DEMONSTRATIO MATHEMATICA Vol.XLIV No2 2011 Ajay K. Sharma GENERALIZED WEIGHTED COMPOSITION OPERATORS ON THE BERGMAN SPACE Abstract. Let ϕ and ψ be holomorphic maps on the open unit disk D such that ϕ(D) ⊂ D and H(D) be the space of holomorphic functions on D. For a non-negative n n (n) integer n, define a linear operator Dϕ,ψ as Dϕ,ψf = ψ ·(f ◦ϕ), f ∈ H(D). In this paper, n 2 we characterize boundedness and compactness of Dϕ,ψ on the Bergman space A . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space. 1. Introduction Throughout this paper, we denote by H(D), the space of holomorphic functions on the open unit disk D = {z ∈ C : |z| < 1} in the complex plane C. For z, w ∈ D, let βz(w)=(z − w)/(1 − zw) be the Möbius transformation (n) of D which interchanges 0 and z. The nth derivative of βz is βz (w) = 2 n−1 n+1 1 1 n!(|z| − 1)(z) /(1 − zw) . Let dA(z) = π dxdy = π rdrdθ be the normalized area measure on D. For 0 < p < ∞, let Lp be the Lebesgue space Ì p which contains measurable functions f on D such that D |f(z)| dA(z) < ∞. Also denote by Ap = Lp ∩ H(D) the weighted Bergman space with the norm defined as \ 1/p p p ||f||Aα = |f(z)| dA(z) < ∞. D Recall that the Bergman space A2(D) = A2 is a functional Hilbert space of 2 analytic functions on D and Kz(w)=1/(1 − zw) is the Bergman kernel. Moreover, 2 2 2 ′ kz(w)=(1 −|z| )/(1 − zw) = (1 −|z| )Kz(w) = −βz(w) 2000 Mathematics Subject Classification: Primary 47B33, 46E10; Secondary 30D55. Key words and phrases: generalized weighted composition operator, Bergman space, Nevanlinna counting function. 360 A. K. Sharma is the normalized kernel function in A2 and (n) 2 n n+2 (n+1) kz (w)=(n + 1)!(1 −|z| )(z) /(1 − zw) = −βz (w). See [6] and [23] for more details on Bergman spaces. Let ϕ and ψ be holomorphic maps on the open unit disk D such that D D n ϕ( ) ⊂ . For a non-negative integer n, we define a linear operator Dϕ,ψ as follows: n (n) D Dϕ,ψf = ψ · (f ◦ ϕ), f ∈ H( ). We call it generalized weighted composition operator, since it include many known operators. For example, if n = 0 and ψ ≡ 1, then we obtain the composition operator Cϕ induced by ϕ, defined as Cϕf = f ◦ ϕ, f ∈ H(D). n n If ψ =1 and ϕ(z) = z, then Dϕ,ψ = D , the differentiation operator defined n (n) as D f = f . If n =0, then we get the weighted composition operator ψCϕ ′ n defined as ψCϕf = ψ · (f ◦ ϕ). If n =1 and ψ(z) = ϕ (z), then Dϕ,ψ reduces n n to DCϕ. When ψ ≡ 1, then Dϕ,ψ reduces to CϕD . If we put ϕ(z) = z, n n then Dϕ,ψ = MψD , the product of multiplication and differentiation oper- ator. We provide a unified way of treating these operators on the Bergman space A2. Composition and weighted composition operators have gained in- creasing attention during the last three decades, mainly due to the fact that they provide, just as, for example, Hankel and Toeplitz operators, ways and means to link classical function theory to functional analysis and operator theory. In fact, some of the well known conjectures can be linked to composi- tion operators. Nordgren, Rosenthal and Wintrobe [11] have shown that the invariant subspace problem can be solved by classifying certain mini- mal invariant subspaces of certain composition operators on H2, whereas, Louis de Branges used composition operators to prove the Bieberbach con- jecture. For general background on composition operators, we refer [2], [15] and references therein. Weighted composition operators also appear natu- rally in different contexts. For example, W. Smith in [20] showed that the Brennan’s conjecture [1], which says that if G is a simply connected planar D Ì ′ p domain and g is a conformal map of G onto , then G |g | dA < ∞ holds for 4/3 < p < 4, is equivalent to the existence of self-maps of D that make cer- tain weighted composition operators compact. The surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [4] and [5]). Singh and Sharma [18] related the boundedness of composition operators on the Hardy space of the upper half-plane with the bounded- ness of weighted composition operators on the Hardy space of the open unit disk D. Weighted composition operators also played an important role in the study of compact composition operators on Hardy spaces and Bergman Generalized weighted composition operators on the Bergman space 361 spaces of unbounded domains (see [10] and [16]). Hibschweiler and Portony [7] defined the linear operators DCϕ and CϕD and investigated the bound- edness and compactness of these operators between Bergman spaces using Carleson-type measures. S. Ohno [12] discussed boundedness and compact- ness of CϕD between Hardy spaces. X. Zhu [22] characterized boundedness n and compactness of Dϕ,ψ from Bergman type spaces to Bers type spaces. The goal of this work is to estimate the essential norm of the linear n operators Dϕ,ψ on the Bergman space. Recall that the essential norm ||T ||e of a bounded linear operator on a Banach space X is given by ||T ||e = inf{||T − K|| : K is compact on X}. It provides a measure of non-compactness of T. Clearly, T is compact if and only if ||T ||e =0. Recently, several authors have computed the upper and the lower bounds of the essential norm of composition and weighted composition operators on different spaces of analytic functions. In fact, essential norm can actually be computed explicitly only in certain cases. For example, J. H. Shapiro [14] in 1987 was able to obtained the general expression for the essential norm of the composition operators on the Hardy space H2. He discovered the connection between the essential norm of the composition operator on the Hardy space H2 and the Nevanlinna counting function for ϕ, thereby, obtained the following expression 2 Nϕ(w) ||Cϕ||e = limsup . − 1 |w|→1 log |w| Rodriguez [13] computed the essential norm of composition operators on weighted Bloch spaces. Essential norm of weighted composition operators on the Bergman space and Bloch type spaces was computed by Cuckovic and Zhao [3] and MacCluer and Zhao [9], respectively. The notation A ≈ B means that there is a positive constant C such that B/C ≤ A ≤ CB. 2. Preliminaries In this section, we collect some essential facts that will be needed through- out the paper. Next result is an alternate characterization of the Bergman space A2 (see [21]). Lemma 2.1. Let f ∈ H(D). Then f ∈ A2 if and only if for all n ∈ N, f (n)(z)(1 −|z|2)n ∈ L2, and n−1 \ 2 (k) (n) 2 2 2n ||f||A2 ≈ |f (0)| + |f (z)| (1 −|z| ) dA(z) . Xk=0 D 362 A. K. Sharma 2 (n) 2 (n) Thus if f ∈ A , then f ∈ A and ||f || 2 ≤ C||f|| 2 . Moreover, for 2n A2n A every z in D, we have (n) ||f ||A2 ||f|| 2 (2.1) |f (n)(z)| ≤ C 2n ≤ CC′ A (1 −|z|2)1+n (1 −|z|2)1+n We now incorporate some results from Shapiro’s paper [14] (see [15] and [19] also). For a holomorphic self-map ϕ of D, the Nevanlinna counting function Nϕ(·) is defined by: 1 N (w) = log , w ∈ D \{ϕ(0)}, ϕ |z| z∈ϕX−1(w) where ϕ−1(w) denotes the set of ϕ-preimages of w counting the multiplicity, and Nϕ(w)=0 if w 6∈ ϕ(D). In [14], Shapiro also introduced the generalized Nevanlinna counting func- tion Nϕ,γ, defined for γ > 0 by 1 γ N (w) = log , w ∈ D \{ϕ(0)}. ϕ,γ |z| z∈ϕX−1(w) This counting function provides a change of variable formula [14]. We need the following special case of the change of variable formula. \ 2 2 ′ 2 ||f ◦ ϕ||A2 ≈|f(ϕ(0))| + |f (z)| Nϕ,2dA(z). (2.2) D The next lemma shows that these counting functions, while not subharmonic themselves, satisfy a sub-mean value property. Lemma 2.2. [19] Let τ be a holomorphic map of D such that τ(D) ⊂ D, and let γ > 0. If τ(0) 6=0 and 0 < r < |τ(0)|, then 1 \ N (0) ≤ N (z)dA(z). τ,γ r2 τ,γ rD We also make extensive use of Carleson measure techniques, so we give a short introduction to Carleson sets and Carleson measures. Let S(I) denotes the Carleson set: z S(I) = z ∈ D :1 −|I|≤|z| ≤ 1, ∈ I , |z| where I runs through arcs on the unit circle. Finally, we need a special case of the Luecking’s result [8] for p = q =2 and α =0 in which he characterized positive measures µ with the property (n) ||f ||L2(µ) ≤ C||f||A2 . Theorem 2.3. For a non-negative integer n and a positive Borel measure µ on D the following are equivalent Generalized weighted composition operators on the Bergman space 363 (1) There is a constant C1 > 0 such that, 2(1+n) µ(S(I)) ≤ C1|I| . 2 (2) There is a constant C2 > 0 such that, for every f ∈ A , \ (n) 2 2 |f (w)| dµ(w) ≤ C2||f||A2 .