Vol. LXXX, 1 (2011), pp. 107–117
PRODUCTS OF INTEGRAL-TYPE AND COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH
SPACES
HAIYING LI and TIANSHUI MA
Abstract. Letϕbe a holomorphic self-map of the open unit diskDon the complex plane and 0 < α, β < +∞. The boundedness and compactness of products of integral-type and composition operators between generally weighted Bloch spaces are investigated.
1. Introduction and preliminaries
LetDbe the unit disc on the complex plane andϕa holomorphic self-map ofD. We denote byH(D) the space of all holomorphic functions onD, denote by dm(z) the normalized Lebesgue area measure and define the composition operatorCϕon H(D) byCϕf =f◦ϕ.
The space of analytic functions onDsuch that kfkBlog =|f(0)|+ sup
z∈D
|f0(z)|(1− |z|2) log 2
1− |z|2 <∞
is called weighted Bloch space Blog. Blog and BM OAlog first appeared in the study of boundedness of the Hankel operators on the Bergman space
A1={f ∈H(D) : Z
D
|f(z)|dm(z)<∞}
and the Hardy spaceH1, respectively. BM OAlog also appeared in the study of a Volterra type operator (see e.g. [1, 2, 3, 4, 9, 10]). In [11], Yoneda studied the composition operators from Blog to BM OAlog. In [5, 6, 7], we introduced the spaceBlogα , α <0, the space of analytic functions on Dsuch that
kfkBαlog =|f(0)|+ sup
z∈D
|f0(z)|(1− |z|2)αlog 2
1− |z|2 <∞ that is called generally weighted Bloch spaceBαlog.
Received May 15, 2010; revised November 9, 2010.
2001Mathematics Subject Classification. Primary 47B38.
Key words and phrases. holomorphic self-map; composition operator; generally weighted Bloch space.
HAIYING LI and TIANSHUI MA
Letg∈H(D), forf ∈H(D) be the integral-type operatorIgandJgrespectively, defined by
Igf(z) =
z
Z
0
f0(ζ)g(ζ)dζ,
Jgf(z) =
z
Z
0
f(ζ)g0(ζ)dζ, z∈D.
The importance of the operatorsIg andJg comes from the fact that Iφf(z) +Jφf(z) =Mφf(z)−f(0)φ(0), z∈D, whereMg is the multiplication operator
(Mgf)(z) =g(z)f(z), f ∈H(D), z∈D.
The products of composition operators and integral-type operators are defined by
CϕJgf(z) =
ϕ(z)
Z
0
f(ξ)g0(ξ)dξ, JgCϕf(z) =
z
Z
0
f(ϕ(ξ))g0(ξ)dξ,
CϕIφf(z) =
ϕ(z)
Z
0
f0(ξ)φ(ξ)dξ, IφCϕf(z) =
z
Z
0
(f◦ϕ)0(ξ)φ(ξ)dξ.
In this article, we consider the characterization of boundedness and compactness of products of integral-type and composition operators between generally weighted Bloch spaces on the unit disk. Throughout the remainder of this paper C will denote a positive constant, the exact value of which will vary from one appearance to the next.
2. The boundedness and compactness of CϕJg(CϕIg) :Blogα →Bβlog At the beginning, the following Lemma 2.1 can be seen in [5].
Lemma 2.1. Let f ∈Blogα andz∈D, then (a) For0< α <1, |f(z)| ≤
1 + 1
(1−α) log 2
kfkBlogα ; (b) Forα= 1, |f(z)| ≤ log1−|z|4 2
log 2 kfkBlogα ; (c) Forα >1, |f(z)| ≤
1 + 2α−1 (α−1) log 2
1
(1− |z|2)α−1kfkBlogα .
Lemma 2.2. Assume that ϕ is a holomorphic self-map of D and α, β > 0.
Then CϕJg(orCϕIg) : Blogα → Blogβ is compact if and only if for any bounded sequence (fj)j∈N in Blogα , when fj → 0 uniformly on compact subsets of D, kCϕJgfjkBβ
log
→0 orkCϕIgfjkBβ log
)→0 asj→ ∞.
The result follows from standard arguments similar to those in [4].
It is easy to obtain the following result by a similar method in [8] for 0< α <1.
Lemma 2.3. Assume that ϕis a holomorphic self-map of D and 0 < α <1, β > 0. Then CϕJg : Bαlog → Blogβ is compact if and only if for any bounded sequence (fj)j∈N in Blogα , when fj → 0 uniformly on D, kCϕJgfjkBβ
log
→ 0 as j→ ∞.
Lemma 2.4. Assume thath∈H(D), f ∈Blogα ,α >0for a fixedz0∈D. Then there exists a positive constantC independent of f such that
z0
Z
0
f(ζ)h(ζ)dζ
≤CkfkBlogα max
|ζ|≤|z0||h(ζ)|,
z0
Z
0
f0(ζ)h(ζ)dζ
≤CkfkBlogα max
|ζ|≤|z0||h(ζ)|.
Proof. Forh∈H(D),f ∈Blogα ,then
z0
Z
0
f(ζ)h(ζ)dζ
≤ max
|ζ|≤|z0||f(ζ)| max
|ζ|≤|z0||h(ζ)|
≤
|f(0)|+|z0| max
|ζ|≤|z0||f0(ζ)|
max
|ζ|≤|z0||h(ζ)|
≤max (
1, |z0|
(1− |z0|2)αlog1−|z2
0|2
)
kfkBαlog max
|ζ|≤|z0||h(ζ)|.
Similarly, we have
z0
Z
0
f0(ζ)h(ζ)dζ
≤ |z0| max
|ζ|≤|z0||f0| max
|ζ|≤|z0||h(ζ)|
≤ |z0|
(1− |z0|2)αlog1−|z2
0|2
kfkBα
log max
|ζ|≤|z0||h(ζ)|.
Theorem 2.5. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α∈(0,1),β >0, thenCϕJg:Blogα →Blogβ is bounded if and only if
sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2 <∞.
(2.1)
Proof. Assume that CϕJg:Blogα →Bβlog is bounded. Then by the definition of the operatorCϕJg,
(CϕJgf)0(z) =f(ϕ(z))g0(ϕ(z))ϕ0(z).
(2.2)
HAIYING LI and TIANSHUI MA
Letf0(z) = 1,thenf0∈Blogα .Then by the boundedness of CϕJg
(1− |z|2)β|ϕ0(z)|g0(ϕ(z))|log 2
1− |z|2 ≤ kCϕJgkkf0kBlogα <∞.
(2.3)
Then (2.1) holds by (2.3).
Conversely, assume that (2.1) holds. Then by Lemma 2.1 and (2.2) (1− |z|2)β(CϕJgf)0(z) log 2
1− |z|2
≤CkfkBlogα (1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2 1− |z|2. (2.4)
Then, by Lemma 2.4, withh=g0 andz0=ϕ(0),
|(CϕJgfj)(0)|=
Z ϕ(0)
0
f(ζ)g0(ζ)dζ
≤CkfkBα
log max
|ζ|≤|ϕ(0)||g0(ζ)|.
(2.5)
By (2.4), we have kCϕJgfkBβ
log
≤C sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2 1− |z|2
+ max
|ζ|≤|ϕ(0)||g0(ζ)|
kfkBαlog.
By (2.1) and (2.5), the boundedness ofCϕJg is obtained.
Theorem 2.6. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α∈(0,1),β >0, thenCϕJg:Blogα →Blogβ is compact if and only if
sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2 <∞.
Proof. Assume that CϕJg :Blogα →Blogβ is compact, then it is bounded, hence (2.1) holds by Theorem 2.5.
Conversely, assume that (2.1) holds. Then by Theorem 2.5,CϕJg :Blogα →Blogβ is bounded. By Lemma 2.3 for any bounded sequence (fj)j∈N inBlogα ,whenfj →0 uniformly onD, we need only to prove thatkCϕJgfjkBβ
log
→0 asj→ ∞. Then
j→∞lim sup
z∈D
(1− |z|2)β(CϕJgfj)0(z) log 2 1− |z|2
≤sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2 1− |z|2 lim
j→∞kfjk∞= 0.
|(CϕJgfj)(0)|=
Z ϕ(0)
0
fj(ζ)g0(ζ)dζ
≤Ckfjk∞ max
|ζ|≤|ϕ(0)||g0(ζ)| →0as j→ ∞.
Then the compactness ofCϕJgis completed.
Theorem 2.7. Assume thatϕis a holomorphic self-map ofD,g∈H(D), β >0.
(i) If sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2
1− |ϕ(z)|2 <∞, (2.6)
thenCϕJg:Blog→Blogβ is bounded.
(ii) If CϕJg:Blog→Blogβ is bounded, then sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log log 2
1− |ϕ(z)|2 <∞.
(2.7)
Proof. (i) Forf ∈Blog,by Lemma 2.1, it holds (1− |z|2)β(CϕJgf)0(z) log 2
1− |z|2
≤CkfkBlog(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2 1− |ϕ(z)|2. By (2.6), we have thatCϕJg:Blog →Blogβ is bounded.
(ii) Assume thatCϕJg:Blog→Blogβ is bounded. Forw∈D, set fw(z) = log log 2
1−wz. Then
fw0(z) = 1
log1−wz2 · w 1−wz. Then|fw(0)|= log log 2 and
(1− |z|2)|fw0(z)|log 2
1− |z|2 =(1− |z|2)|w|log1−|z|2 2
|1−wz|log|1−wz|2
≤(1− |z|2) log1−|z|2 2
|1−z|log|1−z|2 <∞.
Thusfw∈Blog. Hence by the boundedness of CϕJg :Blog →Blogβ , we have (1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log log 2 1− |ϕ(z)|2
≤CkCϕJgfϕ(z)kBβ log
≤ kCϕJgk · kfϕ(z)kBlog <∞.
Theorem 2.8. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β >0.
(i) If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2 <∞
HAIYING LI and TIANSHUI MA
and lim
|ϕ(z)|→1(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2
1− |ϕ(z)|2 = 0, (2.8)
thenCϕJg:Blog→Bβlog is compact.
(ii) If CϕJg :Blog →Blogβ is compact, then lim
|ϕ(z)|→1(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log log 2
1− |ϕ(z)|2 = 0.
(2.9)
Proof. (i) By (2.8), we have that for anyε >0 there exists an r0 ∈(0,1) such that
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2
1− |ϕ(z)|2 < ε, (2.10)
for every|ϕ(z)|> r0.
Let (fj)j∈N be a norm bounded sequence inBlog such thatfj →0 uniformly on compact subsets ofDasj→ ∞. By Lemma 2.1, (2.1) and (2.10), we have
(1−|z|2)β(CϕJgfj)0(z) log 2 1− |z|2
≤ sup
|ϕ(z)|≤r0
|fj(ϕ(z))| sup
|ϕ(z)|≤r0
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2 1− |z|2 + CkfjkBlog sup
|ϕ(z)|>r0
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2 1− |ϕ(z)|2
≤C sup
|ζ|≤r0
|fj(ζ)|+CεkfjkBlog.
|(CϕJgfj)(0)|=
ϕ(0)
Z
0
f(ζ)g0(ζ)dζ
≤ max
|ζ|≤|ϕ(0)||fj(ζ)| max
|ζ|≤|ϕ(0)||g0(ζ)| →0 (j→ ∞).
Taking the supremum overz∈Dand lettingj→ ∞, we havekCϕJgfjkBβ log
→0 asj→ ∞. ThusCϕJg :Blog →Blogβ is compact.
(ii) Assume thatCϕJg :Blog →Blogβ is compact and (zn)n∈N is a sequence in Dsuch that limn→∞|ϕ(zn)|= 1. Let
fn(z) =
log log 2 1− |ϕ(zn)|2
−1
log log 2 1−ϕ(zn)z
!2
, n∈N.
Thenfn is a uniformly bounded family onBlog that converges to 0 on compact subsets ofD. ThenkCϕJgfnkBβ
log
→0 asn→ ∞.
kCϕJgfnkBβ log
≥sup
z∈D
(1− |z|2)β(CϕJgfn)0(z) log 2 1− |z|2
≥1− |zn|2)β|ϕ0(zn)||g0(ϕ(zn))|log 2
1− |zn|2log log 2 1− |ϕ(zn)|2. Hence
n→∞lim(1− |zn|2)β|ϕ0(zn)||g0(ϕ(zn))|log 2
1− |zn|2log log 2
1− |ϕ(zn)|2 = 0.
So (2.9) holds.
Theorem 2.9. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >1,β >0. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log1−|z|2 2 (1− |ϕ(z)|2)α−1 <∞, (2.11)
thenCϕJg:Blogα →Blogβ is bounded.
Proof. By Lemma 2.1 and (2.11), for f ∈Blogα , (1− |z|2)β(CϕJgf)0(z) log 2
1− |z|2
≤CkfkBα
log
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)α−1 <∞.
|(CϕJgf)(0)| ≤ max
|ζ|≤|ϕ(0)||f(ζ)| max
|ζ|≤|ϕ(0)||g0(ζ)|
≤max (
1, |ϕ(z0)|
(1− |ϕ(z0)|2) log1−|z2
0|2
)
kfkBlogα max
|ζ|≤|ϕ(0)||g0(ζ)|.
Then the boundedness ofCϕJg is obtained.
Theorem 2.10. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >1,β >0. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2 <∞ and
lim
|ϕ(z)|→1
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)α−1 = 0, (2.12)
thenCϕJg:Bαlog→Blogβ is compact.
HAIYING LI and TIANSHUI MA
Proof. By (2.12), then for any ε >0,there exists anr0∈(0,1) such that (1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)α−1 < ε, for every |ϕ(z)|> r0. Let (fj)j∈N be a norm bounded sequence in Blogα such that fj →0 uniformly on compact subsets ofDasj → ∞. By Lemma 2.1, we have
(1− |z|2)β(CϕJgfj)0(z) log 2 1− |z|2
≤ sup
|ϕ(z)|≤r0
|fj(ϕ(z))| sup
|ϕ(z)|≤r0
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2 1− |z|2 +CkfjkBα
log sup
|ϕ(z)|>r0
(1− |z|2)β|ϕ0(z)||g0(ϕ(z))|log 2
1− |z|2log 2 1− |ϕ(z)|2
≤C sup
|ζ|≤r0
|fj(ζ)|+CεkfjkBlogα .
|(CϕJgfj)(0)|=
ϕ(0)
Z
0
f(ζ)g0(ζ)dζ
≤CkfjkBα
log max
|ζ|≤|ϕ(0)||g0(ζ)|.
Taking the supremum overz ∈ D and letting j → ∞, kCϕJgfjkBβ log
→ 0. Thus
CϕJg:Blogα →Blogβ is compact.
Theorem 2.11. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α∈(0,1),β >0,thenJgCϕ:Blogα →Bβlog is bounded if and only ifJgCϕ: Blogα → Blogβ is compact if and only ifg∈Blogβ .
Theorem 2.12. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β >0,. If
sup
z∈D
(1− |z|2)β|g0(z)|log 2
1− |z|2log 2
1− |ϕ(z)|2 <∞, thenJgCϕ:Blog→Blogβ is bounded.
Theorem 2.13. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β >0, if
sup
z∈D
(1− |z|2)β|g0(z)|log 2
1− |z|2 <∞ and
|ϕ(z)|→1lim (1− |z|2)β|g0(z)|log 2
1− |z|2log 2
1− |ϕ(z)|2 = 0, thenJgCϕ:Blog →Blogβ is compact.
Theorem 2.14. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >1,β >0. If
sup
z∈D
(1− |z|2)β|g0(z)|log1−|z|2 2
(1− |ϕ(z)|2)α−1 <∞, thenJgCϕ:Bαlog→Blogβ is bounded.
Theorem 2.15. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >1,β >0. If
sup
z∈D
(1− |z|2)β|g0(z)|log 2
1− |z|2 <∞ and
lim
|ϕ(z)|→1
(1− |z|2)β|g0(z)|log1−|z|2 2
(1− |ϕ(z)|2)α−1 = 0, thenJgCϕ:Bαlog→Blogβ is compact.
Theorem 2.16. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >0,β >0,. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2 <∞, (2.13)
thenCϕIg: Blogα →Bβlog is bounded.
Proof. By the definition of CϕIg, (CϕIgf)0(z) = ϕ0(z)g(ϕ(z))f0(ϕ(z)). For f ∈Blogα , we have
(1− |z|2)β(CϕIgf)0(z) log 2 1− |z|2
≤ (1− |z|2)β|ϕ0(z)||g(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2
kfkBlogα .
|(CϕIgf)(0)|=
ϕ(0)
Z
0
f0(ζ)g(ζ)dζ
≤CkfkBα
log max
|ζ|≤|ϕ(0)||g(ζ)|.
By (2.13), we haveCϕIg:Bαlog→Bβlog is bounded.
Theorem 2.17. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >0,β >0,. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g(ϕ(z))|log 2
1− |z|2 <∞ (2.14)
and
lim
|ϕ(z)|→1
(1− |z|2)β|ϕ0(z)||g(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2 = 0, (2.15)
thenCϕIg: Blogα →Bβlog is compact.
Proof. By (2.15), for any ε >0,there exists anr∈(0,1) such that (1− |z|2)β|ϕ0(z)||g(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2
< ε (2.16)
for everyr <|ϕ(z)|<1.
HAIYING LI and TIANSHUI MA
Let (fj)j∈N be a norm bounded sequence inBlogα such thatfj →0 uniformly on compact subsets ofDasj→ ∞. Then
kCϕIgfjkBβ log
≤ sup
|ϕ(z)|≤r
(1− |z|2)β|ϕ0(z)||g(ϕ(z))||fj0(ϕ(z))|log 2 1− |z|2 + sup
|ϕ(z)|>r
(1− |z|2)β|ϕ0(z)||g(ϕ(z))||fj0(ϕ(z))|log 2 1− |z|2
+ max
|ζ|≤|ϕ(0)||fj0(ζ)| max
|ζ|≤|ϕ(0)||g(ζ)|
≤ sup
z∈D
(1− |z|2)β|ϕ0(z)||g(ϕ(z))|log 2 1− |z|2 sup
|ζ|≤r
|fj0(ζ)|
+ sup
|ϕ(z)|>r
(1− |z|2)β|ϕ0(z)||g(ϕ(z))|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2 kfjkBlogα
+ max
|ζ|≤|ϕ(0)||fj0(ζ)| max
|ζ|≤|ϕ(0)||g(ζ)|.
(2.17)
Since fj → 0 uniformly on compact subsets of D as j → ∞, by Cauchy’s estimate,fj0 →0 uniformly on compact subsets ofDasj → ∞. Hence by (2.14), (2.16) and (2.17), we havekCϕIgfjkBβ
log
→0 asj→ ∞. HenceCϕIg:Blogα →Blogβ
is compact.
Theorem 2.18. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >0,β >0,. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g(z)|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2 <∞, thenIgCϕ: Blogα →Bβlog is bounded.
Theorem 2.19. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α >0,β >0. If
sup
z∈D
(1− |z|2)β|ϕ0(z)||g(z)|log 2
1− |z|2 <∞ and
lim
|ϕ(z)|→1
(1− |z|2)β|ϕ0(z)||g(z)|log1−|z|2 2
(1− |ϕ(z)|2)αlog1−|ϕ(z)|2 2
= 0, thenIgCϕ: Blogα →Bβlog is compact.
Acknowledgment. This work is supported by the Natural Science Founda- tion of Henan (No. 2008B110006; 2010A110009; 102300410012) and the Foster Foundation of Henan Normal University (No. 2010PL01).
References
1. Attele K. R., M. Toeplitz and Hankel on Bergman one space, Hokaido Math J.,21(1992), 279–293.
2. Benke G., and Chang D. C.,A note on weighted Bergman spaces and the Ces`aro operator, Nagoya Mathematical J.,159(2000), 25–43.
3. Cima J., Stegenda D.,Hankel operators onHp, in: Berkson E. R., Peck N. T.and Ulh J.
(Eds.), Analysis at Urbana 1, London Math. Soc. Lecture Note ser., Cambridge Univ. Press, Cambridge,137(1989), 133–150.
4. Cowen C. C. and MacCluer B. D.,Composition operators on spaces of analytic functions, Boca Roton: CRC Press, 1995.
5. Li H., Liu P. and Wang M.,Composition operators between generally weighted Bloch spaces of polydisk, J. Inequal Pure Appl. Math,8(3)(2007), 1–8.
6. Li H. and Liu P.,Composition operators between generally weighted Bloch space andQqlog space. Banach Journal of Mathematical Analysis,3(1)(2009), 99–110.
7. Li H. and Yang X., Products of integral-type and composition operators from generally weighted Bloch space toF(p, q, s)space. Filomat,23(3)(2009), 231–241.
8. Ohno S., Stroethoff K. and Zhao R.,Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math.,33(1)(2003), 191–215.
9. Ramey W. and Ulrich D.,Bounded mean oscillation of Bloch pull-backs, Math. Ann.,291 (1991), 591–606.
10. Siskakis A. and Zhao R.A Volterra type operator on spaces of analytic functions, Contemp.
Math,232(1999), 299–311.
11. Yoneda R.,The composition operators on the weighted Bloch space, Arch. Math.,78(2002), 310–317.
Haiying Li, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China,e-mail:[email protected]
Tianshui Ma, College of Mathematics and Information Science, Henan Normal University, Xinx- iang 453007, P. R. China,e-mail:[email protected]
