorem.). Let = ν z ) for every z D with ν HD) be a weight as defined in Section.. Then the operator C φ : A v Hv is bounded if and only if sup <. ) φz) )vφz)) Having characterized the boundedness of the composition operator acting between A v and Hv we took the second result of Bonet, Domański, Lindström and Taskinen as a motivation to ask the question: For which weights v are all operators C φ : A v Hv bounded? The answer is as follows: Theorem 3 [4], Theorem 3.). Let = ν z ) for every z D with ν HD) be a weight as defined in Section.. Moreover, let sup <. Then the z ) composition operator C φ : A v Hv is bounded for every analytic self-map φ of D if and only if n n )v n ) inf n N v n > 0. ).4. Geometry of the unit disk Let p D and α p denote the Möbius transformation that interchanges p with 0, i.e. α p z) = z p pz for every z D. As a corollary to the Schwarz lemma, every automorphism α of D is of the form αz) = ξα p z) for every z D, where ξ D and p D. The following facts are well-known: 9

6 . α p = α p.. α p z) = z ) p ) pz. Moreover, we need the following lemma given by Bourdon in [6]. Lemma 4 Bourdon). Suppose that α is a holomorphic automorphism of D. Then for 0 r <, α0) + r max αz) = z =r + α0) r. Next, we need some facts about the behaviour of holomorphic functions in the unit disk. It is well-known that every bounded function in HD) has non-tangential limits at every point of a subset of D having full Lebesgue measure. When f HD) has a non-tangential limit at ξ we denote the value of the limit by fξ). Recall that the holomorphic self-map ϕ of D has angular derivative at ξ D if there is η D such that lim z ξ ϕz) η z ξ =: ϕ z) 3) exists as a complex number, where lim z ξ denotes the non-tangential limit. The limit in 3) is called the angular derivative of ϕ at ξ. Furthermore, for our investigations we need the Julia-Carathéodory-Theorem as well as some of its consequences. For more details including the proofs of the results listed below we refer the reader to [6]. Theorem 5 Julia-Carathéodory-Theorem). Suppose that ϕ is a holomorphic selfmap of D and ξ D. The following are equivalent: a) there is η D with ϕ ξ) = lim z ξ ϕz) η z ξ <. b) ϕ and ϕ have finite non-tangential limits at ξ and ϕξ) = η has modulus. c) lim inf z ξ ϕz) z = δ <. Moreover, with the given assumptions, δ > 0, ϕ ξ) = lim z ξ ϕ z) = ηξδ and ϕz) finally lim z ξ z = ϕ ξ). Corollary 6. Suppose that ϕ has no angular derivatives having modulus. Then there is a positive number r < such that ϕz) < z whenever r < z <. 30

7 Lemma 7 Julia-Carathéodory inequality). Suppose that lim inf z ξ ϕz) z = δ < and η is the non-tangential limit of ϕ at ξ, then for all z D, η ϕz) ξ z δ ϕz) z. Another important classical result which plays a main role in this article is the Denjoy-Wolff-Theorem. The n-th iterate of an analytic self-map φ of D is denoted by φ n. Theorem 8 Denjoy-Wolff Theorem). Let φ be an analytic self-map of D. If φ is not the identity and not an automorphism with exactly one fixed point, then there is a unique point p D such that φ n ) n converges to p uniformly on the compact subsets of D. 3. Results Theorem 9. Let v be a weight as defined in Section.. Moreover we assume that v satisfies L) and sup <. Then the following are equivalent: z ) a) C φ : A v H v is bounded for every analytic self-map φ of D. b) C ψa : A v H v with is bounded for every a 0, ). ψ a z) = az + a for every z D c) inf n N a n+ ) )v a n+ ) v a n ) > 0 for every a 0, ). d) inf n N a n+ ) )v a n+ ) v a n ) > 0 for some a 0, ). e) inf t 0,] at) )v at) v t) > 0 for some a 0, ). f) C α : A v H v is bounded for some automorphism α of D with α0) 0. g) C α : A v H v is bounded for every automorphism α of D. 3

8 Proof. The implication a) = b) is trivial. Next, we show that c) follows from b). To do this, we fix a 0, ). Then obviously ψa n 0) = a n. Since the operator C ψa is bounded, we can find M > 0 such that sup ψ a z) )vψ a z)) <. Now, we pick z = ψ n a 0), n N arbitrary, and obtain using the inequality above v a n ) a n+ ) )v a n+ ) < M which is equivalent to a n+ ) )v a n+ ) v a n ) > M > 0. The implication c) = d) is obvious. Thus, we proceed with showing that e) a follows from d). Here, we suppose that β := inf n+ ) )v a n+ ) n N v a n ) > 0. We have to distinguish the following cases: First, we assume that a t. Then 0 < γ := a ) )v a ) v0) at) )v at) v t). Now, if a k+ t < a k we obtain at) )v at) v t) = ak+ ) )v a k+ ) v a k+ ) ak+ ) )v a k+ ) v a k ) v a k+ ) v a k ) > δβ > 0 v a where inf k+ ) n N > δ since v has L). v a k ) Next, we prove that e) implies f). First, we suppose that e) holds: at) λ := inf )v at) t 0,] v t) > 0. Now, let p = a +a so that p is positive and p +p = a. We show that the automorphism α pz) = z p pz induces a bounded composition operator. Note that α p 0) 0 since p 0. Using Lemma and the fact that 3

9 v is non increasing and radial we find that for z D: α p z) )vα p z)) v p+ z +p z ) v z )) = )) v p+ z +p z = v p) z ) +p z v z ) p+ z +p z ) ) ) p+ z +p z v z )) ) p) z ) ) +p z v z )) ) v p +p z ) ) p +p z ) v z )) = v a z )) a z ) ) λ. Hence the operator C αp must be bounded. Next, we show that f) implies g). We suppose that f) holds. C α : A v Hv is bounded for α = ξα p where ξ = and p D\{0}. Since C α is bounded, we have sup vαz)) αz) ) = sup vα p z)) α p z) ) < since v is a radial weight. Thus, C αp is also bounded. Note that α p has Denjoy-Wolff point p p D. Hence αp n 0) as n. Let τz) = ξ z q qz be an arbitrary disk automorphism. We choose the positive integer n such that αp n 0) > q and let s = αp n 0). We know that C α n p is bounded. Thus, there is a constant C such that for every z D: vαp n z)) αp n z) ) C. Let z D be arbitrary, let r = z and choose z 0 with z 0 = r such that αp n z 0 ) = s+r x+r +sr. Since x +xr is increasing for r < x <, q < s and v and z are non-increasing vτz)) τz) ) ) v q +r + q r q +r + q r 33 ) = ) v s+r +sr s+r +sr ) vz 0 ) vα n z 0 )) α n z 0 ) ) C

10 and it follows that C τ is bounded. Last, we prove that g) implies a). Let φ : D D be an analytic map. Let p = φ0). Then C αp is bounded by hypothesis. Next, we consider ψ = α p φ. Then C ψ is bounded since ψ0) = α p φ0)) = α p 0) = 0. Then, by the Schwarz Lemma we have that ψz) z for every z D and thus sup ψz) )vψz)) sup z ) = sup z ) < by hypothesis. Now, C φ = C ψ C α p operators. must be bounded as composition of bounded Next, we want to compare condition L) with the following condition n ) )v n ) inf n N v n ) First, we reformulate equation 4) and obtain > 0. 4) n v n ) inf n N v n ) > 0. 5) Now, we will show that L) does not imply 5): To do this we consider the standard weight = z for every z D. It is well-known that this weight satisfies L). But it does not enjoy 5) as the following calculation shows: n v n ) inf n N v n ) = inf n N n n n = inf n 3 = 0. n N Conversely, L) does not follow from 5). We pick the weight = z )e z for every z D. Then v does not satisfy L) since for every n N we have v n ) v n ) = n e n+ n e n = 4 e n 0 if n. Hence L) is not satisfied. It remains to show that 5) is fulfilled. For every n N: n v n ) v n ) = n n e n+ n = e n 4 > 0. Thus, the claim follows. 34

11 Example. a) Let v α z) = z ) α, α. Then v α satisfies all the required conditions: i) It is well-known, that v α has condition L). ii) Moreover, sup z ) = sup z ) α + z sup z ) α since α 0 by hypothesis. iii) inf n v n ) n N = inf n n ) α v n ) n N = inf n ) α n N α )n α = 4 > 0 z ) b) We consider the weights v α,β z) = α, α 3, β > 0. These weights log z )) β also satisfy all the required conditions. i) v α,β has the condition L) since both, the standard and the logarithmic weights, have this condition. ii) sup z ) = sup z ) α <, since α + z ) log z )) β > 0. iii) inf n v n ) n N = inf n α ) α +n log ) β v n ) n N > 0 by hypthesis. +n+) log ) β c) Finally, we look at the exponential weights v γ z) = e z ) γ, γ > 0. > 0 since α i) It is well-known that the exponential weights do not satisfy condition L). ii) We have sup z ) = sup e z ) γ z ) <. iii) inf n N n v n ) v n ) = inf n N n e γn e γn = inf n N n = 0. Theorem 0. Let v be a weight as defined in Section.. Moreover we assume that v satisfies L) and sup z ) <. Suppose that C φ : A v Hv is bounded and that φ has an agular derivative less than at some point ξ D. Then every composition operator acting between A v and Hv is bounded. 35

12 Proof. Applying the Julia-Carathéodory theorem we can find η D such that φ has nontangential limit η at ξ. By Theorem the composition operators C ηz and C ξz are bounded since C ηz = sup C ξz = sup ηz )vηz) ξz )vξz) = sup z ) < resp. = sup z ) < by hypothesis. Hence the composition operator with the symbol ψ defined by ψz) = ηφξz) is also bounded. Moreover, ψ) = ηφξ) = and ψ ) = ηφ ξ) = φ ξ) <. Hence ψ has Denjoy-Wolff point. We put a := ψ ) <. Using the Julia- Carathéodory inequality inductively we obtain for every n N ψ n z) z ψ n an z) z. Hence, for every n N: ψ n 0) a n. Thus, for every n N we arrive at ψ n 0) = gn)a n where g is a positive bounded function on N. Since ψ n 0)) n converges non tangentially to we can apply the Julia-Carathéodory theorem to conclude that ψ lim n+ 0) gn+) n ψ n 0) = a or, equivalently, lim n gn) =. For each n N set e n = n + log a gn)). Then ) gn + ) e n+ e n = + log a. gn) )) Since log gn+) a gn) tends to 0 as n we can find K N such that e n+ e n n for every n K. Because C ψ is bounded, C ψ ) 3 = C ψ 3 is also bounded and there is a constant M such that Thus, for every n N, ψ 3 z) )vψ 3 z)) M for every z D. M v ψ n 0) )) ψ n+3 0) ) )v ψ n+3 0) ) = v a en ) a e n+3 ) )v a e n+3 ). 36

13 Let j N such that j K + log a gk)) = e K. Now, let n 0 K be the greatest positive integer such that e n0 j. Note that e n0 + > j. Since e n+ e n for every n K we have that e n0 +3 > e n0 + j +. Since v and the standard weights are decreasing we arrive at v a j ) a j+ ) )v a j+ ) v a en 0 ) a e n 0 +3 ) )v a e n 0 +3 ) M. Since j e K is arbitrary, it follows that v a n+ ) a n+ ) ) inf n N v a n > 0. ) Thus, all composition operators acting between A v and H v are bounded. References [] J.M. Anderson, J. Duncan, Duals of Banach spaces of entire functions, Glasgow Math. J. 3, 990), 5-0. [] K.D. Bierstedt, J. Bonet, J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math ), [3] K.D. Bierstedt, R. Meise, W.H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 7 98), [4] J. Bonet, P. Domański, M. Lindström, J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Serie A) ), 0-8. [5] J. Bonet, M. Friz, E. Jordá, Composition operators between weighted inductive limits of spaces of holomorphic functions, Publ. Math. 67, ), [6] P.S. Bourdon, Bellwethers for boundedness of composition operators on weighted Banach spaces of analytic functions, J. Aust. Math. Soc ), [7] C. Cowen, B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 995. [8] W. Lusky, On the structure of Hv 0 D) and hv 0 D), Math. Nachr ), [9] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer,

14 [0] A.L. Shields, D.L. Williams, Bounded projections, duality, and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 99/ ), [] E. Wolf, Bellwethers of composition operators for typical growth spaces, Asian- European J. Math. 7, 3 04), 3pp [] E. Wolf, Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Rev. Mat. Complut., 008), [3] E. Wolf, Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Bull. Acad. Stiinte Repub. Mold. Mat. 04), [4] E. Wolf, Composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Irish Math. Soc. Bull ), Elke Wolf Institut für Mathematik, Universität Paderborn, Paderborn, Germany lichte@math.uni-paderborn.de 38