COMPOSITION OPERATORS ON ANALYTIC WEIGHTE HILBERT SPACES K. KELLAY Abstract. We consider composition operators in the analytic weighted Hilbert space. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established.. Composition operators on the Hardy space In this expository paper, we consider composition operators acting on the weighted Hilbert spaces of analytic functions on the unit disc. This paper is based on [5, 6]. A comprehensive study of composition operators in function spaces and their spectral behavior could be found in [3,, 5]. See also [, 8, 9, 3,, 6] for a treatment of some of the questions addressed in this paper. Let ϕ be an analytic map of the unit disk of the complex plane into itself. We define the composition operator C ϕ by C ϕ (f) = f ϕ for f analytic in. We recall that the Hardy space H 2 consists of those analytic functions f : C for which the norm ( ) /2 f 2 = sup f(rζ) 2 dζ 0<r< 2π T is finite. Every function f H 2 has non tangential limits almost everywhere on T. We denote by f(ζ) the non tangential limit of f at ζ T if it exists... Boundedness. It is a consequence of Littlewoods subordination principle [, 5] that every composition operator C ϕ restricts to a bounded operator on H 2. Furthermore C ϕ (f) 2 2 + ϕ(0) ϕ(0) f 2 2. 2000 Mathematics Subject Classification. 7B38, 30H05, 30C85, 7A5. Key words and phrases. Weighted analytic space, composition operators, generalized Nevanlinna counting function, capacity. Research partially supported by ANR ynop and GR 2753 AFHA.
2 KELLAY.2. Compactness. Shapiro gives in [0] a complete characterization of compact composition operators in the Hardy space in term of Nevanlinna counting functions Let ϕ Hol() such that ϕ(). The Nevanlinna counting function is defined for every z \ {ϕ(0)} by N ϕ (z) = log w. ϕ(w)=z w where each pre-image w is counted according to its multiplicity. Note that N ϕ (z) = O( log z ). Shapiro s characterization is as follow C ϕ is compact in H 2 N ϕ (z) lim z log(/ z ) = 0. Given E T, we write E for the Lebesgue measure of E. Note also that if C ϕ is compact, then the level set of ϕ E ϕ () = {ζ T : ϕ(ζ) = } has Lebesgue measure zero. Indeed, suppose that E ϕ () has positive measure, we will show that C ϕ is not compact. Since the sequence (z n ) n converges weakly to zero and C ϕ (z n ) 2 2 = ϕ n 2 2 = ϕ(ζ) 2n dζ 2π T ϕ(ζ) 2n dζ 2π = E ϕ() 2π {ζ T : ϕ(ζ) =} > 0, C ϕ (z n ) 2 not converge to zero. Hence C ϕ is not compact..3. Hilbert-Schmidt class. For s (0, ), the level set E ϕ (s) of ϕ is given by E ϕ (s) = {ζ T: ϕ(ζ) s}. One can completely describe the membership of C ϕ in the Hilbert-Schmidt class in terms of the size of the level sets of the inducing map ϕ. Indeed, C ϕ is Hilbert-Schmidt in H 2 if and only if n 0 ϕ n 2 2 = 2π T dζ ϕ(ζ) 2 <. Let f be a measurable function on T, the associated distribution function m f is given by m f (λ) = {ζ T : f(ζ) > λ}, λ > 0.
T COMPOSITION OPERATORS 3 It then follows that C ϕ is in the Hilbert-Schmidt class of H 2 if and only if dζ = m ϕ(ζ) 2 ( ϕ 2 ) (λ) dλ 0 E ϕ (s) ds <. ( s) 2 It was shown by Gallardo Gonzalez in [9] that there exists a a mapping ϕ : such that C ϕ is compact in H 2, and that the level set E ϕ () has Hausdorff measure equal to one. Let A() denote the disc algebra. In [6] with El-Fallah, Shabankhah and Youssfi we obtain the following result Theorem.. Let E be a closed subset of T with Lebesgue measure zero. There exists a mapping ϕ :, ϕ A() such that C ϕ is a Hilbert-Schmidt operator on H 2 and that E ϕ () = E. 2. Composition operators on weighted Hilbert spaces of analytic functions Given a positive integrable function ω C 2 [0, ), we extend it by ω(z) = ω( z ), z, and call such ω a weight function. We denote by H ω the weighted Hilbert space consisting of analytic functions f on such that f 2 H ω = f(0) 2 + f (z) 2 ω(z)da(z) <. where da(z) = dxdy/π stands for the normalized area measure in. A simple computation shows that f(z) = n=0 a nz n H ω, if and only if where ω 0 = and f 2 H ω = n 0 a n 2 w n <, ω n = 2n 2 r 2n ω(r)dr, n. 0 Let α >, ω α (r) = ( r 2 ) α and denote H ωα by H α. The Hardy space H 2 can be identified with H. The irichlet space α is precisely H α for 0 α < and H 0 corresponds to classical irichlet space. Finally, the Bergman spaces A 2 α() := Hol() L 2 (, ( z 2 ) α da(z)) can be identified with H α+2. From now we assume that the weight ω satisfies the following properties (W ) ω is non-increasing, (W 2 ) ω(r)( r) (+δ) is non-decreasing for some δ > 0, (W 3 ) lim ω(r) = 0. r Furthermore, we assume that one of the two properties of convexity is fulfilled (W (I) ) ω is convex and lim ω (r) = 0. r (W (II) ) ω is concave.
KELLAY Such a weight function is called admissible. If ω satisfies conditions (W ) (W 3 ) and (W (I) ) (resp. (W (II) )), we shall say that ω is (I)-admissible (resp. (II)-admissible). Note that (I)-admissibility corresponds to the case H 2 H ω A 2 α() for some α >, whereas (II)-admissibility corresponds to the case H ω H 2. The weight ω 0 = is not an admissible weight, so the results of this section do not apply to the irichlet space. The Nevanlinna counting functions play a key role here (see [7] for recent results on this topic). Let ϕ Hol() such that ϕ(). The generalized counting Nevanlinna function associated to ω is defined for every z \ {ϕ(0)} by N ϕ,ω (z) = ω(a). ϕ(a)=z a Let us point out two crucial properties the generalized counting Nevanlinna function : If f is positive mesurable function on and ϕ is a holomorphic self-map on, then (f ϕ)(z) ϕ (z) 2 w(z)da(z) = f(z)n ϕ,w (z)da(z). () If ω is an admissible weight and ϕ is a holomorphic self-map of. Then the generalized Nevanlinna counting function satisfies the sub-mean value property, that is for every r > 0 and for every z such that (z, r) \ (0, /2) N ϕ,α (z) 2 N r 2 ϕ,α (ζ)da(ζ). (2) ζ z <r If ϕ is a holomorphic map on the unit disk into itself, it is an easy consequence of Littlewood s subordination principle that the composition operator with ϕ, induces a bounded operator C ϕ on H ω for (I) admissible weight ω. In [5] with Lefèvre, see also [], we obtain the following results. For the case of (II) admissible weight we have Theorem 2.. Let ω be a (II) admissible weight and ϕ H ω. Then C ϕ is bounded on H ω if and only if N ϕ,ω (z) sup < (3) z < ω(z) The characterization of compactness in the weighted Bergman spaces H α+2 is given by ϕ(z) 2 C ϕ is compact in H α+2 lim = +. z z 2 [0, Theorem 2.3, Corollary 6.] or []. By the classical Julia-Caratheodory theorem (see []), this last condition is equivalent to nonexistence of the angular derivative of ϕ at any point ξ T. The following theorem obtained in [5] generalizes the previous on Hardy and Bergman spaces. Theorem 2.2. Let ω be an admissible weight and ϕ H ω. Then C ϕ is compact on H ω if and only if N ϕ,ω (z) lim = 0 () z ω(z)
COMPOSITION OPERATORS 5 In particular, Theorem 2.2 asserts that C ϕ is compact on α := H α for 0 < α < if and only if N ϕ,α (z) := ( w 2 ) α = o(( z 2 ) α ). ϕ(w)=z The characterization of compact composition operators on the irichlet spaces α in terms of Carleson measures for the Bergman spaces A 2 α can be found in [3, 3, 6]. A positive Borel measure µ given on is called a Carleson measure for the Bergman space A 2 α if the identity map i α : A 2 α L 2 (µ) is a bounded operator. Such a measure has the following equivalent properties (see [2] Theorem.2 ): A positive Borel measure µ is a Carleson measure for A 2 α if and only if sup( λ 2 ) 2+α λ dµ(z) < sup λz 2(2+α) I T for any subarc I T with arclengh I, and the Carleson box S(I) = {z : z/ z I, I z < }. µ(s(i)) <, I 2+α The measure µ is called vanishing (or compact) Carleson measure for A 2 α if the identity map i α : A 2 α L 2 (µ) is a compact operator. This happens if and only if lim ( λ 2 ) 2+α λ dµ(z) = 0 lim λz 2(2+α) I 0 µ(s(i)) = 0. I 2+α Therefore, as a consequence of Theorem 2. and Theorem 2.2, C ϕ is bounded in α sup N I T I 2+α ϕ,α (z) da(z) < ; S(I) C ϕ is compact in α lim N I 0 I 2+α ϕ,α (z) da(z) = 0. That is C ϕ is bounded on α if and only if N ϕ,α (z)da(z) is a Carleson measure for A 2 α and C ϕ is compact on α if and only if N ϕ,α (z)da(z) is a vanishing Carleson measure for A 2 α. Note that by the change of variable formula (), J α (λ) := ( λ 2 ) α+2 ϕ (z) 2 λϕ(z) da α(z) 2(α+2) = ( λ 2 ) 2+α S(I) N ϕ,α (z) da(z). λz 2(2+α) Hence, we can also characterize, boundedness and compactness of C ϕ on α, 0 < α <, by sup λ J α (λ) < and lim λ J α (λ) = 0. The Proposition 3. is devoted to the case where α = 0 corresponding to the classical irichlet space.
6 KELLAY 3. composition operators on the irichlet space We first give the following result, which was established in a general case in [3,, 5]. For the sake of completeness, we give here a simple proof Proposition 3.. Let ϕ : such that ϕ. Then (a) C ϕ is bounded in sup( λ 2 ) 2 ϕ (z) 2 da(z) <. λ λϕ(z) (b) C ϕ is compact in lim ( λ 2 ) 2 ϕ (z) 2 da(z) = 0. λ λϕ(z) Proof. (a). We only have to prove the converse; since the necessary condition easily follows from the fact that z ( λ 2 ) is bounded (uniformly relatively to λ ). Without ( λz) loss of generality, we may assume that ϕ(0) = 0. Let f C ϕ (f) 2 = f(ϕ(0)) 2 + ϕ (z) 2 f (ϕ(z)) 2 da(z) ( f(0) 2 + ϕ (z) 2 f (λ) 2 ) λϕ(z) da(λ) da(z) ( = f(0) 2 + f (λ) 2 ϕ (z) 2 ) λϕ(z) da(z) da(λ) c f 2. (b). We only have to prove the converse. Assume that the limit is equal to zero and let (f n ) n be a sequence which converges weakly to 0 in. Since f n 0 uniformly on compact sets, it follows the proof of part (a) and for r close to that C ϕ (f n ) 2 f n (0) 2 f n(λ) 2 ( λ 2 ) 2 ϕ (z) 2 da(z) da(λ) r λϕ(z) + f n(λ) 2 ( λ 2 ) 2 ϕ (z) 2 \r λϕ(z) da(z) da(λ) 0 as n. For f, the irichlet integral of f is given by (f) = f (z) 2 da(z). The following result is an immediate consequence of Proposition 3.. Corollary 3.2. Let ϕ : such that ϕ. (a) If sup (ϕ n ) <, then C ϕ is bounded; n
(b) If lim n (ϕ n ) = 0, then C ϕ is compact. COMPOSITION OPERATORS 7 Proof. Both (a) and (b) follow from the following inequality: ( λ 2 ) 2 ϕ (z) 2 λϕ(z) da(z) c( λ 2 ) 2 ϕ (z) 2 ( λ 2 ϕ(z) 2 ) da(z) c ( λ 2 ) 2 (n + ) 3 λ 2n ϕ (z) 2 ϕ n (z) 2 da(z) n 0 = c ( λ 2 ) 2 n 0 ( + n) λ 2n (ϕ n+ ) c lim sup (ϕ n+ ). n We are interested herein in describing the spectral properties of the composition operator C ϕ, such as compactness and Hilbert-Schmidt class membership, in terms of the size of the level set of ϕ. In order to state our second result, we introduce the notion of logarithmic capacity. Given a (Borel) probability measure µ on T, we define its energy by µ(n) 2 I(µ) =. n For a closed set E T, its logarithmic capacity cap(e) is defined by n= cap(e) := / inf{i(µ) : µ is a probability measure on E}. Since the irichlet space is contained in the Hardy space H 2 (), every function f has non-tangential limits f almost everywhere on T. In this case, however, more can be said. According to well-known result of Beurling [2], for each function f then the radial limits f(ζ) = lim r f(rζ) exists q.e on T, that is cap ( {ζ T : f(ζ) does not exist} ) = 0. The weak type inequality for capacity [2] states that, for f and t f 2, cap({ζ : f(ζ) t}) 6 f 2 t 2. As a result of this inequality, we see that if lim inf ϕ n = 0, then cap(e ϕ ()) = 0. Indeed, since E ϕ () = E ϕ n(), the weak capacity inequality implies that cap(e ϕ ()) = cap(e ϕ n()) 6 ϕ n 2. Now let n, we can conclude. Hence, in particular, if the operator C ϕ is in the Hilbert- Schmidt class in, n 0 ϕn 2 <, then cap(e ϕ ()) = 0. This result was first obtained by Gallardo-Gutiérrez and González [8] using a completely different method. Theorems 3.3 and 3. give quantitative versions of this result. The following theorem is similar to the Theorem. for the irichlet space
8 KELLAY Theorem 3.3. If C ϕ is a Hilbert-Schmidt operator in, then 0 cap(e ϕ (s)) s The following theorem shows that condition (5) is optimal. log ds <. (5) s Theorem 3.. Let h : [, + [ [, + [ be a function such that lim x h(x) = +. Let E be a closed subset of T such that cap(e) = 0. Then there is ϕ A(), ϕ() such that : () E ϕ () = E; (2) C ϕ is in the Hilbert Schmidt class in ; cap(e ϕ (s)) e ( ) (3) log s s h ds = +. s 0 Acknowledgement. The author is grateful to Pascal Lefèvre for his valuable remarks and suggestions. References [] U. Boettger, Kompositionsoperatoren auf gewichteten Bergmanrumen. issertation Karlsruhe (998). http://digbib.ubka.uni-karlsruhe.de/volltexte/20798. [2] L. Carleson, Selected Problems on Exceptional Sets. Van Nostrand, Princeton NJ, 967. [3] C.C. Cowen, B.. MacCluer, Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL, 995. [] B. MacCluer, J. H. Shapiro. Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canadien J. Math 38 (986) 878 906 [5] K. Kellay, P. Lefèvre, Compact composition operators on weighted Hilbert spaces of analytic functions. J. Math. Anal. App. 386 (2) (202) 78-727 [6] O. El-Fallah, K. Kellay, M. Shabankhah, H. Youssfi, Level sets and Composition operators on the irichlet space. J. Funct. Anal 260 (20) 72-733. [7] P. Lefèvre,. Li, H. Queffélec, L. Rodriguez-Piazza, Nevanlinna counting function and Carleson function of analytic maps. Math Ann. 35 (2) (20) 305-326. [8] E. A. Gallardo-Gutiérrez and J. M. González, Exceptional sets and Hilbert-Schmidt composition operators. J. Funct. Anal. 99 (2003) 287-300. [9] E. A. Gallardo-Gutiérrez, J. M. González, Hausdorff measures, capacities and compact composition operators. Math. Z., 253 (2006), 63 7. [0] J. H. Shapiro, The essential norm of a composition operator. Annals of Math. 25 (987) 375 0. [] J. H. Shapiro, Composition operators and classical function theory, Springer Verlag, New York 993. [2]. Stegenga, Multipliers of the irichlet space, Illinois. J. Math. 2 (980), 3 39. [3] M. Tjani, Compact composition operators on Besov spaces. Trans. Amer. Math. Soc., 355 (2003), no., 683 698. [] K.-J. Wirths, J. Xiao, Global integral criteria for composition operators. J. Math. Anal. Appl., 269 (2002), 702 75. [5] Zhu, K., Operator theory in function spaces. Monographs and textbooks in pure and applied mathematics, 39, Marcel ekker, Inc (990). [6] N. Zorboska, Composition operators on weighted irichlet spaces. Proc. Amer. Math. Soc., 26 (998), no. 7, 203 2023.
COMPOSITION OPERATORS 9 IMB, Université Bordeaux I, 35 cours de la Libration, 3305 Talence, France E-mail address: Karim.Kellay@math.u-bordeaux.fr