Composition operators on the weighed Bergman-Nevanlinna classes

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 COMPOSITION OPERATORS ON THE WEIGHTED BERGMAN-NEVANLINNA CLASSES SOMDATT D. SHARMA AND AJAY K. SHARMA Abstract. In this paper we use an α-Carleson measure and a vanishing Carleson measure to characterize bounded and compact composition operators on weighted Bergman-Nevanlinna spaces. 1. Introduction Let Dbe the open unit disk in the complex plane C. For each analytic selfmap φof D, the composition operator is defined by Cφf=f◦φfor all fanalytic on D. It is well known that Cφis a bounded linear operator on the Hardy spaces Hpof the unit disk, 0 < p < ∞, as well as on the weighted Bergman spaces Ap αof the unit disk, 0 < p < ∞. Compact composition operators are among the most studied composition operators on these spaces. The study of compact composition operators on H2of Dwas initiated by H. J. Schwartz [S] in his unpublished thesis in the late sixties. He proved that if Cφis compact, then |φ∗|<1 a.e. on the unit circle. In other words, Cφis not compact whenever the set {|φ∗|= 1}has positive measure. Schwartz also proved that this condition is not sufficient by showing that the composition operator induced by φ(z) = 1+z 2 is not compact, even though the range of φtouches the unit disk just at one point. The complete characterization of φfor which Cφis compact on H2have been given by Shapiro [Sh1-2] and McCluer [Mac]. As operators on the Nevanlinna class composition operators were first studied by Masri in his thesis [Mas], where he obtained several necessary conditions 2000 Mathematics Subject Classification. 47B33, 46E22. Revised December 17, 2000. Supported by C.S.I.R. grant ]F.No.9/100(100)2002EMR −1. 123 124 S. D. SHARMA AND A. K. SHARMA and sufficient conditions on φfor the operator Cφto be compact. The compactness of Cφas an operator on the Nevanlinna classes Nand Nphas been studied by Choa and Kim, see [ChK1] and [ChK2]. Our goal in the present work is to characterize those holomorphic self-maps φof Dthat induce compact composition operators on the weighted Bergman-Nevanlinna class A0 α. Our criterion provides a complete characterization of those φfor which Cφis compact on weighted Bergman-Nevanlinna classes. We will show that, like in the case of Hardy spaces and weighted Bergman spaces, every analytic self-map φof D induces a bounded composition operator on the weighted Bergman-Nevanlinna class A0 αand that Cφis compact on the Bergman-Nevanlinna class A0 αif and only if it is compact on any of the weighted Bergman space Ap α, 0 < p < ∞. It is known that Cφis compact on the Nevanlinna class Nif and only if it is compact on H2[ChK1] and that for an arbitrary φthe compactness problem for Cφon Hpspaces is quite different from the one on weighted Bergman spaces [MaS]. MacCluer and Shapiro [MaS] gave a nice example of analytic self-map of D, which induces a compact composition operator on weighted Bergman space, but does not induce a compact composition operator on the Hardy spaces. As a matter of fact, they established the existence of an inner function φ(holomorphic on Dwith modulus ≤1 everywhere on Dand radial limits of modulus 1 almost everywhere on ∂D) such that Cφis compact on Ap αfor all 0<p<∞and α > −1. However, it is well known that no inner function can induce a compact composition operator on any of the Hpspaces [MaS] and, therefore, the space N. In fact, they cited the following example. Example 1.1. Let φ(z) = exp Z∂D z+ζ z−ζdµ(ζ), where µis a Borel measure on ∂Dthat is singular with respect to linear Lebesgue measure and Z∂D dµ(ζ) |ζ−ω|2=∞ at every ω∈∂D. Then φis a singular inner function which induces compact composition operator on Ap αbut not on the Hardy spaces Hp. 2. Preliminaries Let H(D) denote the space of all holomorphic functions on the unit disk Dof the complex plane C. Let dA(z) be the area measure on Dnormalized so that the area of Dis 1. For each α∈(−1,∞), we set dνα(z) = (α+1)(1−|z|2)αdA(z), z∈D. Then dναis a probability measure on D. For 0 < p < ∞the weighted Bergman space Ap αis defined as Ap α={f∈H(D) : ||f||p,α =µZD |f(z)|pdvα(z)¶1/p <∞}. COMPOSITION OPERATORS ON WEIGHTED BERGMAN-NEVANLINNA CLASSES 125 Note that ||f||p,α is a true norm only if 1 ≤p < ∞. When 0 < p < 1, Ap αis an F-space with respect to the translation invariant metric defined by dp(f, g) = ||f−g||p p,α. The growth restrictions of functions in the Bergman space is essential in our study. To this end, the following sharp estimate will be useful. Lemma 2.1 [HKZ] Let f∈Ap α. Then for every zin D, we have |f(z)| ≤ ||f||p,α (1 − |z|2)(2+α)/2 with equality if and only if fis a constant multiple of the function ka(z) = µ1− |z|2 (1 −az)2¶ 2+α p . It can be easily shown that ||ka||p p,α ≈1 with constant depending only on αand p[Sh1, p. 400]. The weighted Bergman- Nevanlinna class A0 αis defined by A0 α={f∈H(D) : ||f||0,α =ZD log+|f(z)|dνα(z)<∞}, where log+x= max(log x, 0). The space A0 αappears in the limit as p→0 of the weighted Bergman space Ap α, in the sense of lim p→0 tp−1 p= log+t, 0< t < ∞. Of course, we are abusing of the term norm since it fails to satisfy the properties of a norm, but in this case (f, g)→ ||f−g||0,α defines a translation invariant metric on A0 αand this turns A0 αinto a complete metric space. Obviously, the inequality log+x≤log(1 + x)≤1 + log+x, x ≥0 implies that f∈A0 αif and only if ZD log(1 + |f(z)|)dνα(z)<∞ for fholomorphic on D. α-Carleson measures. For ζ∈∂Dand 0 < δ < 2, let S(δ, ζ) = {z∈D: |z−ζ|< δ}. A positive Borel measure µon Dis called α-Carleson measure if sup δ>0 sup ζ∈∂D µ(S(δ, ζ)) δα+2 <∞, and a vanishing Carleson measure if lim δ→0sup ζ∈∂D µ(S(δ, ζ)) δα+2 = 0. 126 S. D. SHARMA AND A. K. SHARMA We use the sets S(δ, ζ) as the Carleson sets along with a more convenient choice of pseudo-hyperbolic disks. We now incorporate a few lines from Axler’s paper [Ax] for the sake of a more self-contained exposition. For w∈D, let τwbe the function defined by τw(z) = w−z 1−wz for z∈D. The function τwis an automorphism of D. For wand zin D, the pseudo-hyperbolic distance dbetween wand zis defined by d(w, z) = |τw(z)|. For 0 < r < 1 and w∈D, denote by D(w, r) the disk whose pseudo-hyperbolic center is wand whose pseudo-hyperbolic radius is r, that is, D(w, r) = ½z∈D:¯¯¯¯ w−z 1−wz ¯¯¯¯ < r¾. Since τwis a linear fractional transformation, the pseudo-hyperbolic disk D(w, r) is also a Euclidean disk. Except for the special case D(0, r) = rD, the Euclidean center and Euclidean radius of D(w, r) do not coincide with its pseudohyperbolic center and pseudo-hyperbolic radius. The Euclidean centre and Euclidean radius of D(w, r) are 1−r2 1−r2|w|2wand 1− |w|2 1−r2|w|2r, respectively. For w∈D, it is easy to verify that τwis its own inverse under composition (τw◦τw)(z) = zfor all z∈D. Another simple calculation shows that τwpreserves pseudo-hyperbolic distances, that is, d(λ, z) = d(τw(λ), τw(z)) for all λ, z ∈D. Thus τwmaps a pseudo-hyperbolic disk centered at the point λto the pseudo-hyperbolic disk centered at τw(λ): τw(D(λ, r)) = D(τw(λ), r) for all λ∈Dand r∈(0,1). By |D(w, r)|Awe denote the area of D(w, r). Lemma 2.2. [Ax] If w∈Dand 0< r < 1, then (i) |D(w, r)|A=πr2(1 − |w|2)2(1 −r2|w|2)−2. (ii) inf ½(1 − |w|2)2 |1−wz|4:z∈D(w, r)¾=(1 −r|w|)4 (1 − |w|2)2. (iii) sup ½(1 − |w|2)2 |1−wz|4:z∈D(w, r)¾=(1 + r|w|)4 (1 − |w|2)2. We also have |D(w, r)|A≈(1 − |w|2)2≈(1 − |z|2)2≈ |D(z, r)|A, COMPOSITION OPERATORS ON WEIGHTED BERGMAN-NEVANLINNA CLASSES 127 for z∈D(w, r), where ≈means that the two quantities are bounded above and below by constants independent of w. Also for each D(w, r),there is a ζ∈∂D so that D(w, r)⊂S(δ, ζ)for δ≈1− |w|and for fixed r,0< r < 1, (α+ 1) ZD(w,r) (1 − |z|2)αdA(z)≈(1 − |w|2)α+2. Lemma 2.3. [Ax] For a fixed r,0< r < 1, there exists a positive constant C depending upon rsuch that |f(w)|p≤C |D(w, r)|AZD(w,r) |f(z)|pdA(z) for fanalytic in Dand w∈D. Lemma 2.4. [Ax] Let 0< r < 1. Then there is a sequence {an}in Dand a positive integer Msuch that ∪∞ n=1D(an, r) = Dand each z∈Dis in, at most, Mof the pseudo-hyperbolic disks Dµa1,1 + r 2¶, D µa2,1 + r 2¶, D µa3,1 + r 2¶, . . . . 3. Boundedness of composition operators on A0 α In this section we provide a necessary and sufficient condition for boundedness of Cφ:A0 α→A0 αin terms of a Carleson measure condition satisfied by the pull back measure να◦φ−1on D. We need the following lemma. Lemma 3.1. If µis an α-Carleson measure on D, then there is a constant C such that ZD log(1 + |f(w)|)dµ(w)≤CZD log(1 + |f(w)|)dνα(w) for any f∈A0 α. Proof. Fix 0 < r < 1.Pick a sequence {an}in Dsatisfying the conditions of Lemma 2.4. There are constants C0,C00 and C000, such that for any f∈A0 α, we have ZD log(1+|f(w)|)dµ(w)≤ 128 S. D. SHARMA AND A. K. SHARMA ≤ ∞ X n=1 ZD(an,r) log(1 + |f(w)|)dµ(w) ≤ ∞ X n=1 µ(D(an, r)) sup{log(1 + |f(w)|) : w∈D(an, r)} ≤C0 ∞ X n=1 µ(D(an, r)) |D(an, r)|AZD(an,1+r 2) log(1 + |f(w)|)dν(w) ≤C0C00 ∞ X n=1 µ(S(1 − |an|, ζ)) (1 − |an|2)2(1 − |an|2)αZD(an,1+r 2) log(1 + |f(w)|)dνα(w) =C0C00 ∞ X n=1 µ(S(1 − |an|, ζ)) (1 − |an|2)α+2 ZD(an,1+r 2) log(1 + |f(w)|)dνα(w) ≤C0C00C000 ∞ X n=1 ZD(an,1+r 2) log(1 + |f(w)|)dνα(w) ≤CM ZD log(1 + |f(w)|)dνα(w),where C=C0C00C000. ¤ Theorem 3.2. Let φbe a holomorphic self-map of D. Then Cφ:A0 α→A0 αis bounded if and only if the pull back measure να◦φ−1is an α-Carleson measure on D. Proof. Suppose Cφis bounded. Assume 0 < δ < 1 and ζ∈∂D. Take fa(z) = exp µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶, where a= (1 −δ)ζ. Now ||fa||0,α =ZD log+¯¯¯¯ exp µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶¯¯¯¯ dνα(z) ≤ZD (1 − |a|2)(α+2) |1−az|2(α+2) dνα(z) ≤ ||ka||2 2,α ≈1. Since Cφis bounded, there is a constant Ksuch that ||Cφfa||0,α ≤K||fa||0,α ≤C. COMPOSITION OPERATORS ON WEIGHTED BERGMAN-NEVANLINNA CLASSES 129 That is, C≥ZD log+|fa◦φ(z)|dνα(z) =ZD <µ(1 − |a|2)(α+2) (1 −aφ(z))2(α+2) ¶dνα(z) =ZD <µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶dνα◦φ−1(z). Now <µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶=(1 − |a|2)(α+2) (1 − |a|)2(α+2) <µ1− |a| 1−az ¶2(α+2) =(1 − |a|2)(α+2) (1 − |a|)2(α+2) <µ1 + |a|(1 −zζ) (1 − |a|)¶−2(α+2) ,¡ζ=a |a|¢ >(1 − |a|2)(α+2) (1 − |a|)2(α+2) 1 2α+2 ≥1 (2δ)α+2 if |1−zζ| 1−|a|< γ0for some fixed γ0>0, that is, if z∈S(γ0δ, ζ).Thus for ζ∈∂D and 0 < δ < 1, we have C≥1 2α+2 ZS(γ0δ,ζ) 1 δα+2 dνα◦φ−1(z) = 1 2α+2 1 δα+2 να◦φ−1(S(γ0δ, ζ)), that is, να◦φ−1(S(γ0δ, ζ)) ≤Cδα+2 and so να◦φ−1is an α-Carleson measure on D. Conversely, suppose να◦φ−1is an α-Carleson measure. Then, by Lemma 3.1, we have, for each f∈A0 α, ||Cφf||0,α =ZD log(1 + |(f◦φ)(w)|)dνα(w) =ZD log(1 + |f(w)|)dνα◦φ−1(w) ≤CZD log(1 + |f(w)|)dνα(w) =C||f||0,α. This completes the proof. ¤ Remark. Theorem 3.2 above and Theorem 4.3 of MacCluer and Shapiro [MaS] assert that Cφis bounded on Ap αas well as on A0 αif and only if να◦φ−1is an α-Carleson measure on D. But in view of Theorem 3.4 of MacCluer and Shapiro [MaS], every analytic self-map φof Dinduces a bounded composition 130 S. D. SHARMA AND A. K. SHARMA operator on Ap αfor all 0 < p < ∞and α > −1.Hence we conclude that every analytic self-map φof Dinduces a bounded composition operator on A0 α. 4. Compactness of composition operators on A0 α Before proving the main result of this section, we recall that Cφis compact on A0 αif and only if for every sequence {fn}which is bounded in A0 αand converges to 0 uniformly on compact subsets of D, we have ||Cφfn||0,α →0. We now characterize compact composition operators on A0 α. Theorem 4.1. Let φbe a holomorphic self-map of D. Then Cφ:A0 α→A0 αis compact if and only if the measure να◦φ−1is a vanishing Carleson measure on D. Proof. Suppose Cφis compact. Let ζ∈∂Dand 0 < δ < 1 2. Consider the family of functions fa(z) = (1 − |a|)α+2 exp µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶, where a= (1 −δ)ζfor some ζ∈∂D. Clearly fa→0 uniformly on compact subsets of Das |a| → 1. As in the proof of Theorem 3.2, there exists a positive constant Csuch that ||fa||0,α ≤C. Again as in the proof of Theorem 3.2, there exist γ0>0 such that if z∈ S(γ0δ, ζ), then <µ(1 − |a|2)(α+2) (1 −az)2(α+2) ¶≥1 2α+2δα+2 and so log+|fa(z)| ≥ log+µ(1 − |a|)α+2 exp µ<(1 − |a|α+2) (1 −az)2(α+2) ¶¶ ≥log+³δα+2 exp 1 2α+2δα+2 ´. Therefore, we have for any ζ∈∂Dand 0 < δ < 1 log+³δα+2 exp 1 2α+2δα+2 ´να◦φ−1(S(γ0δ, ζ)) ≤ZS(γ0δ,ζ) log+|fa(z)|dνα◦φ−1(z) ≤ZD log+|fa◦φ(z)|dνα(z) =||Cφfa||0,α. But compactness of Cφforces ||Cφfa||0,α to tend to zero as δ→0, which implies that lim δ→0³log+³δα+2 exp 1 2α+2δα+2 ´να◦φ−1(S(γ0δ, ζ))´= 0 COMPOSITION OPERATORS ON WEIGHTED BERGMAN-NEVANLINNA CLASSES 131 uniformly on ζ∈∂D. Now, since lim δ→0δα+2³log+δα+2 exp 1 2α+2δα+2 ´= lim t→∞ 1 tα+2 ³tα+2 2α+2 −(α+ 2) log t´ =1 2α+2 >0, it follows that lim δ→0 να◦φ−1(S(γ0δ, ζ)) δα+2 = 0 uniformly on ζ. Thus να◦φ−1is a vanishing Carleson measure on D. Conversely, suppose that να◦φ−1is a vanishing Carleson measure on D. Then να◦φ−1(S(γ0δ, ζ)) δα+2 →0 uniformly in ζas δ→0, that is, να◦φ−1(D(w, r)) (1 − |w|)α+2 →0 uniformly as |w| → 1. Thus, for every ε > 0,we can choose r0>0 such that φ−1(D(w, r)) < ε(1 − |w|)α+2 for all w∈Dfor |w|> r0. Suppose {fm}converges to zero weakly in A0 α.Let {wn}be a sequence as in Lemma 2.4 such that |w1|<|w2|<· · · . Then φ−1(D(wn, r)) < ε(1 − |wn|)α+2 for all wn∈Dsuch that |wn|> r0. Thus ||Cφfm||0,α =ZD log(1 + |(fm◦φ)(z)|)dνα(z) =ZD log(1 + |fm(z)|)dνα◦φ−1(z) =Z|z|≤r0 log(1 + |fm(z)|)dνα◦φ−1(z) +Z|z|>r0 log(1 + |fm(z)|)dνα◦φ−1(z). Since {fm}converges to zero on each compact subset of D, lim m→∞ Z|z|≤r0 log(1 + |fm(z)|)dνα◦φ−1(z) = 0, whereas the second term in the above expression is bounded by ∞ X n=k+1 ZD(wn,r) log(1+|fm(z)|)dνα◦φ−1(z)≤