DEMONSTRATIO MATHEMATICA Vol.XLIV No2 2011

Ajay K. Sharma

GENERALIZED WEIGHTED COMPOSITION OPERATORS ON THE BERGMAN

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 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 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 and . 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 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 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 . D

(3) There is a constant C3 > 0 such that, for every z ∈ D, \ (1+n) 2 |βz (w)| dµ(w) ≤ C3. D A positive Borel measure µ which satisfies the above equivalent conditions is called a 2(n + 1)-Carleson measure for the Bergman space A2. If we define µ(S(I) ||µ|| = sup , 2(n+1) I⊂∂D |I| then ||µ|| and the constants in Theorem 2.3 are comparable. A positive Borel measure µ on D is called a vanishing 2(n + 1)-Carleson measure if µ(S(I)) lim sup =0. 2(n+1) |I|→0 I⊂∂D |I| Theorem 2.4. For a non-negative integer n and a positive Borel measure µ on D the following are equivalent; (1) The measure µ on D is a vanishing 2(n + 1)-Carleson measure.

(2) For all a ∈ D, we have \ (1+n) 2 lim |βz (w)| dµ(w)=0. |a|→1 D

n 3. Boundedness of Dϕ,ψ The following result characterizes bounded generalized weighted compo- n sition operator Dϕ,ψ on the Bergman space. Theorem 3.1. Let ϕ and ψ be holomorphic maps on D such that ϕ(D) ⊂ D and n be a non-negative integer. Then the following are equivalent

n 2 (1) Dϕ,ψ is bounded on A . −1 (2) The pull-back measure µϕ,ψ = νψ ◦ ϕ of νψ induced by ϕ is a 2(n +1)- 2 Carleson measure. Here dνψ = |ψ| dA.

\ 2 2 (1 −|z| ) 2 (3) sup D |ψ(w)| dA(w) < ∞. z∈ 2(n+2) D |1 − zϕ(w)| 364 A. K. Sharma

n 2 Proof. By definition, Dϕ,ψ is bounded on A if and only if there is a constant

C > 0 such that for any f ∈ A2, \ (n) 2 2 2 |f (ϕ(z))| |ψ(z)| dA(z) ≤ C||f||A2 . D 2 −1 Let dνψ(z) = |ψ(z)| dA(z) and µϕ,ψ = νψ ◦ ϕ be the pull-back measure of

νψ induced by ϕ. If we change variable w = ϕ(z), then we get

\ \ (n) 2 2 (n) 2 |f (ϕ(z))| |ψ(z)| dA(z) = |f (ϕ(z))| dνψ(z)

D D \ (n) 2 = |f (w)| dµϕ,ψ(w). D

Thus, by Theorem 2.3, (1) is equivalent to \ (n) 2 2 |f (w)| dµϕ,ψ(w) ≤ C||f||A2 . D Hence (1) and (2) are equivalent. By Theorem 2.3, the condition that µϕ,ψ is a 2(n +1)-Carleson measure

is equivalent to \ (n+1) 2 |βa (w)| dµϕ,ψ(w) ≤ C. D Changing the variable, we get

\ (1 −|z|2)2|z|2n sup |ψ(w)|2dA(w) < ∞. (3.1) 2(n+2) z∈D D |1 − zϕ(w)| Thus (1), (2) and (3.1) are equivalent. Clearly, (3) =⇒ (3.1). Suppose that (3.1) holds. Then for any 0 < r0 < 1, we have

\ (1 −|z|2)2 sup |ψ(w)|2dA(w) < ∞. (3.2) 2(n+2) r0<|z|<1 D |1 − zϕ(w)| n 2 n 2

By (1), Dϕ,ψ is bounded on A . Thus by taking f(z) = z /n! in A , we get \ 2 1 |ψ(z)| dA(z) ≤ 2 D (n!) (n + 1) and so

\ (1 −|z|2)2 (3.3) sup |ψ(w)|2dA(w) 2(n+2) 0≤|z|≤r0 D |1 − zϕ(w)|

1 \ ≤ |ψ(w)|2dA(w) < ∞. 2(n+2) (1 − r0) D Combining (3.2) and (3.3), we have (3.1) =⇒ (3). Generalized weighted composition operators on the Bergman space 365

Corollary 3.2. Let ϕ and ψ be holomorphic maps on D such that D D n 2 ϕ( ) ⊂ and n be a non-negative integer. If Dϕ,ψ is bounded on A , then (1 −|z|2) sup |ψ(z)| < ∞. 2 (n+1) z∈D (1 −|ϕ(z)| ) n 2 Proof. Suppose that Dϕ,ψ is bounded on A . By Theorem 3.1, we have

\ (1 −|z|2)2 sup |ψ(w)|2dA(w) < ∞. 2(n+2) z∈D D |1 − zϕ(w)| In particular,

\ (1 −|ϕ(a)|2)2 (3.4) sup |ψ(w)|2dA(w) < ∞. 2(n+2) a∈D D |1 − ϕ(a)ϕ(w)| For a fixed a ∈ D, let Ω(a) = {z ∈ D : |z−a| ≤ (1−|a|2)/2}. Then Ω(a) ⊂ D. By subharmonicity of the function |ψ(w)|2/|1 − ϕ(a)ϕ(w)|2(n+2), we get

4 \ (1 −|ϕ(a)|2)2 |ψ(w)|2dA(w) (1 −|a|2)2 |1 − ϕ(a)ϕ(w)|2(n+2) Ω(a) |ψ(a)|2 ≥ . (1 −|ϕ(a)|2)2(n+1) Thus (1 −|a|2) (3.5) |ψ(a)| (1 −|ϕ(a)|2)(n+1)

\ (1 −|ϕ(a)|2)2 1/2 ≤ 2 |ψ(w)|2dA(w) .  2(n+2)  Ω(a) |1 − ϕ(a)ϕ(w)| The result follows by (3.4) and (3.5). Theorem 3.3. Let ϕ be a holomorphic map on D such that ϕ(D) ⊂ D. Then the following are equivalent: n 2 (1) CϕD is bounded on A . −1 (2) The pull-back measure µϕ = A◦ϕ induced by ϕ is a 2(n + 1)-Carleson measure. Ì (1−|z|2)2 (3) sup D dA(w) < ∞. z∈ D |1−zϕ(w)|2(n+2) 2(n+1) (4) Nϕ,2(w) = O log(1/|w|) as |w|→ 1. Proof. In view of Theorem 3.1, we need only to show that (1)⇒(4) and (4)⇒(1). n 2 To prove (1)⇒(4), let us suppose that CϕD is bounded on A . For 2 2 2 λ ∈ D, we consider the function fλ(z)=(1 −|λ| )/(1 − λz) . Then fλ ∈ A 366 A. K. Sharma

′ and ||fλ|| 2 =1. Using Lemma 2.1, we have ||f|| 2 ≈|f(0)| + ||f || 2 . Thus A A A2 by the change of variable formula (2.2), we have

n 2 (n) 2

||CϕD fλ||A2 = ||fλ ◦ ϕ||A2 \ (n) 2 (n+1) 2 ≈|fλ (ϕ(0))| + |fλ (z)| Nϕ,2(z)dA(z) D

\((n + 1)!)2|λ|2(n+1)(1 −|λ|2)2 ≥ N (z)dA(z). 2(n+3) ϕ,2 D |1 − λz|

′ 2 Substituting z = βλ(ζ)=(λ − ζ)/(1 − λζ) and using the fact that |βλ(ζ)| = (1 −|λ|2)2/(1 − λz)4, we get

\ 2 2(n+1) 2(n+3) 2 ((n + 1)!) |λ| |1 − λz| ||C Df || 2 ≥ N (β (ζ))dA(ζ) ϕ λ A 2 2(n+1) ϕ,2 λ D (1 −|λ| )

\ ((n + 1)!)2|λ|2(n+1)|1 − λz|2(n+3) ≥ Nϕ,2(β (ζ))dA(ζ). (1 −|λ|2)2(n+1) λ D/2

Note that |1 − λζ| ≥ 1/2 for ζ ∈ D/2. Using sub-averaging property of the Nevanlinna counting function, we obtain

2 4 2 4 2 ((n + 1)!) |λ| Nϕ,2(βλ(0)) ((n + 1)!) |λ| Nϕ,2(λ) ||CϕDf || 2 ≥ = λ A 22(n+1) (1 −|λ|2)2(n+1) 22(n+1) (1 −|λ|2)2(n+1) for λ ∈ D \{ϕ(0)}. Since log(1/|λ|) is comparable to 1 −|λ| as |λ|→ 1−, we obtain 2(n+1) − Nϕ,2(λ) = O([log(1/|λ|)] ) as |λ|→ 1 .

(4)⇒(1). Suppose that for some r, 0 < r < 1, there is a constant M such that 2(n+1) sup [Nϕ,2(z)/[log(1/|z|)] ] ≤ M. r<|z|<1

By Lemma 2.1 and the change of variable formula, we have \ n 2 (n) 2 (n+1) 2 ||CϕD f||A2 ≈|f (ϕ(0))| + |f (z)| Nϕ,2(z)dA(z)

D

\ \ (n) 2 (n+1) 2 = |f (ϕ(0))| + + |f (z)| Nϕ,2(z)dA(z).  rD D\rD  Generalized weighted composition operators on the Bergman space 367

(n) 2 1+n By Lemma 2.1, |f (ϕ(0))| ≤ ||f||A2 /((1 −|ϕ(0)| ) ). Thus we need to

estimate the second and the third term

\ \ (n+1) 2 (n+1) 2 |f (z)| Nϕ,2(z)dA(z) ≤ sup |f (z)| Nϕ,2(z)dA(z) rD |z|≤r  rD

||f||2 \ ≤ sup A2 N (z)dA(z).  (1 −|z|2)2+n  ϕ,2 |z|≤r rD Again by the change of variable formula the right side of the above inequality is dominated by

2 \ 2 ||f||A2 ′ 2 2 ||f||A2 2 sup 2 2+n |ϕ (z)| [log(1/|z|)] dA(z) = 2 2+n ||ϕ||A2 .  |z|≤r (1 −|z| )  D (1 −|r| )

Finally, we consider the third term

\ \ (n+1) 2 (n+1) 2 2(n+1) |f (z)| Nϕ,2(z)dA(z) ≤ M |f (z)| [log(1/|z|)] dA(z). D\rD D\rD

Since log(1/|λ|) is comparable to 1−|λ| as |λ|→ 1−, by Lemma 2.1, we have \ (n+1) 2 2(n+1) 2 |f (z)| [log(1/|z|)] dA(z) ≤ C||f||A2 . D\rD 2 Hence CϕD is bounded on A . Corollary 3.4. Let ϕ and ψ be a holomorphic maps on D such that ϕ(D) ⊂ D. Then the following are equivalent 2 (1) DCϕ is bounded on A . −1 (2) The pull-back measure µϕ,ϕ′ = νϕ′ ◦ ϕ of νϕ′ induced by ϕ is a 2- ′ 2 Carleson measure. Here dνϕ′ = |ϕ | dA.

\ 2 2 (1 −|z| ) ′ 2 (3) supz∈D 6 |ϕ (w)| dA(w) < ∞. D |1 − zϕ(w)|

Proof. As DCϕ = Mϕ′ CϕD, the proof follows by Theorem 3.1. n Corollary 3.5. Let ψ be a holomorphic map on D. Then MψD is bounded on A2 if and only if ψ ∈ X, where H∞ if n =0, X = {0} if n ≥ 1. n 2 Proof. First suppose that MψD is bounded on A . Then by taking ϕ(z) = z in Corollary 3.2, we have (3.6) sup |ψ(z)|/(1 −|z|2)n < ∞ z∈D and so ψ ∈ H∞ if n = 0. Again if n ≥ 1, then (3.6) implies that there is a positive constant C such that |h(z)| ≤ C(1 −|z|2)n for all z ∈ D. It follows 368 A. K. Sharma that |ψ(z)|→ 0 as |z|→ 1−, so by the maximum modulus theorem, we have ψ ≡ 0. Conversely, suppose that h ∈ X. Now if n ≥ 1, then the result is

obvious. If n =0, then \ \ 2 2 2 2 (1 −|z| ) 2 2 (1 −|z| ) 2 ∞ ∞ 4 |ψ(w)| dA(w) ≤ ||ψ||H 4 dA(w) = ||h||H D |1 − zw| D |1 − zw| n 2 and so by Theorem 3.1, MψD is bounded on A .

4. Essential norms ψ Define Λϕ, where ϕ and ψ are holomorphic maps of D such that ϕ(D) ⊂ D, as

\(1 −|a|2)2|ψ(z)|2 Λψ(a) = limsup dA(z). ϕ 2(n+2) |a|→1 D |1 − aϕ(z)| Theorem 4.1. Let ϕ and ψ be holomorphic maps on D such that ϕ(D) ⊂ D. n 2 Let Dϕ,ψ be bounded on A . Then there is an absolute constant C ≥ 1 such that ψ n 2 ψ lim sup Λϕ(a) ≤ ||Dϕ,ψ||e ≤ C lim sup Λϕ(a). |a|→1 |a|→1 In order to prove the Theorem 4.1, we need several lemmas. The following two lemmas are proved in [17]. Lemma 4.2. Let 1/2

Let \ ∗ (n+1) 2 Mr = sup |βa (z)| dµ(z). |a|≥r D Then, if µ is a 2(n + 1)-Carleson measure for the Bergman space A2, so is ∗ µr = µ|D\D(0,r). Moreover, ||µr|| ≤ NMr , where N is a constant depending upon n only. e e ∞ k D m−1 k Let f(z) = k=0 akz be holomorphic on , Qmf(z) = k=0 akz and Rmf(z) = I − QPm, where I is the identity map. Then Rm isP the orthogonal 2 m 2 ∞ k projection of A onto z A and Rmf(z) = k=m akz . Lemma 4.3. For each r, 0 < r < 1 and f ∈P A2, we have ∞ 2 1/2 n k! 2(k−n) |(R f) (z)|≤||f|| 2 r (k + 1) m A (k − n)!  k=maxX{m,n}  for |z| ≤ r. Finally, we need the following lemma of Cowen and MacCluer [4, p. 134] n to estimate the essential norm of Dϕ,ψ. Generalized weighted composition operators on the Bergman space 369

Lemma 4.4. Let T be a bounded operator on A2. Then

||T ||e = lim ||T Rm||. m→∞ Proof of Theorem 4.1. Upper bound. By Lemma 4.4, n 2 n 2 n 2 ||D || = lim ||D Rm|| = lim sup ||(D Rm)f|| 2 . ϕ,ψ e m→∞ ϕ,ψ e m→∞ ϕ,ψ A ||f||A2 ≤1 Thus

n 2

||(Dϕ,ψRm)f||A2

\ \ 2 (n) 2 (n) 2 = |ψ(z)| |(Rmf) (ϕ(z))| dA(z) = |(Rmf) (z)| dµψ,ϕ(z)

D D

\ \ (n) 2 (n) 2 = |(Rmf) (z)| dµψ,ϕ(z) + |(Rmf) (z)| dµψ,ϕ(z) D\D(0,r) D(0,r)

= I1 + I2. n 2 Since Dϕ,ψ is bounded on A , dµψ,ϕ is a 2(n + 1)-Carleson measure for the

Bergman space. So \ (n) 2 I2 ≤ sup |(Rmf) (z)| dµψ,ϕ(z) |z|≤r  D(0,r) ∞ 2 2 k! 2(k−n) ≤ C||f|| 2 r (k + 1) . A  (k − n)!  k=maxX{m,n}

Thus for fixed r as m →∞, we have sup||f||≤1 I2 → 0. On the other hand, if

we denote by µψ,ϕr = µψ,ϕ|D\D(0,r), then by Lemma 4.2, we have \ (n) 2 ∗ ∗ I1 ≤ g |(Rnf) (z)| dµψ,ϕ(z) ≤ lim KNM = KNM . n→∞ r r D\D(0,r) Therefore, n 2 ∗ ||D Rm|| ≤ KN lim M

ϕ,ψ e n→∞ r \ (n+1) 2 = KN lim sup |βa (z)| dµψ,ϕ(z) |a|→1 D

\ (1 −|a|2)2 = KN lim sup |ψ(z)|2dA(z), 2(n+2) |a|→1 D |1 − aϕ(z)| which gives the desired upper bound. Lower bound. Consider the normalized kernel function 2 2 ka(z)=(1 −|a| )/(1 − az) . 370 A. K. Sharma

D Then ||ka||A2 =1 and ka → 0 uniformly on compact subsets of as |a|→ 1. 2 Fix a compact operator K on A . Then ||Kka||A2 → 0 as |a|→ 1. Therefore, n n ||Dϕ,ψ − K|| ≥ lim sup ||(Dϕ,ψ − K)ka||A2 |a|→1 n ≥ lim sup(||Dϕ,ψka||A2 − ||Kka||A2 ) |a|→1 n = limsup(||Dϕ,ψka||A2 ). |a|→1 Thus

\ (1 −|a|2)2 ||Dn ||2 ≥ ||Dn − K||2 ≥ lim sup |ψ(z)|2dA(z). ϕ,ψ e ϕ,ψ 2(n+2) |a|→1 D |1 − aϕ(z)| Routine calculations yield the following Theorems. Theorem 4.5. Let ψ and ϕ be holomorphic maps on D such that ϕ(D) ⊂ D. Then the following are equivalent n 2 (1) Dϕ,ψ is compact on A . −1 (2) The pull-back measure µϕ,ψ = νψ ◦ϕ on νψ induced by ϕ is a vanishing 2 2(n + 1)-Carleson measure. Here dνψ = |ψ| dA.

\ (1 −|z|2)2 (3) lim |ψ(w)|2dA(w)=0. |a|→1 2(n+2) D |1 − zϕ(w)| Theorem 4.6. Let ϕ and ψ be holomorphic maps on D such that ϕ(D) ⊂ D. Then the following are equivalent n 2 (1) CϕD is compact on A . −1 (2) The pull-back measure µϕ = A◦ϕ induced by ϕ is a vanishing 2(n + 1)- Carleson measure. \ (1 −|z|2)2 (3) lim dA(w)=0. |a|→1 2(n+2) D |1 − zϕ(w)| 2(n+1) (4) Nϕ,2(w) = o([log(1/|w|)] ) as |w|→ 1. Theorem 4.7. Let ϕ and ψ be holomorphic map of D such that ϕ(D) ⊂ D. Then the following are equivalent 2 (1) DCϕ is compact on A . −1 (2) The pull-back measure µϕ,ϕ′ = νϕ′ ◦ϕ of νϕ′ induced by ϕ is a vanishing ′ 2 2-Carleson measure. Here dνϕ′ = |ϕ | dA.

\ 2 2 (1 −|z| ) ′ 2 (3) lim|z|→1 6 |ϕ (z)| dA(z)=0. D |1 − zϕ(w)| γ γ γ Example 4.8. Let ϕγ(z)=1−(1−z) , or ϕγ(z)=(β(z) −1)/(β(z) +1), 2 where β(z)=(1+ z)/(1 − z)0 < γ < 1. Then for z near to 1, 1 −|ϕγ(z)| ≈ Generalized weighted composition operators on the Bergman space 371

γ n (1 − z) . Thus by Theorem 3.3 and 4.6, CϕD is bounded (respectively compact) on A2, when 0 < γ ≤ 1/(n + 1) (respectively 0 < γ < 1/(n + 1)). Acknowledgement. The author thanks the referee for his valuable comments and suggestions. The author is also thankful to NBHM/DAE, India for partial support (Grant No. 48/4/2009/R&D-II/426).

References

[1] J. E. Brennan, The integrability of the derivative in conformal mapping, J. London Math. Soc. 18 (1978), 261–272. [2] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Func- tions, CRC Press Boca Raton, New York, 1995. [3] Z. Cuckovic, R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499–511. [4] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746–765. [5] F. Forelli, The isometries of Hp spaces, Canad. J. Math. 16 (1964), 721–728. [6] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer, New York, 2000. [7] R.A. Hibschweiler, N. Portnoy, Composition followed by differentiation between Berg- man and Hardy spaces, Rocky Mountain J. Math. 35 (2005), 843–855. [8] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. Math. J. 40 (1985), 85–111. [9] B. D. MacCluer, R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437–1458. [10] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999), 1483–1491. [11] E. Nordgren, P. Rosenthal, F. S. Wintrobe, Invertible composition operators on Hp, J. Funct. Anal. 73 (1987), 324–344. [12] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc. 73 (2006), 235–243. [13] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London. Math. Soc. 61 (2000), 872–884. [14] J. H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404. [15] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993. [16] J. H. Shapiro, W. Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal. 205 (2003), 62–89. [17] A. Sharma, A. K. Sharma, Carleson measures and a class of generalized Integration operators on the Bergman space, Rocky Mountain J. of Math., to appear. [18] R. K. Singh, S. D. Sharma, Composition operators on a functional Hilbert space, Bull. Austral. Math. Soc. 20 (1979), 377–384. [19] W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), 2331–2348. 372 A. K. Sharma

[20] W. Smith, Brennan’s conjecture for weighted composition operators. Recent advances in operator-related function theory, Contemp. Math. 393, 209–214. [21] R. Zhao, Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), 139–150. [22] X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type space, Integral Transforms. Spec. Funct. 18 (3) (2007), 223– 231. [23] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

SCHOOL OF APPLIED PHYSICS AND MATHEMATICS SHRI MATA VAISHNO DEVI UNIVERSITY UDHAMPUR-182121, INDIA

E-mail: aksju−[email protected]

Received April 14, 2009; revised version March 3, 2010.