Volume 2008, Article ID 619525,14pages doi:10.1155/2008/619525
Research Article
Weighted Composition Operators from Generalized Weighted Bergman Spaces to Weighted-Type Spaces
Dinggui Gu
Department of Mathematics, JiaYing University, Meizhou, GuangDong 514015, China
Correspondence should be addressed to Dinggui Gu,[email protected]
Received 3 November 2008; Revised 22 November 2008; Accepted 24 November 2008 Recommended by Kunquan Lan
Letϕbe a holomorphic self-map and let ψ be a holomorphic function on the unit ballB. The boundedness and compactness of the weighted composition operatorψCϕfrom the generalized weighted Bergman space into a class of weighted-type spaces are studied in this paper.
Copyrightq2008 Dinggui Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetBbe the unit ball ofCnand letHBbe the space of all holomorphic functions onB. For f∈HB, let
Rfz n
j1
zj
∂f
∂zjz 1.1
represent the radial derivative off∈HB. We writeRmfRRm−1f.
For anyp >0 andα∈R, letNbe the smallest nonnegative integer such thatpNα >
−1. The generalized weighted Bergman spaceApαis defined as follows:
Apα
f∈HB| fApα f0
B
RNfzp
1− |z|2pNα
dvz 1/p<∞
. 1.2
Heredvis the normalized Lebesgue measure ofBi.e.,vB 1. The generalized weighted Bergman space Apα is introduced by Zhao and Zhu see, e.g., 1. This space covers the
classicalweighted Bergman spaceα > −1, the Besov spaceAp−n1, and the Hardy space H2. See1,2for some basic facts on the weighted Bergman space.
Letμbe a positive continuous function on0,1. We say thatμis normal if there exist positive numbersαandβ, 0< α < β,andδ∈0,1such thatsee3
μr
1−rα is decreasing on δ,1, lim
r→1
μr 1−rα 0;
μr
1−rβ is increasing onδ,1, lim
r→1
μr 1−rβ ∞.
1.3
Anf ∈HBis said to belong to the weighted-type space, denoted byHμ∞Hμ∞B, if
fHμ∞ sup
z∈B μ
|z|fz<∞, 1.4
whereμis normal on0,1. The little weighted-type space, denoted byHμ,0∞, is the subspace ofHμ∞consisting of thosef∈Hμ∞such that
|z| →lim1μ
|z|fz0. 1.5
See4,5for more information onHμ∞.
Let ϕ be a holomorphic self-map of B. The composition operator Cϕ is defined as follows:
Cϕf
z f◦ϕz, f∈HB. 1.6
Letψ∈HB. Forf ∈HB, the weighted composition operatorψCϕis defined by ψCϕf
z ψzf ϕz
, z∈B. 1.7
The book6contains a plenty of information on the composition operator and the weighted composition operator.
In the setting of the unit ball, Zhu studied the boundedness and compactness of the weighted composition operator between Bergman-type spaces and H∞ in 7. Some extensions of these results can be found in 8. Some necessary and sufficient conditions for the weighted composition operator to be bounded or compact between the Bloch space andH∞are given in 9. In the setting of the unit polydisk, some necessary and sufficient conditions for a weighted composition operator to be bounded and compact between the Bloch space andH∞are given in10,11 see also12for the case of composition operators.
In13, Zhu studied the boundedness and compactness of the Volterra composition operators from generalized weighted Bergman space toμ-Bloch-type space. Other related results can be found, for example, in4,5,14–22.
In this paper, we study the weighted composition operatorψCϕ from the generalized weighted Bergman space to the spaces Hμ∞ and Hμ,0∞. Some necessary and sufficient conditions for the weighted composition operator ψCϕ to be bounded and compact are given.
Throughout the paper, constants are denoted byC, they are positive and may differ from one occurrence to the other.
2. Main results and proofs
Before we formulate our main results, we state several auxiliary results which will be used in the proofs. They are incorporated in the lemmas which follow.
Lemma 2.1see1. iSuppose thatp >0 andαn1>0. Then there exists a constantC >0 such that
fz≤ CfApα
1− |z|2nα1/p 2.1
for allf∈Apαandz∈B.
iiSuppose thatp >0 andαn1<0 or 0< p≤1 andαn10. Then every function inApαis continuous on the closed unit ball. Moreover, there is a positive constantCsuch that
f∞≤CfAp
−n1, 2.2
for everyf ∈Ap−n1.
iiiSuppose thatp >1, 1/p1/q1,andαn10. Then there exists a constantC >0 such that
fz≤C
ln e 1− |z|2
1/q 2.3
for allf∈Apαandz∈B.
The following criterion for compactness of weighted composition operators follows from standard arguments similar to those outlined in6, Proposition 3.11 see also12, proof of Lemma 2. We omit the details of the proof.
Lemma 2.2. Assume thatψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ :Apα → Hμ∞ is compact if and only ifψCϕ :Apα → Hμ∞is bounded and for any bounded sequencefkk∈NinApα which converges to zero uniformly on compact subsets ofBas k → ∞, one hasψCϕfkHμ∞ → 0 ask → ∞.
Note that whenp > 0 andαn1<0, the functions inApαare Lipschitz continuous see1, Theorem 66. By Lemma 2.1and Arzela-Ascoli theorem, similarly to 19, proof of Lemma 3.6, we have the following result.
Lemma 2.3. Letp >0 andαn1<0. Letfkbe a bounded sequence inApαwhich converges to 0 uniformly on compact subsets ofB, then
klim→ ∞sup
z∈B
fkz0. 2.4
The following lemma is from21 one-dimensional case is20, Lemma 2.1.
Lemma 2.4. Assume thatμis a normal function on0,1. A closed setKinHμ,0∞ is compact if and only if it is bounded and satisfies
|z| →lim1sup
f∈Kμ
|z|fz0. 2.5
We will consider three cases:n1α >0,n1α0,andn1α <0.
2.1. Casen1α >0
Theorem 2.5. Assume thatp > 0,αis a real number such thatnα1 > 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ :Apα → Hμ∞is bounded if and only if
M:sup
z∈B
μ
|z|ψz
1−ϕz2n1α/p <∞. 2.6
Proof. Assume thatψCϕ:Apα → Hμ∞is bounded. Let
t > nmax
1,1 p
α1
p . 2.7
Fora∈B, set
faz
1− |a|2t−n1α/p
1− z, a t . 2.8
It follows from1, Theorem 32thatfa∈Apαand supa∈BfaApα <∞.Hence CψCϕ
Apα→H∞μ ≥ψCϕfϕb
H∞μ
sup
z∈Bμ
|z|ψCϕfϕb z
≥ μ
|b|ψb 1−ϕb2n1α/p,
2.9
from which we get2.6.
Conversely, suppose that 2.6 holds. Then for arbitrary z ∈ B and f ∈ Apα, by Lemma 2.1we have
μ
|z|ψCϕf
zμ
|z|f
ϕzψz≤CfApα μ
|z|ψz
1−ϕz2n1α/p. 2.10
In light of condition2.6, the boundedness of the operatorψCϕ : Apα → Hμ∞follows from 2.10by taking the supremum overB. This proof is completed.
Theorem 2.6. Assume thatp > 0,αis a real number such thatnα1 > 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ∞is compact if and only ifψ ∈Hμ∞and
|ϕz| →1lim μ
|z|ψz
1−ϕz2n1α/p 0. 2.11
Proof. Assume thatψCϕ :Apα → Hμ∞is compact, thenψCϕ :Apα → Hμ∞is bounded. Taking fz≡1, we get thatψ∈Hμ∞. Letzkk∈Nbe a sequence inBsuch that|ϕzk| → 1 ask → ∞ if such a sequence does not exist that condition2.11is vacuously satisfied. Set
fkz
1−ϕ
zk2t−nα1/p
1− z, ϕ
zk
t , k∈N, 2.12
wheretsatisfies2.7. From1, Theorem 32, we see thatfkk∈Nis a bounded sequence in Apα. Moreover, it is easy to see thatfkconverges to zero uniformly on compact subsects ofB.
ByLemma 2.2, lim supk→ ∞ψCϕfkHμ∞ 0.On the other hand, we have ψCϕfk
Hμ∞ sup
z∈B μ
|z|ψCϕfk
z| ≥ μzkψ zk 1−ϕ
zk2n1α/p. 2.13
Hence
lim sup
k→ ∞
μzkψ zk 1−ϕ
zk2n1α/p 0, 2.14
from which2.11follows.
Conversely, assume thatψ ∈Hμ∞and2.11holds. Then, it is easy to check that2.6 holds. HenceψCϕ : Apα → Hμ∞is bounded. According to2.11, for givenε > 0, there is a constantδ∈0,1such that
sup
{z∈B:δ<|ϕz|<1}
μ
|z|ψz
1−ϕz2n1α/p < ε. 2.15
Letfkk∈Nbe a bounded sequence inApαsuch thatfk → 0 uniformly on compact subsets of Bask → ∞. LetδD{w∈B:|w| ≤δ}. From2.15andψ∈Hμ∞, we have
ψCϕfk
Hμ∞ sup
z∈B μ
|z|fk
ϕz ψz
sup
{z∈B:|ϕz|≤δ} sup
{z∈B:δ<|ϕz|<1}
μ
|z|ψzfk
ϕz
ψH∞μ sup
w∈δD
fkwCfk
Apα sup
{z∈B:δ<|ϕz|<1}
μ
|z|ψz 1−ϕz2n1α/p
≤ ψH∞μ sup
w∈δD
fkwCε.
2.16
SinceδDis a compact subset ofB, we have limk→ ∞supw∈δD|fkw| 0. Using this fact and lettingk → ∞in2.16, we obtain
lim sup
k→ ∞
ψCϕfk
Hμ∞≤Cε. 2.17
Sinceεis an arbitrary positive number, we obtain lim supk→ ∞ψCϕfkHμ∞ 0.ByLemma 2.2, the implication follows.
Theorem 2.7. Assume thatp > 0,αis a real number such thatnα1 > 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ,0∞ is bounded if and only ifψCϕ :Apα → Hμ∞is bounded andψ ∈Hμ,0∞.
Proof. Assume thatψCϕ : Apα → Hμ,0∞ is bounded. Then it is clear thatψCϕ :Apα → Hμ∞is bounded. Takingfz 1 and employing the boundedness ofψCϕ :Apα → Hμ,0∞, we see that ψ∈Hμ,0∞.
Conversely, assume thatψCϕ : Apα → Hμ∞ is bounded andψ ∈ Hμ,0∞. Suppose that f∈ApαwithfApα ≤L, using polynomial approximations we obtainsee, e.g.,1
|z| →lim1
1− |z|2n1α/pfz0. 2.18
From the above equality andψ ∈Hμ,0∞, we have that for everyε > 0, there exists aδ ∈0,1 such that whenδ <|z|<1,
1− |z|2n1α/pfz< ε
M, 2.19
μ
|z|ψz< ε
1−δ2n1α/p
L , 2.20
whereMis defined in2.6. Therefore, ifδ <|z|<1 andδ <|ϕz|<1, from2.6and2.19 we have
μ
|z|ψCϕf
z μ
|z|ψz 1−ϕz2n1α/p
1−ϕz2n1α/pf
ϕz
≤M
1−ϕz2n1α/pf
ϕz< ε.
2.21
Ifδ <|z|<1 and|ϕz| ≤δ, usingLemma 2.1and2.20we have
μ
|z|ψCϕf
z μ
|z|ψz 1−ϕz2n1α/p
1−ϕz2n1α/pf
ϕz
≤CfApα μ
|z|ψz 1−ϕz2n1α/p
≤ CfApα 1−δ2
n1α/pμ
|z|ψz< ε.
2.22
Combining2.21and2.22, we obtain thatψCϕf ∈Hμ,0∞. Sincef is an arbitrary element of Apαwe see that
ψCϕ
Apα
⊂Hμ,0∞, 2.23
which, along with the boundedness ofψCϕ :Apα → Hμ∞, implies the result.
Theorem 2.8. Assume thatp > 0,αis a real number such thatnα1 > 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ,0∞ is compact if and only if
|z| →lim1
μ
|z|ψz
1−ϕz2n1α/p 0. 2.24
Proof. Assume that2.24holds. For anyf ∈ApαwithfApα ≤1, by2.10we have
μ
|z|ψCϕf
z≤CfApα μ
|z|ψz
1−ϕz2n1α/p. 2.25
Using2.24, we get
|z| →lim1 sup
fAp
α≤1μ
|z|ψCϕf
z≤C lim
|z| →1
μ
|z|ψz
1−ϕz2n1α/p 0. 2.26
From this andLemma 2.4, we see thatψCϕ :Apα → Hμ,0∞ is compact.
Conversely, assume that ψCϕ : Apα → Hμ,0∞ is compact. ThenψCϕ : Apα → Hμ,0∞ is bounded andψCϕ :Apα → Hμ∞is compact. By Theorems2.6and2.7, we obtain
|ϕz| →1lim μ
|z|ψz
1−ϕz2n1α/p 0, 2.27
|z| →1lim μ
|z|ψz0. 2.28
Ifϕ∞<1, it holds that
|z| →lim1
μ
|z|ψz
1−ϕz2n1α/p ≤ 1
1− ϕ2∞n1α/p lim
|z| →1μ
|z|ψz0, 2.29
from which the result follows in this case.
Hence, assume thatϕ∞1. In terms of2.27, for everyε >0, there exists aδ∈0,1, such that whenδ <|ϕz|<1,
μ
|z|ψz
1−ϕz2n1α/p < ε. 2.30
According to2.28, for the aboveε, there exists anr∈0,1, such that whenr <|z|<1, μ
|z|ψz< ε
1−δ2n1α/p
. 2.31
Therefore, whenr <|z|<1 andδ <|ϕz|<1, we have that μ
|z|ψz
1−ϕz2n1α/p < ε. 2.32
Ifr <|z|<1 and|ϕz| ≤δ, we obtain μ
|z|ψz
1−ϕz2n1α/p ≤ 1
1−δ2n1α/pμ
|z|ψz< ε. 2.33
Combining2.32with2.33we get2.24, as desired.
2.2. Casen1α0
Theorem 2.9. Assume thatp > 1,αis a real number such thatnα1 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ :Apα → Hμ∞is bounded if and only if
M1:sup
z∈Bμ
|z|ψz
ln e
1−ϕz2 1−1/p
<∞. 2.34
Proof. Assume that2.34holds. Then for arbitraryz∈Bandf ∈Apα, byLemma 2.1we have μ
|z|ψCϕf
zμ
|z|f
ϕzψz
≤CfApαμ
|z|ψz
ln e
1−ϕz2 1−1/p
. 2.35
From2.34and2.35, the boundedness ofψCϕ :Apα → Hμ∞follows.
Now assume thatψCϕ :Apα → Hμ∞is bounded. Fora∈B, set
faz
ln e 1− |a|2
−1/p
ln e
1− z, a
. 2.36
By using2, Theorem 1.12, we easily check thatfa∈Ap−n1. Therefore, CψCϕ
Apα→Hμ∞ ≥ψCϕfϕb
Hμ∞
sup
z∈B μ
|z|ψCϕfϕb z
≥μ
|b|ψb
ln e
1−ϕb2 1−1/p
.
2.37
From the last inequality, we get the desired result.
Theorem 2.10. Assume thatp >1,αis a real number such thatnα1 0,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ∞is compact if and only ifψ ∈Hμ∞and
|ϕz| →lim 1μ
|z|ψz
ln e
1−ϕz2 1−1/p
0. 2.38
Proof. First assume that2.38holds andψ ∈Hμ∞. In this case, the proof ofTheorem 2.6still works with minor changes, hence we omit the details.
Now we assume thatψCϕ :Apα → Hμ∞is compact, then it is clear thatψCϕ :Apα → Hμ∞is bounded. Similarly to the proof ofTheorem 2.6, we see thatψ ∈ Hμ∞. Letzkk∈Nbe a sequence inBsuch that |ϕzk| → 1 ask → ∞if such a sequence does not exist that condition2.38is vacuously satisfied. Set
fkz
ln e
1−ϕ zk2
−1/p
ln e
1− z, ϕ
zk
, k∈N. 2.39
From2, Theorem 1.12, we see thatfkk∈Nis a bounded sequence inApα. Moreover,fk → 0 uniformly on compact subsets ofBask → ∞. It follows fromLemma 2.2thatψCϕfkHμ∞ → 0 ask → ∞.Because
ψCϕfk
Hμ∞ sup
z∈B μ
|z|ψCϕfk
z
≥μzkψ zk
ln e
1−ϕ zk2
1−1/p ,
2.40
we obtain
klim→ ∞μzkψ zk
ln e
1−ϕ zk2
1−1/p
0, 2.41
from which we get the desired result. The proof is completed.
Theorem 2.11. Assume thatp >1,αis a real number such thatnα1 0,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ,0∞ is bounded if and only ifψCϕ :Apα → Hμ∞is bounded andψ ∈Hμ,0∞.
Proof. First assume thatψCϕ : Apα → Hμ,0∞ is bounded. Then clearly ψCϕ : Apα → Hμ∞ is bounded. Takingfz 1, then employing the boundedness ofψCϕ :Apα → Hμ,0∞, we have thatψ∈Hμ,0∞, as desired.
Conversely, assume that ψCϕ : Apα → Hμ∞ is bounded and ψ ∈ Hμ,0∞. For each polynomialp, we have
μ
|z|ψCϕp
zμ
|z|p
ϕzψz≤ p∞μ
|z|ψz, 2.42
from which we have thatψCϕp∈Hμ,0∞.
Since the set of all polynomials is dense inApαsee2, for everyf ∈ Apα there is a sequence of polynomialspkk∈Nsuch that
pk−f
Apα −→0 ask−→ ∞. 2.43
From the boundedness ofψCϕ:Apα → Hμ∞, we have that ψCϕpk−ψCϕf
H∞μ ≤ψCϕpk−f
Apα −→0 ask−→ ∞. 2.44 SinceHμ,0∞ is a closed subset ofHμ∞, we obtain
ψCϕf lim
k→ ∞ψCϕpk∈Hμ,0∞. 2.45
Therefore,ψCϕ:Apα → Hμ,0∞ is bounded.
Using Theorems 2.10 and 2.11, similarly to the proof of Theorem 2.8we obtain the following result. We omit the proof.
Theorem 2.12. Assume thatp >1,αis a real number such thatnα1 0,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ:Apα → Hμ,0∞ is compact if and only if
|z| →lim1μ
|z|ψz
ln e
1−ϕz2 1−1/p
0. 2.46
Theorem 2.13. Assume that 0< p≤1,αis a real number such thatnα10,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ :Apα → Hμ∞is bounded if and only ifψ∈Hμ∞.
Proof. Assume thatψ∈Hμ∞. For anyf∈Apα, byLemma 2.1we have
sup
z∈Bμ
|z|ψCϕf
z≤CfApαsup
z∈Bμ
|z|ψz. 2.47
From the above inequality, we obtain thatψCϕ:Apα → Hμ∞is bounded.
Conversely, assume thatψCϕ :Apα → Hμ∞is bounded. Takingfz 1 and using the boundedness ofψCϕ:Apα → Hμ∞, we getψ∈Hμ∞, as desired.
Theorem 2.14. Assume that 0< p≤1,αis a real number such thatnα10,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. ThenψCϕ :Apα → Hμ∞is compact if and only ifψ∈Hμ∞and
|ϕz| →lim 1μ
|z|ψz0. 2.48
Proof. First assume that ψ ∈ Hμ∞ and 2.48 holds. In this case, the proof is similar to the corresponding part of the proof ofTheorem 2.6and hence will be omitted.
Now we suppose thatψCϕ :Apα → Hμ∞is compact. It follows fromTheorem 2.13and the boundedness ofψCϕ :Apα → Hμ∞thatψ ∈Hμ∞. Letzkk∈Nbe a sequence inBsuch that
|ϕzk| → 1 ask → ∞. Set
fkz 1−ϕ zk2 1−
z, ϕ zk
, k∈N. 2.49
From 2, Theorem 6.6, we see that fkk∈N is a bounded sequence in Apα. Moreover, fk
converges to zero uniformly on compact subsets ofB. Hence byLemma 2.2it follows that lim sup
k→ ∞
ψCϕfk
Hμ∞ 0. 2.50
On the other hand, we obtain ψCϕfk
Hμ∞ sup
z∈B μ
|z|ψCϕfk
z≥μzkψ
zk. 2.51
Combining2.50with2.51we obtain that2.48holds.
Theorem 2.15. Assume that 0< p≤1,αis a real number such thatnα10,ψ ∈HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. Then the following statements are equivalent:
iψCϕ :Apα → Hμ,0∞ is bounded;
iiψCϕ :Apα → Hμ,0∞ is compact;
iiiψ ∈Hμ,0∞.
Proof. ii⇒i. This implication is clear.
i⇒iii. Takingfz 1 and employing the boundedness ofψCϕ :Apα → Hμ,0∞,we obtain thatψ ∈Hμ,0∞.
iii⇒ii. For anyf ∈ApαwithfApα ≤1, we have μ
|z|ψCϕf
z≤CfApαμ
|z|ψz≤Cμ
|z|ψz, 2.52
from which we obtain
|z| →lim1 sup
fAp
α≤1μ
|z|ψCϕf
z≤C lim
|z| →1μ
|z|ψz0. 2.53
UsingLemma 2.4, we obtain thatψCϕ :Apα → Hμ,0∞ is compact.
2.3. Casen1α <0
Theorem 2.16. Assume thatp > 0,αis a real number such thatnα1 < 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. Then the following statements are equivalent:
iψCϕ :Apα → Hμ∞is bounded;
iiψCϕ :Apα → Hμ∞is compact;
iiiψ ∈Hμ∞.
Proof. ii⇒i. This implication is obvious.
i⇒iii. Takingfz 1, then using the boundedness ofψCϕ :Apα → Hμ∞,we obtain thatψ∈Hμ∞.
iii⇒ii. Iff∈Apα, byLemma 2.1we obtain μ
|z|ψCϕf
z≤CfApαμ
|z|ψz, 2.54
from which it follows thatψCϕ:Apα → Hμ∞is bounded. Letfkk∈Nbe any bounded sequence inApαandfk → 0 uniformly onBask → ∞. ByLemma 2.3, we have
ψCϕfk
H∞μ sup
z∈Bμ
|z|fk
ϕz
ψz≤ ψH∞μ sup
z∈B
fk
ϕz−→0, 2.55
ask → ∞. The result follows fromLemma 2.2.
Similarly to the proof ofTheorem 2.15, we have the following result. We omit the proof here.
Theorem 2.17. Assume thatp > 0,αis a real number such thatnα1 < 0,ψ ∈ HB,ϕis a holomorphic self-map ofB,andμis a normal function on0,1. Then the following statements are equivalent:
iψCϕ :Apα → Hμ,0∞ is bounded;
iiψCϕ :Apα → Hμ,0∞ is compact;
iiiψ ∈Hμ,0∞.
Acknowledgment
The author is supported partly by the NSF of Guangdong Province of Chinano. 07006700.
References
1 R. Zhao and K. Zhu, “Theory of Bergman spaces on the unit ball,” to appear M´emoires de la Soci´et´e Math´ematique de France.
2 K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
3 A. L. Shields and D. L. Williams, “Bonded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287–302, 1971.
4 S. Stevi´c, “Essential norms of weighted composition operators from theα-Bloch space to a weighted- type space on the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 279691, 10 pages, 2008.
5 S. Stevi´c, “Norm of weighted composition operators from Bloch space toHμ∞on the unit ball,” Ars Combinatoria, vol. 88, pp. 125–127, 2008.
6 C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
7 X. Zhu, “Weighted composition operators betweenH∞and Bergman type spaces,” Communications of the Korean Mathematical Society, vol. 21, no. 4, pp. 719–727, 2006.
8 S. Stevi´c, “Weighted composition operators between mixed norm spaces andHα∞spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.
9 S. Li and S. Stevi´c, “Weighted composition operators betweenH∞andα-Bloch spaces in the unit ball,” Taiwanese Journal of Mathematics, vol. 12, pp. 1625–1639, 2008.
10 S. Li and S. Stevi´c, “Weighted composition operators fromH∞to the Bloch space on the polydisc,”
Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.
11 S. Li and S. Stevi´c, “Weighted composition operators fromα-Bloch space toH∞on the polydisc,”
Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911–925, 2007.
12 S. Stevi´c, “Composition operators betweenH∞andα-Bloch spaces on the polydisc,” Zeitschrift f ¨ur Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457–466, 2006.
13 X. Zhu, “Volterra composition operators from generalized weighted Bergman spaces toμ-Bloch type spaces,” to appear in Journal of Function Spaces and Applications.
14 D. D. Clahane and S. Stevi´c, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006.
15 X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,”
Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.
16 S. Li and S. Stevi´c, “Weighted composition operators from Bergman-type spaces into Bloch spaces,”
Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 3, pp. 371–385, 2007.
17 S. Li and S. Stevi´c, “Products of Volterra type operator and composition operator fromH∞and Bloch spaces to Zygmund spaces,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 40–52, 2008.
18 K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995.
19 S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,”
The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191–215, 2003.
20 A. Montes-Rodr´ıguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872–884, 2000.
21 S. Stevi´c, “Essential norms of weighted composition operators from the Bergman space to weighted- type spaces on the unit ball,” to appear in Ars Combinatoria.
22 X. Zhu, “Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces,” Indian Journal of Mathematics, vol. 49, no. 2, pp. 139–150, 2007.
