Weighted composition operators via Berezin transform and Carleson measure

Seria I: PRACE MATEMATYCZNE XLVI (1) (2006), 63-78

Romesh Kumar, Kanwar Jatinder Singh

Weighted composition operators via Berezin transform and Carleson measure

Abstract. In this paper, we study the boundedness and the compactness of weigh- ted composition operators on Hardy spaces and weighted Bergman spaces of the unit polydisc in Cⁿ.

1991 Mathematics Subject Classification: Primary 47B33, 46E30; Secondary 47B07, 46B70.

Key words and phrases: Berezin transform, Bergman spaces, Carleson measure, com- pact operators, Hardy spaces, unit polydisc, weighted compositon operators.

1. Introduction. Let Dⁿbe the unit polydisc in Cⁿ and let ϕ, ψ be analytic functions defined on Dⁿsuch that ϕ(Dⁿ) ⊆ Dⁿ. Then a weighted composition ope- rator W_ϕ,ψ is defined as W_ϕ,ψ(f )(z) = ψ(z)f (ϕ(z), for all f ∈ H(Dⁿ), the space of holomorphic functions on the unit polydisc Dⁿ. For composition operators on spaces of analytic functions one can refer to [1], [6] and [20], whereas for weighted composition operator we refer to [4],[5],[7],[14],[17],[21] and references there in. The weighted composition operators appears naturally in the study of isometries on most of the function spaces, see [9] and [19]. Operators of this kind also appears in many branches of analysis; the theory of dynamical systems, semigroups, the theory of operator algebras and so on.

Let Tⁿdenotes the distinguished boundary of Dⁿ, and we use m_nto denote the n–dimensional Lebesgue area measure on Tⁿ, normalised so that m_n(Tⁿ) = 1. Also we use the notation f_r for the function f_r(z) = f (rz), where z = (z₁, z₂, . . . , z_n) ∈ Tⁿ. Take p > 0. Then f ∈ H(Dⁿ) is said to be a member of the Hardy space H^p(Dⁿ) if and only if

kf k_H^p_(Dⁿ₎= ( sup

0<r<1

Z

Tⁿ

| f_r|^pdm_n)^1p < ∞.

For p = ∞, H^∞(Dⁿ) is the set of bounded analytic functions on Dⁿwith the supre- mum norm.

If f ∈ H^p(Dⁿ), then the limit function f^∗defined as f^∗(z) = lim

r→1f (rz) exists a.e.

on Tⁿand f^∗∈ L^p(m_n), the space of measurable functions f on Tⁿfor which Z

Tⁿ

| f (z) |^pdm_n< ∞.

Let σ_n denote the volume measure on Dⁿ, so that σ_n(Dⁿ) = 1 and σ_n,α given by Qn

i=1(1− | z_i|²)^ασ_ndenotes the weighted measure on Dⁿ, where z = (z₁, z₂, . . . , z_n)

∈ Dⁿ. Then f ∈ H(Dⁿ) is said to be in the weighted Bergman space A^p_α(Dⁿ) if and only if

kf k_A^p_α(Dn)=

Z

Dⁿ

| f (z) |^pdσ_n,α

¹_p

< ∞.

It is well known that the spaces H^p(Dⁿ) and A^p_α(Dⁿ) are Banach spaces with the above norms,see [18] and [6].

Also, any mapping ϕ : Dⁿ→ Dⁿ can be described by n functions ϕ₁, ϕ₂, . . . , ϕ_nas ϕ(z) = (ϕ₁(z), ϕ₂(z), . . . , ϕ_n(z)),

where z = (z₁, z₂, . . . , z_n) ∈ Dⁿ. The mapping ϕ will be called holomorphic if the n functions ϕ₁, ϕ₂, . . . , ϕ_n are holomorphic functions in Dⁿ.

If R is a rectangle on Tⁿ, then let S(R) denotes the corona associated to R. If R = I₁× I2× . . . × In⊂ Tⁿ, where I_j is the interval on T of length δ_j and centre e^{i(θj◦+δj/2)} for j = 1, 2, . . . , n, then

S(R) = S(I₁) × S(I₂) × . . . S(I_n), where

S(I_j) = {re^iθ∈ T : 1 − δj< r < 1 and θ_j^◦< θ < θ_j^◦+ δ_j}

If V is any open subset in Tⁿ, then S(V ) is defined as S(V ) = ∪{S(R) : R is a rectangle in V }.

For a ∈ D, let ϕ_a be the linear fractional transformation on D given by ϕ_a(z) = a − z

1 − a z.

Then each ϕ_a is an automorphism on D and ϕ⁻¹_a = ϕ_a. For ω = (ω₁, ω₂, . . . , ω_n) ∈ Dⁿ, let ϕ_ω(z) = (ϕ_ω1(z₁), ϕ_ω2(z₂), . . . , ϕ_ωn(z_n)). Then ϕ_ω is an automorphism on Dⁿ that exchanges 0 and ω. Since every point evaluation is a bounded linear func- tional on A²_α(Dⁿ). So, for every a ∈ Dⁿ, there exists a unique function K_a in

A²_α(Dⁿ) such that f (a) =< f, K_a>, f A²_α(Dⁿ), where hi denotes the inner product in A²_α(Dⁿ).. The explict formula for the reproducing kernel in A²_α(Dⁿ) is given by

K_a^α(z) =

n

Y

i=1

1

(1 − a_iz_i)^α+2, z, a ∈ Dⁿ. Also, using the reproducing property of K_a^α, we have

kK_a^αk²₂=< K_a^α, K_a^α> = K_a(a) =

n

Y

i=1

1 (1− | a_i|²)^α+2. Thus the normalised reproducing kernel k_a^α is given by

n

Y

i=1

(1− | a_i|²)^1+α/2 (1 − a_iz_i)^α+2 .

Also, we know that A²₋₁(Dⁿ) = H²(Dⁿ), see [12]. Then, for α = −1, k_a^α is the normalised reproducing kernel for H²(Dⁿ). For an holomorphic self map ϕ on Dⁿ and a function f ∈ L¹(Dⁿ), the weighted ϕ− Berezin transform of f is defined as

B_ϕ,α(f )(a) = Z

Dⁿ

f (z)

n

Y

i=1

(1− | a_i|²)^α+2

| 1 − a_iϕ_i(z) |^{4+2 α}dσ_n,α, for α > −1 and

B_ϕ,−1(f )(a) = Z

Tⁿ

f (z)

n

Y

i=1

(1− | a_i|²)

| 1 − aiϕ_i(z) |²dm_n. If ϕ_i(z) = z_i, then B_ϕ,α is just the usual Berezin transform.

2. Boundedness. In this section, we characterize the boundedness of W_ϕ,ψ using the Carleson measure criterion.

Definition 2.1 Let 1 ≤ p < ∞. A finite, non-negative, Borel measure µ on Dⁿ is said to be a p-Carleson measure if µ(S(V )) ≤ C m_n(V )^pfor all connected open sets V ⊂ Tⁿ and µ is said to be a compact p-Carleson measure if

lim

mn(V )→0

sup

V ⊂Tⁿ

µ(S(V )) m_n(V )^p = 0.

Definition 2.2 Let 1 ≤ p < ∞. A finite, non-negative, Borel measure µ on Dⁿ is said to be a (p, α) - Carleson measure if µ(S(R)) ≤ C Qn

i=1δ_i^{p (2+α)} for all R ⊂ Tⁿ and µ is said to be a compact (p, α) - Carleson measure if

lim

δi→0 sup

θ∈Tⁿ

µ(S(R)) Qn

i=1δ^p(2+α)_i

= 0.

Take ψ ∈ H^q(Dⁿ). Define the measure ν_non Dⁿ by ν_n(E) =

Z

ϕ⁻¹(E)∩Tⁿ

| ψ |^q d m_n,

where E is a measurable subset of the closed unit polydisc Dⁿ. Again for ψ ∈ A^q_α(Dⁿ), let the measure ν_n,αon Dⁿ, be defined by

ν_n,α(E) = Z

ϕ⁻¹(E)

| ψ |^q d σ_n,α, where E is a measurable subset of the unit polydisc Dⁿ. Using Halmos [10], we can easily prove the following Lemmas.

Lemma 2.3 Suppose ϕ, ψ ∈ H^q(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. Then Z

Dⁿ

g dν_n= Z

Tⁿ

| ψ |^q(g ◦ ϕ) dm_n, where g is an arbitrary measurable positive function in Dⁿ.

Lemma 2.4 Suppose ϕ, ψ ∈ A^qα(Dⁿ) and let ϕ(Dⁿ) ⊆ Dⁿ. Then Z

Dⁿ

g dν_n,α= Z

Dⁿ

| ψ |^q(g ◦ ϕ) dσ_n,α, where g is an arbitrary measurable positive function in Dⁿ.

We will need the following results

Theorem 2.5 ([11], [12]) Let µ be a nonnegative, Borel measure on Dⁿ. Then the following statements are equivalent:

(i) The inclusion map I : H^p(Dⁿ) → L^p(Dⁿ, µ) is bounded.

(ii) The measure µ is p - Carleson measure.

(iii) There is a constant M > 0 such that, for every a ∈ Dⁿ, sup

a∈Dⁿ

Z

Dⁿ

| ka|^pdµ < M < ∞,

where k_a is the normalised reproducing kernel function for H^p(Dⁿ) A similar result holds for the weighted Bergman spaces A^p_α(Dⁿ).

Corollary 2.6 ([12]) Let µ be a finite, non-negative, Borel measure on Dⁿ, and β ≥ 1. Then the following statements are equivalent:

(i) The identity map I : H^p(Dⁿ) → L^pβ(Dⁿ, µ) is bounded.

(ii) There exists C > 0, such that the measure µ satisfies µ(S(V )) ≤ C m_n(V )^β.

Theorem 2.7 ([11] Theorem 2.5) Take 1 < p ≤ q < ∞, α > −1 and let µ be a finite positive measure on Dⁿ. Then the identity map I_α : A^p_α(Dⁿ) → L^q(µ) is bounded if and only if µ is (^q_p, α) - Carleson measure, that is, there exists C > 0 such that

µ(S(R)) ≤ C

n

Y

i=1

δ^q(α+2)/p_i , for all connected open subsets V in Tⁿ.

Corollary 2.8 Let µ be a finite, non-negative, Borel measure on Dⁿ. If there exists C > 0 and β > 1 such that µ(S(V )) ≤ C m_n(V )^β, then µ(E) = 0 for all measurable subsets E of Dⁿ.

In the following theorem we obtain a lower bound for the norm of weighted composition operator by using the reproducing kernel function .

Theorem 2.9 If Wϕ,ψ is the weighted composition operator on H^p(Dⁿ), then sup

z∈Dⁿ n

Y

i=1

(1− | z_i|²)

(1− | ϕ_i(z) |²) | ψ(z) |≤ kW_ϕ,ψk^p_p

Proof For the sake of convenience, we shall prove the theorem for the case p = 2.

For z ∈ Dⁿ, let K_z be the reproducing kernel in H²(Dⁿ). Also, we have kK_zk²₂= < K_z, K_z > = K_z(z) =

n

Y

i=1

1 1− | z_i|². Again, we have

k(W_ϕ,ψ)^∗(K_z)k²₂= kψ(z) K_ϕ(z)k²₂=

n

Y

i=1

| ψ(z) |² 1− | ϕ_i(z) |². Further,

k(W_ϕ,ψ)^∗(K_z)k²₂ = | ψ(z) | kK_ϕ(z)k²

= | ψ(z) | K_ϕ(z)(ϕ(z))

= | ψ(z) | (K_ϕ(z)◦ ϕ)(z)

≤ | (ψCϕK_ϕ(z))(z) |

≤ kWϕ,ψ(K_ϕ(z))k₂(

n

Y

i=1

1

1− | z_i|²)^1/2.

In the above expression the last inequality holds, since | f (z) |≤ kf k₂(Qn i=1 1

1−|zi|²)^1/2, for every z ∈ Dⁿ

Hence for every z ∈ Dⁿ, we have

kWϕ,ψk²₂ ≥ k(Wϕ,ψ)^∗(K_z)k²₂

n

Y

i=1

(1− | z_i|²)

=

n

Y

i=1

1− | z_i|²

1− | ϕ_i(z) |² | ψ(z) | . Taking supremum over all z ∈ Dⁿ, we have

kW_ϕ,ψk²₂≥ sup

z∈Dⁿ n

Y

i=1

(1− | z_i|²)

(1− | ϕ_i(z) |²) | ψ(z) | .

. _

Corollary 2.10 For 1 < p < ∞, Wϕ,ψinduces an unbounded weighted composition operator on H^p(Dⁿ) if

sup

z∈Dⁿ n

Y

i=1

(1− | z_i|²)

(1− | ϕ_i(z) |²) | ψ(z) |= ∞.

Theorem 2.11 Let 1 < p ≤ q < ∞. If ϕ ∈ H^p(Dⁿ) is such that ϕ(Dⁿ) ⊆ Dⁿ and ψ ∈ H^q(Dⁿ), then W_ϕ,ψ : H^p(Dⁿ) → H^q(Dⁿ) is bounded if and only if ν_n is ^q_p - Carleson measure, that is there exists C > 0 such that

ν_n(S(V )) ≤ Cm_n(V )^q/p.

Proof By Corollary 2.4, the measure νn is ^q_p - Carleson if and only if there is a constant M > 0 such that

Z

Dⁿ

| f |^q dν_n≤ M kf k^q_pfor all f ∈ H^p(Dⁿ) Again by using Lemma 2.3 with g = | f |^q, we have

Z

Dⁿ

| f |^q dν_n= Z

Tⁿ

| ψ |^q| f ◦ ϕ |^qdm_n= kW_ϕ,ψ(f )k^q_p.

Hence ν_nis ^q_p- Carleson measure if and only if there is a constant M > 0 such that kW_ϕ,ψ(f )k_q≤ M^1/qkf k_q, for all f ∈ H^p(Dⁿ).

. _

Now, we give a characterization for the boundedness of W_ϕ,ψ on H^p(Dⁿ) by using the weighted ϕ - Berezin transform of the function | ψ |^p.

Theorem 2.12 Suppose ϕ, ψ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. Then the weigh- ted composition operator W_ϕ,ψis bounded on H^p(Dⁿ) if and only if the weighted ϕ - Berezin transform of the function | ψ |^pis bounded, that is, B_ϕ,−1(| ψ |) ∈ L^∞(Dⁿ).

Proof The proof follows by using Theorem 2.3 and Theorem 2.9. 

Theorem 2.13 Let 1 < p ≤ q < ∞, and ϕ ∈ A^pα(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. If ψ ∈ A^q_α(Dⁿ), then W_ϕ,ψ : A^p_α(Dⁿ) → A^q_α(Dⁿ) is bounded if and only if ν_n,α is (^q_p, α)) - Carleson measure,that is, there exists C > 0 such that

ν_n,α(S(R)) ≤ C

n

Y

i=1

δ_i^q(α+2)/p.

Proof Using Theorem 2.5, νn,αis (^q_p, α) - Carleson measure if and only if there is a constant C > 0 such that

Z

Dⁿ

| f |^q dν_n≤ Ckf k^q_pfor all f ∈ A^p_α(Dⁿ).

Again by Lemma 2.4 with g = | f |^q, we have Z

Dⁿ

| f |^q dν_n= Z

Dⁿ

| ψ |^q| f ◦ ϕ |^qd σ_n,α= kW_ϕ,ψ(f )k^q_p.

Hence ν_n,αis (^q_p, α) - Carleson measure if and only if there is a constant C > 0 such that

kW_ϕ,ψ(f )k_q≤ C^1/qkf k_p, for all f ∈ A^p_α(Dⁿ).

. _

Corollary 2.14 Let 1 < p < q < ∞, ϕ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ and ψ ∈ H^q(Dⁿ). If the operator W_ϕ,ψ : H^p(Dⁿ) → H^q(Dⁿ) is bounded, then

m_n({z = (z₁, z₂, . . . , z_n) ∈ Tⁿ:| ϕ_j(z) |_j=1,...,n= 1}) = 0.

Proof By Theorem 2.9, the measure νn satisfies ν_n(S(V )) ≤ Cm_n(V )^q/p in Dⁿ. Since _p^q > 1, by Corollary 2.6, the measure ν_n|Tⁿ= 0. Take

E = {z = (z₁, z₂, . . . , z_n) ∈ Tⁿ:| ϕ_j(z) |_j=1,...,n= 1}.

Then

0 = ν_n(ϕ(E)) = Z

ϕ⁻¹(ϕ(E))∩Tⁿ

| ψ |^q d m_n≥ Z

E∩Tⁿ

| ψ |^q d m_n= Z

E

| ψ |^q dm_n. Thus | ψ |= 0 almost everywhere on E. Since ψ ∈ H₁,we have m_n(E) = 0. _

Theorem 2.15 Let ϕ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. Then

(i) Suppose 1 ≤ q ≤ ∞ and ψ ∈ H(Dⁿ). Then W_ϕ,ψ : H^∞(Dⁿ) → H^q(Dⁿ) is bounded if and only if ψ ∈ H^q(Dⁿ).

(ii) For 1 ≤ p < ∞ and ψ ∈ H^p(Dⁿ), we have W_ϕ,ψ : H^p(Dⁿ) → H^∞(Dⁿ) is bounded if and only if

sup

z∈Dⁿ n

Y

i=1

| ψ(z) |

(1− | ϕ_i(z) |²)^1−1/p < ∞.

Proof Proof (i) is obvious .

(ii) For any z ∈ Dⁿ, the reproducing kernel K_z can be considered as an element of the dual of H^p(Dⁿ), given by < K_z, f > = f (z). If W_ϕ,ψ : H^p(Dⁿ) → H^∞(Dⁿ) is bounded, then there exists a constant M > 0 such that k(W_ϕ,ψ)^∗K_zk_(H^p₎∗ ≤ M kK_zk_(H∞)^∗= M , for all z ∈ Dⁿ. Again, W_ϕ,ψ^∗ (K_z) = ψ(z)K_ϕ(z), so we have k(W_ϕ,ψ)^∗K_zk_(H^p₎∗ = kψ(z)K_ϕ(z)k_(H^p₎∗ =

n

Y

i=1

| ψ(z) |

(1− | ϕ_i(z) |²)^1−1/p ≤ M, for all z ∈ Dⁿ. Conversely, suppose M = sup_z∈Dn

Qn i=1

|ψ(z)|^p

(1−|ϕi(z)|²) < ∞. If f ∈ H^p(Dⁿ), then

| Wϕ,ψ(f )(z) | = | ψ(z)f (ϕ(z)) |

≤ | ψ(z) | kK_ϕ(z)k_(H^p₎∗kf k_p

=

n

Y

i=1

| ψ(z) |

(1− | ϕ_i(z) |²)^1−1/pkf kp

≤ M^1/pkf k_p.

. _

In the following theorem we give a necessary condition for the boundedness of W_ϕ,ψ on H^p(Dⁿ).

Theorem 2.16 Suppose ϕ, ψ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. If the weighted composition operator W_ϕ,ψ is bounded on H^p(Dⁿ), then

sup

z∈Dⁿ n

Y

i=1

(1− | z_i|²)^1/p

(1− | ϕ_i(z) |²)^1/p | ψ(z) |< ∞.

We can easily prove the following characterization for the closed range of W_ϕ,ψ. Theorem 2.17 Take 1 < p ≤ q < ∞ Let ϕ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ and ψ ∈ H^q(Dⁿ). Suppose W_ϕ,ψ : H^p(Dⁿ) → H^q(Dⁿ) is bounded. Then W_ϕ,ψ has closed range if and only if there exists a constant M > 0 such that

Z

Dⁿ

| f |^q dν_n≥ M kf k^q_pfor all f ∈ H^p(Dⁿ)

The following theorem shows that the question of studying the composition ope- rators between H²(C₊× C₊) can be reduced to study the weighted composition operators between H²(D²).

Take W = H²(C₊× C₊) → H²(D × D) given by (W f )(z, w) = π( 2i

1 − z)( 2i

1 − w)f (τ₁(z), τ₁(w)), where τ₁(z) = i_1−z^1+z and

(W⁻¹g)(z, w) = π⁻¹( 1 i + z)( 1

i + w)f (τ₂(z), τ₂(w)), where τ₂(z) = ^z−i_z+i. Then clearly W and W⁻¹are isometries [8].

Theorem 2.18 Let Ψ : C+× C₊ → C₊× C₊ be holomorphic mapping. Then C_Ψ : H²(C₊× C₊) → H²(C₊× C₊) is unitarily equivalent to the operator L_Φ defined by

L_Φ= (i + Φ₁(z)

i + z )(i + Φ₂(w)

i + w )f ◦ Φ(z, w).

where Φ(z, w) = (Φ₁(z), Φ₂(w)), Φ₁(z) = τ₁◦ Ψ₁◦ τ₂(z) and Φ₂(w) = τ₁◦ Ψ₂◦ τ₂(w).

Proof We have

(W ◦ C_Ψ◦ W⁻¹f )(z, w)

= π⁻¹( 1 i + z)( 1

i + w)(C_Ψ◦ W f )(τ₂(z), τ₂(w))

= π⁻¹( 1 i + z)( 1

i + w)(W f )(Ψ(τ₂(z), τ₂(w)))

= π⁻¹( 1 i + z)( 1

i + w)(W f )(Ψ₁◦ τ₂(z), Ψ₂◦ τ₂(w))

= π⁻¹( 1 i + z)( 1

i + w)( 2i

1 − Ψ₁◦ τ₂(w))( 2i 1 − Ψ₂◦ τ₂(w)) f (τ₁◦ Ψ₁◦ τ₂(z), τ₁◦ Ψ₂◦ τ₂(w)).

Hence

L_Φ= (i + Φ₁(z)

i + z )(i + Φ₂(w)

i + w )f ◦ Φ(z, w).

. _

For more examples of weighted composition operators, see [14] and [21].

3. Compactness. The following lemma is easy to establish.

Lemma 3.1 Let 1 ≤ p, q ≤ ∞ and Wϕ,ψ : H^p(Dⁿ) → H^q(Dⁿ) be bounded. Then W_ϕ,ψ is compact if and only if whenever {f_n} is a bounded sequence in H^p(Dⁿ) converging to zero uniformly on compact subsets of Dⁿ, we have kW_ϕ,ψ(f_n)k_q→ 0.

The above lemma is also true for the weighted Bergman spaces

Theorem 3.2 [11], [12] Take 1 < p < ∞. Let µ be a nonnegative, Borel measure on Dⁿ. Then the following statements are equivalent:

(i) The inclusion map I : H^p(Dⁿ) → L^p(Dⁿ, µ) is compact.

(ii) The measure µ is a compact p - Carleson measure.

(iii) For every a ∈ Dⁿ, we have lim

kak→1

Z

Dⁿ

| k_a|^pdµ = 0.

A similar result holds for the weighted Bergman spaces A^p_α(Dⁿ).

Theorem 3.3 Let ϕ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ, and ψ ∈ H^∞(Dⁿ), and 1 ≤ p < ∞. Then W_ϕ,ψ : H^p(Dⁿ) → H^∞(Dⁿ) is compact if and only if either kϕk_∞< 1 or

lim

|ϕi(z)|→1 n

Y

i=1

| ψ(z) |^p

(1− | ϕ_i(z) |²) = 0.

Proof Suppose the operator Wϕ,ψ is compact, kϕk_∞< 1 and lim

|ϕi(z)|→1 n

Y

i=1

| ψ(z) |^p

(1− | ϕ_i(z) |²) 6= 0.

So we can find a sequence {z_n} in Dⁿand  > 0 such that | ϕ_i(z_n) |→ 1 and

Qn

i=1(1− | ϕ_i(z_n) |²) ≤ | ψ(z_n) |^pfor all natural number n. Let us define a function f_nin H^p(Dⁿ) by

f_n(z) =

n

Y

i=1

(1− | ϕ_i(z_n) |²)^1/p (1 − ϕ_i(z_n)z)^2/p .

Then f_n is a bounded sequence in H^p(Dⁿ) and tends to 0 uniformly on compact subsets of Dⁿ. So, by Lemma 3.1 ,kW_ϕ,ψ(f_n)k∞→ 0. Also, we have that

kW_ϕ,ψ(f_n)k∞≥| ψ(z_n) || f_n(ϕ_i(z_n)) |=

n

Y

i=1

| ψ(z_n) | (1− | ϕ_i(z_n) |²)^−1/p≥ ^1/p, for all n, a contraciction.

Conversely, suppose that lim

|ϕi(z)|→1 n

Y

i=1

| ψ(z) |^p

(1− | ϕ_i(z) |²) = 0.

Let {f_n} be a bounded sequence in H^p(Dⁿ) that tends to 0 uniformly on compact subsets of Dⁿ. Also , take C = sup_nkf_nk_pand  > 0. Again by hypothesis, we can find r₀ < 1 such that if | ϕ_i(z) |> r₀, then | ϕ_i(z) |^p≤ (_2C^ )^PQn

i=1(1− | ϕ_i(z) |²).

Also, there is a natural number n◦such that if n ≥ n◦, then sup

|zi|≤r_◦

| f_n(z) |≤  2kψk_∞. Then for n ≥ n◦, we have

kW_ϕ,ψ(f_n)k_∞ ≤ sup

|ϕiz)|≤r◦

| ψ(z)f_n(ϕ_i(z)) | + sup

|ϕi(z)|>r◦

| ψ(z)f_n(ϕ_i(z)) |

≤ kψk_∞ sup

|wi|≤r_◦

| f_n(w) | +  2C sup

|ϕi(z)|>r◦ n

Y

i=1

| f_n(ϕ_i(z)) | (1− | ϕ_i(z) |²)^1/p

≤ kψk∞



2kψk_∞+ 

2Ckf_nk_p≤ .

If we assume that kϕk_∞ < 1 the compactness of W_ϕ,ψ follows from the fact that ϕ(Dⁿ) is a compact subset of Dⁿ and ψ ∈ H^∞(Dⁿ). _ In the following theorem we give a characterization for the compactness of W_ϕ,ψ on A^p_α(Dⁿ)

Theorem 3.4 Suppose ϕ : Dⁿ→ Dⁿbe such that ϕ(Dⁿ) ⊆ Dⁿ and ψ is bounded.

Then the weighted composition operator W_ϕ,ψ is compact on A²_α(Dⁿ), if and only if lim

kzk→1 n

Y

i=1

(1− | z_i|²)

(1− | ϕ_i(z) |²) | ψ(z) |²= 0.

(1)

Proof The proof follows on the similar lines as in [16, Corollary 1, page-75].. _

Theorem 3.5 Let 1 ≤ q < ∞, ϕ ∈ H^p(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ and ψ ∈ H^q(Dⁿ). Then the operator W_ϕ,ψ: H^∞(Dⁿ) → H^q(Dⁿ) is compact if and only if m_n({z ∈ Tⁿ :| ϕ_i(z) |= 1, for i = 1, 2, . . . , n}) = 0.

Proof Suppose that mn({z ∈ Tⁿ :| ϕ_i(z) |= 1, for i = 1, 2, . . . , n}) = 0. and take a bounded sequence f_n in H^∞ that converges to zero uniformly on compact subsets of Dⁿ. Fix z ∈ Tⁿ such that | ϕ_i(z) |< 1. Then f_n(ϕ_i(z)) → 0 and so ψ(z)f_n(ϕ_i(z)) → 0 almost everywhere in Tⁿ. Moreover,

| ψ(z)f_n(ϕ_i(z)) |^q≤| ψ(z) |^qkf_nk^q_∞≤ C^q | ψ(z) |^q, where C = sup kfnk∞. Since | ψ |^q∈ L1(Dⁿ, m_n), we have that

n→∞lim Z

Tⁿ

| ψf_n◦ ϕ |^q dm_n= 0.

. _

Lemma 3.6 Let 0 < r < 1 and let Dⁿr = {z = (z₁, z₂, . . . , z_n) ∈ Dⁿ: | z_i |< r, i = 1, 2, . . . , n}. Supppose µ is a positive Borel measure on Dⁿ and let µ_r denotes the restriction of measure µ to the set Dⁿ_r. Take

kµk = sup

R⊂Tⁿ

µ(S(R)) Qn

i=1δ_i^p(2+α)

and kµk_r= sup

Qn i=1δi≤1−r

µ(S(R)) Qn

i=1δ_i^p(2+α) .

Then if µ is Carleson measure for the weighted Bergman space A^p_α(Dⁿ), so is µ_r. Moreover kµ_rk ≤ K kµk_r, where K > 0 is a constant.

Lemma 3.7 Let 0 < r < 1. Take kµk^∗_r = sup

kak≥r

Z

Dⁿ

| k^α_a(z) |^pdµ(z).

If µ is a Carleson measure for the weighted Bergman space A^p_α(Dⁿ), then kµ_rk ≤ M kµk^∗_r, where M is an absolute constant.

Any holomorphic function f on Dⁿhas a power series expansion f (z) =P C(α)z^α, where the sum is over all multi-indices α = (α₁, α₂. . . , α_n) of non-negative integers and z^αdenotes the monomial z^α₁¹, z^α₂², . . . , z^α_nⁿ. The series converges absolutely and uniformly in every compact subset of the plydisk Dⁿ.

For s = 0, 1, . . . , let F_s(z) be the sum of those terms C(α)z^α for which | α |=

α₁+ α₂+ . . . α_n = s. Then F_s is a polynomial which is homogeneous of degree s.Thus we can write f (z) =P∞

s=0F_s(z), where F_s is the homogeneous polynomial P

|α|=sC(α)z^α.

For a positive integer n, define the operators R_n(f ) = R_n(P∞

s=0F_s) =P∞ s=n+1F_s and Q_n= I − R_nacting from A²_α(Dⁿ) to A²_α(Dⁿ), where I is the identity map.

Recall that the essential norm of an operator T is defined as : kT k_e= inf{kT − Kk, where K is a compact operator}.

Now we have the following lemma.

Lemma 3.8 Suppose Wϕ,ψ is bounded on A²_α(Dⁿ). Then kW_ϕ,ψk_e= lim

n→∞kW_ϕ,ψR_nk₂.

The proof is similar to the proof of Lemma given in [6, page–134].

In the following theorem we give the upper and lower estimates for the essential norm of a weighted composition operator.

Theorem 3.9 Let ϕ be a holomorphic self–map of Dⁿ and ψ ∈ A²_α(Dⁿ). Suppose W_ϕ,ψ is bounded on A²_α(Dⁿ). Then there is an absolute constant M ≥ 1 such that

lim sup

kak→1

B_ϕ,α(| ψ |²)(a) ≤ kW_ϕ,ψk²_e ≤ M lim sup

kak→1

B_ϕ,α(| ψ |²)(a) Proof First we find the upper estimate.

Upper estimate: By Lemma 3.8, we have kW_ϕ,ψk²_e= lim

n→∞kW_ϕ,ψR_nk₂= lim

n→∞ sup

kf k2≤1

k(W_ϕ,ψR_n)f k₂.

Also, by using Lemma 2.1, we have k(W_ϕ,ψR_n)f k²₂ =

Z

Dⁿ

| ψ(z) |²| (R_nf )(ϕ(z)) |²dσ_n,α.

= Z

Dⁿ

| (Rnf )(ω) |² dν_n,α(ω).

= Z

Dⁿ\Dⁿ_r

| (R_nf )(ω) |²dν_n,α(ω) + Z

Dⁿr

| (R_nf )(ω) |²dν_n,α(ω).

= I₁+ I₂. (2)

Also, the measure ν_n,α is a Carleson measure, because the operator W_ϕ,ψ is bounded on A^p_α(Dⁿ). Again by using [6, page–133], we can show that, for a fixed r,

sup

kf k2≤1

Z

Dⁿ_r

| (R_nf )(ω) |²dν_n,α(ω) → 0 as n → ∞.

Let ν_n,α,r denotes the restriction of measure ν_n,αto the set Dⁿ\ Dⁿ_r. So by using Lemma 3.7 and Theorem 3.2, we have

Z

Dⁿ\Dⁿ_r

| (Rnf )(ω) |²dν_n,α,r(ω) ≤ K kν_n,α,rkk(Rnf )k²₂≤ K M kνn,α,rk^∗_rkf k²₂

≤ K M kν_n,α,rk^∗_r,

where K and M are absolute constants and kν_n,α,rk^∗_r is defined as in Lemma 3.7 Therefore,

n→∞lim sup

kf k2≤1

k(W_ϕ,ψR_n)f k₂≤ lim

n→∞K M kν_n,α,rk^∗_r= K M kν_n,α,rk^∗_r Thus, kW_ϕ,ψk²_e≤ K M kν_n,α,rk^∗_r. Taking r → 1, we have

kW_ϕ,ψk²_e ≤ K M lim

r→1kν_n,α,rk^∗_r.

= K M lim sup

kak→1

Z

Dⁿ

| k_a^α(ω) |² dν_n,α(ω)

= K M lim sup

kak→1

B_ϕ,α(| ψ |²)(a)

which is the desired upper bound.

Lower estimate : We know that kk_a^αk = 1 and k_a^α → 0 uniformly on compact subsets of Dⁿas kak → 1. Also, for a compact operator K on A²_α(Dⁿ), kKk^α_ak₂→ 0 as kak → 1. Therefore,

kWϕ,ψ− Kk2 ≥ lim sup

kak→1

(k(W_ϕ,ψ− K)k^α_ak2)

≥ lim sup

kak→1

(k(W_ϕ,ψ)k_a^αk₂− kK k^α_ak₂)

= lim sup

kak→1

k(Wϕ,ψ)k_a^αk2. Thus,

kW_ϕ,ψk²_e≥ kW_ϕ,ψ− Kk²₂≥ lim sup

kak→1

k(W_ϕ,ψ)k_a^αk²₂= lim sup

kak→1

B_ϕ,α(| ψ |²)(a).

and hence the proof. _

Corollary 3.10 Suppose ϕ, ψ ∈ A²α(Dⁿ) be such that ϕ(Dⁿ) ⊆ Dⁿ. Then the weighted composition operator W_ϕ,ψ is compact on A²_α(Dⁿ) if and only if the the weighted ϕ - Berezin transform of the function | ψ |² tends to zero as kak → 1, that is,

lim

kak→1B_ϕ,α(| ψ |²)(a) = 0.

Remark 3.11 All the results in the paper which we have proved for the weighted Bergman spaces are also true for Hardy spaces and vice-versa. Also all the results in this paper after slight modifications can also be proved for Hardy spaces and some weighted Bergman spaces of the unit ball in Cⁿ.

References

[1] P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Memoirs of the A.M.S. No.516, Amer. Math. Soc. Providence. RI 1997.

[2] I. Chalender and J. R. Partington,On structure of invariant subspaces for isometric composi- tion operators on H²(D) and H²(C+), Archiv der Mathematik 81 (2003), 193–207.

[3] B. R. Chao, Y. J. Lee, K. Nam and D. Zheng, Products of Bergman space Toeplitz operators on the polydisk, (preprint).

[4] M. D. Contreras and A. G. Hernandez-Diaz,Weighted composition operator on Hardy spaces, J. Math. Anal. Appl., 263 (2001), 224–233.

[5] M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operator between different Hardy spaces, Integral Equation and Operator Theory, 46 (2003), 105–188.

[6] C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton FL, 1995.

[7] Z. Cuckovic and R. Zhao, Weighted composition operators on the Bergman spaces, J. London Math. Soc., (2) 70 (2004), 499–511.

[8] S. H. Ferguson and M. T. Lacey, A characterization of product BMO by commutantors, Acta Math., 189 (2002), 143–160.

[9] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces, CRC, Boca Raton, New York, 2003.

[10] P. R. Halmos, Measure Theory, Graduate Texts in Mathematics, 18 Springer–Verlag, New York, 1974.

[11] F. Jafari, On bounded and compact composition operators in polydiscs, Can. J. Math., XLII No.5 (1990), 869–889.

[12] F. Jafari, Carleson measure in Hardy and weighted Bergman spaces of polydiscs, Proc. Amer.

Math. Soc., 112 No. 3 (1991), 771–781.

[13] H. Kamowitz, Compact operators of the form uCϕ, Pacific J. Math., 80 (1979), 205–211.

[14] R. Kumar and J. R. Partington, Weighted composition operartors on Hardy and Bergman spaces, J. Operator Theory: Advances and Applications, Birkhauser Verlag Basel, 153, (2004), 157–167.

[15] V. Matache , Composition operators on Hardy spaces of a half plane , Proc. Amer Math. Soc., 127 (1999),1483 -1491.

[16] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), 70–92.

[17] S. Ohno and H. Takagi , Some properties of weighted composition operators on algebra of analytic functions , J. Nonlinear Convex Anal., 2 (2001), 369 -380.

[18] W. Rudin, Function Theory in The Unit Ball of Cⁿ, Springer-Verlag, New-York, 1980.

[19] R. B. Schneider, Isometries of H^p(Uⁿ), Can. J. Math., XXV No.1 (1973), 92–95.

[20] J. H. Shapiro, Composition operators and classical function theory , Springer-Verlag, New York 1993.

[21] J. H. Shapiro and W. Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal. 205, No.1, (2003), 62–89.

[22] W. Smith, Composition operators between Bergman and Hardy Spaces,Trans. Amer. Math.

Soc., 348, No.6 (1996).

[23] R. K. Singh and S. D. Sharma Composition operators in several complex variables, Bull.

Austral. Math. Soc., 23 (1981), 237-247.

Romesh Kumar

Department of Mathematics, University of Jammu Jammu-180 006, INDIA

E-mail: romesh_jammu@yahoo.com Kanwar Jatinder Singh

Department of Mathematics, University of Jammu Jammu- 180 006, INDIA

E-mail: kunwar752000@yahoo.co.in

(Received: 16.12.2005)