Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 5 (2012), 412–417
Research Article
Volterra composition operators from generally
weighted Bloch spaces to Bloch-type spaces on the
unit ball
Haiying Li∗, Haixia Zhang
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P.R. China.
Dedicated to George A Anastassiou on the occasion of his sixtieth birthday
Communicated by Professor R. Saadati
Abstract
Let ϕ be a holomorphic self-map of the open unit ball B, g ∈ H(B). In this paper, the boundedness and
compactness of the Volterra composition operator Tgϕ from generally weighted Bloch spaces to Bloch-type
c
spaces are investigated.2012
NGA. All rights reserved.
Keywords: Volterra composition operator; generally weighted Bloch space; boundedness; compactness.
2010 MSC: Primary 47B38.
1. Introduction and preliminaries
Let B be the unit ball in Cn and H(B) the class
Let z = (z1 , z2 , · · · , zn ), w =
P of all holomorphic functions
P on B.
∂f
(w1 , w2 , · · · , wn ) be points in Cn and hz, wi = nj=1 zj wj . Let Rf (z) = nj=1 zj ∂z
(z)
be the radial derivaj
tive of f ∈ H(B), see for more details in [14].
α of holomorphic functions such
For any 0 < α < ∞, we define the generally weighted Bloch space Blog
that
4
2 α
α = |f (0)| + sup(1 − |z| ) |Rf (z)| log
kf kBlog
< ∞.
1 − |z|2
z∈B
∗
Corresponding author
Email address: tslhy2001@yahoo.com.cn (Haiying Li)
Received 2011-6-4
H. Li, H. Zhang, J. Nonlinear Sci. Appl. 5 (2012), 412–417
413
When α = 1, for the case of the unit disk the logarithmic Bloch space has appeared for the first time in
characterizing of the multipliers of the Bloch space in [1], for more details in [12] and [13]. In [3], [4] and
[5], we studied composition operator on generally weighted Bloch spaces.
A positive continuous function µ on the interval [0,1) is called normal ([8]) if there are δ ∈[0,1) and a
and b, 0 < a < b such that
µ(r)
µ(r)
is decreasing on [0,1) and lim
= 0;
r→1 (1 − r)a
(1 − r)a
µ(r)
µ(r)
is increasing on [0,1) and lim
= 0.
b
r→1
(1 − r)
(1 − r)b
If we say that a function µ : B → [0, ∞) is normal, then we will also assume that µ(z) = µ(|z|), z ∈ B.
The Bloch-type space Bµ (B) consists of analytic functions f : B → C such that
kf kµ = sup µ(z)|Rf (z)| < ∞,
z∈B
where µ is normal.
In [9] and [10], it was shown that Bµ (B) is a Banach space with the norm kf kBµ = |f (0)| + kf kµ .
The little Bloch-type space Bµ,0 (B) consists of analytic functions f : B → C such that
lim µ(z)|Rf (z)| = 0.
|z|→1
Let ϕ be a holomorphic self-map of B. The composition operator Cϕ as usual is defined by
(Cϕ f )(z) = (f ◦ ϕ)(z), f ∈ H(B), z ∈ B.
For some results on composition operators, see [2] or [7].
Suppose that g : B → C is a holomorphic map, define
Z
Tg f (z) =
1
f (tz)
0
dg(tz)
=
dt
Z
1
f (tz)Rg(tz)
0
dt
, f ∈ H(B), z ∈ B.
t
(1.1)
This operator is called the Riemann-Stieltjes operator or extended Cesàro operator, see for example [11].
The Volterra composition operator is defined by
Z 1
Z 1
dg(tz)
dt
ϕ
=
(1.2)
Tg f (z) =
f (ϕ(tz))
f (ϕ(tz))Rg(tz) , f ∈ H(B), z ∈ B.
dt
t
0
0
When ϕ(z) = z, by (1.1) and (1.2), then Tgϕ f (z) = Tg f (z). The Volterra composition operator is a
natural extension of the Riemann-Stieltjes or extended Cesàro operator. The Volterra composition operator
on the unit disk is considered in [6]. The Volterra composition operators on logarithmic Bloch spaces on B
are studied in [15].
In this paper, we give the characterization of the boundedness and compactness of Volterra composition
operator Tgϕ from generally weighted Bloch spaces to Bloch-type spaces. 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.
α → B
2. The boundedness and compactness of Tgϕ : Blog
µ
In the beginning, we introduce some auxiliary results which will be needed in our proofs of the theorems.
H. Li, H. Zhang, J. Nonlinear Sci. Appl. 5 (2012), 412–417
414
α (B) and z ∈ B, then we have
Lemma 2.1. Let f ∈ Blog
1
α , when 0 < α < 1;
)kf kBlog
(1 − α) log 4
4 kf kB1 , when α = 1;
(b) |f (z)| ≤ C log log
log
1 − |z|2
α , when α > 1,
(c) |f (z)| ≤ (1 + A(|z|))kf kBlog
Z |z|
dt
where A(|z|) =
4 .
(1 − s2 )α log 1−s
0
2
(a) |f (z)| ≤ (1 +
Proof. Using the integral representation for R differential operator, we have
|f (z)| =
≤
≤
≤
Z 1
Rf (tz)dt f (0) +
t
0
Z 1
|z|dt
α
|f (0)| +
· kf kBlog
4
2
α
0 (1 − |tz| ) log 1−|tz|2
Z |z|
ds
α
· kf kBlog
|f (0)| +
4
2
(1 − s )α log 1−s
0
2
Z |z|
ds
α
α .
· kf kBlog
kf kBlog +
4
2
(1 − s )α log 1−s
0
2
For 0 < α < 1, α > 1, then (a) and (c) hold.
For α = 1,
Z
|z|
ds
1
4 · kf kBlog
log
(1 − s)α log 1−s
0
4
≤
2 log(2 log
) + 1 − 2 log log 4 kf kB1
log
1 − |z|2
4
≤
2 log log
+ log 4 + 1 − 2 log log 4 kf kB1
log
1 − |z|2
4
)kf kB1 .
≤ C(log log
log
1 − |z|2
|f (z)| ≤ kf kB1 + 2
Proposition 2.2. Let ϕ be a holomorphic self-map of the open unit ball B, g ∈ H(B), and α > 0. Then
α (B) → B (B) is compact if and only if for any bounded sequence (f )
α
Tgϕ : Blog
µ
j j∈N in Blog (B) which converges
ϕ
to zero uniformly on compact subsets of B as j → ∞, kTg fj kBµ → 0 as j → ∞.
α with f → 0 uniformly on
Proof. Assume that Tgϕ is compact and that (fj )j∈N is a bounded sequence in Blog
j
ϕ
ϕ
compact subsets of B. By the compactness of Tg , we have that the sequence (Tg fj )j∈N has a subsequence
α , then for any compact K ⊂ B,
(Tgϕ fjm )m∈N which converges to f in Bµ . By Lemma 2.1 and |f (0)| ≤ kf kBlog
there is a C ≥ 0 such that
|Tgϕ fjm (z) − f (z)| ≤ CkTgϕ fjm − f kBµ , ∀ z ∈ K.
H. Li, H. Zhang, J. Nonlinear Sci. Appl. 5 (2012), 412–417
415
This implies that Tgϕ fjm (z) − f (z) → 0 uniformly on compact subsets of B as m → ∞. Since fjm → 0
on compact subsets of B, by the definition of the operator Tgϕ , it is easy to see that for each z ∈ B,
limm→∞ Tgϕ fjm (z) = 0. Hence f = 0. By the arbitrary of (fj )j∈N , we obtain that Tgϕ fj → 0 in Bµ as j → ∞.
α (0, M ) (at the center of zero with the radius
Conversely, let {hj } be any sequence in the ball KM = BBlog
α
α
M ) of the space Blog . Since khj kBlog
≤ M < ∞, by Lemma 2.1, {hj } is uniformly bounded on compact
subsets of B and hence normal by Montel’s theorem. Hence we may extract a subsequence {hjm } which
α , moreover h ∈ B α and khk α ≤ M . It
converges uniformly on compact subsets of B to some h ∈ Blog
Blog
log
α
follows that (hjm − h) is that khjm − hkBlog
≤ 2M < ∞ and converges to zero on compact subsets of B,
by the hypothesis, we have that Tgϕ hjm → Tgϕ h in Bµ . Thus the set Tgϕ (K) is relatively compact. Hence
α → B is compact.
Tgϕ : Blog
µ
Here we only consider respectively the following two cases: 0 < α < 1; α > 1. Obviously, R(Tgϕ f )(z) =
f (ϕ(z))Rg(z).
Theorem 2.3. Let ϕ be a holomorphic self-map of the open unit ball B, g ∈ H(B), 0 < α < 1. Then the
following statements are equivalent.
α (B) → B (B) is bounded;
(i) Tgϕ : Blog
µ
α (B) → B (B) is compact;
(ii) Tgϕ : Blog
µ
(iii) g ∈ Bµ .
α (B) → B (B), then (i) holds.
Proof. (ii)⇒(i) By (ii) and the compactness of Tgϕ : Blog
µ
α . By taking the
(i)⇒(iii) By (i), then there exists a positive constant C such that kTgϕ f kBµ ≤ Ckf kBlog
α
test function f = 1 which is in Blog , then
sup µ(z)|Rg(z)| ≤ C.
z∈B
Then (iii) holds.
α and f → 0 uniformly on compact subsets of B,
(iii)⇒(i) For any bounded sequence (fk )k∈N in Blog
k
kTgϕ fk k = sup µ(z)|R(Tgϕ fk )(z)|
z∈B
= sup µ(z)|fk (ϕ(z))||Rg(z)|
z∈B
1
α
)|Rg(z)|kfk kBlog
(1 − α) log 4
z∈B
1
α }.
= sup µ(z)|Rg(z)| · {(1 +
)kfk kBlog
(1 − α) log 4
z∈B
≤ sup µ(z)(1 +
α , then (i) holds.
By (iii), g ∈ Bµ , moreover, fk ∈ Blog
Theorem 2.4. Let ϕ be a holomorphic self-map of the open unit ball B, g ∈ H(B), α > 1. Then the following
statements are equivalent.
α (B) → B (B) is bounded;
(i) Tgϕ : Blog
µ
(ii) g ∈ Bµ (B) and
sup µ(z)A(|ϕ(z)|)|Rg(z)| < ∞.
z∈B
(2.1)
H. Li, H. Zhang, J. Nonlinear Sci. Appl. 5 (2012), 412–417
416
α
Proof. (ii)⇒(i) Assume that (ii) holds. Then for f ∈ Blog
kTgϕ f k = sup µ(z)|R(Tgϕ f )(z)|
z∈B
= sup µ(z)|f (ϕ(z))||Rg(z)|
z∈B
α < ∞.
≤ sup µ(z)(1 + A(|ϕ(z)|))|Rg(z)|kf kBlog
z∈B
α (B) → B (B) is bounded.
By (ii), then we have Tgϕ : Blog
µ
Conversely, let
hz,ϕ(zk )i
Z
fk (z) =
0
dt
α
4 , k ∈ N, then fk ∈ Blog .
(1 − t)α log 1−t
∞ > kTgϕ fk k = sup µ(z)|fk (ϕ(z))||Rg(z)|
z∈B
≥ µ(zk )|fk (ϕ(zk ))||Rg(zk )|
≥ Cµ(zk )A(|ϕ(zk )|)|Rg(zk )|.
α , then g ∈ B .
Then (2.1) holds. By taking the test function f = 1 which is in Blog
µ
Theorem 2.5. Let ϕ be a holomorphic self-map of the open unit ball B, g ∈ H(B), α > 1. Then the following
statements are equivalent.
α (B) → B (B) is compact;
(i) Tgϕ : Blog
µ
(ii) g ∈ Bµ (B) and
lim µ(z)A(|ϕ(z)|)|Rg(z)| = 0.
(2.2)
|ϕ(z)|→1
α → B is compact. Let (ϕ(z ))
Proof. Assume that Tgϕ : Blog
µ
k k∈N be a sequence in B such that |ϕ(zk )| → 1
as k → ∞. Let
Z hz,ϕ(zk )i
2 Z |ϕ(zk )|2
−1
dt
dt
fk (z) =
, z ∈ B.
4
4
(1 − t)α log 1−t
(1 − t)α log 1−t
0
0
α and f → 0 uniformly on compact subsets of B,
Then (fk )k∈N is a bounded sequence in Blog
k
kTgϕ fk k ≥ µ(zk )|fk (ϕ(zk ))||Rg(zk )| ≥ Cµ(zk )A(|ϕ(zk )|)|Rg(zk )|.
α , then g ∈ B .
Then (2.2) holds by letting k → ∞. By taking the test function f = 1 which is in Blog
µ
Conversely, by (2.2), for any given ε > 0 there exists a positive number δ (δ < 1) such that
µ(z)A(|ϕ(z)|)|Rg(zk )| < ε wheneverδ < |ϕ(z)| < 1.
α and f → 0 uniformly on compact subsets of B,
For any bounded sequence (fk )k∈N in Blog
k
kTgϕ fk kBµ
= sup µ(z)|fk (ϕ(z))||Rg(z)|
z∈B
≤
sup µ(z)|fk (ϕ(z))||Rg(z)|
|ϕ(z)|≤δ
+ sup µ(z)|fk (ϕ(z))||Rg(z)|
|ϕ(z)|>δ
≤
sup µ(z)|Rg(z)| · sup |fk (ϕ(z))|
|ϕ(z)|≤δ
+Ckfk k
|ϕ(z)|≤δ
α
Blog
sup µ(z)|Rg(z)|(1 + A(|ϕ(z)|)).
|ϕ(z)|>δ
Then kTgϕ fk kBµ → 0 as k → ∞.
H. Li, H. Zhang, J. Nonlinear Sci. Appl. 5 (2012), 412–417
417
Acknowledgements
This work is supported by the Young Core Teachers Program of Henan Province (No.2011GGJS-062)
and the Natural Science Foundation of Henan Province(No.122300410110; 2010A110009).
References
[1] L. Brown, A. L. Shields. Multipliers and cyclic vectors in the Bloch space, Michigan Math. J. 38(1991), 141-146.
1
[2] C. C. Cowen and B. D. MacCluer. Composition operators on spaces of analytic functions, CRC Press, Boca
Roton, 1995. 1
[3] H. Li, P. Liu. Weighted compositioin operators between H∞ and generally weighted Bloch spaces of polydisk.
International Journal of Mathematics, 21(5) (2010), 687-699. 1
[4] H. Li, P. Liu. Composition operators between generally weighted Bloch space and Qqlog space, Banach Journal of
Mathematical Analysis, 3(1) (2009), 99-110. 1
[5] H. Li and X. Yang. Products of integral-type and composition operators from generally weighted Bloch space to
F (p, q, s) space, Filomat, 23(3) (2009), 231-241. 1
[6] S. Li. Volterra composition operators between weighted Bergman spaces and Bloch type spaces. J. Korean Math.
Soc. 45(1)(2008), 229-248. 1
[7] J. H. Shapiro. Composition operators and classical function theory, Springer-Verlag, New York, 1993. 1
[8] A. L. Shields, D. L. Williams. Bounded proje ctions,duality, and multipliers in spaces of analytic functions, Trans.
Amer. Math. Soc. 162 (1971), 287-302. 1
[9] S. Stević. Norm of weighted composition operators from Bloch space to Hµ∞ on the unit ball, Ars. Combin. 88
(2008), 125-127. 1
[10] X. Tang. Extended Cesàro operators between Bloch-type spaces in the unit ball of Cn , J. Math. Anal. Appl. 326(2)
(2007), 1199-1211. 1
[11] J. Xiao. Riemann-Stieltjes operators between weighted Bloch and Bergman spaces of the unit ball, J. London.
Math. Soc. 70(2) (2004), 199-214. 1
[12] S. Ye. Weighted composition operators between different weighted Bloch-type spaces, Acta Mathematica Sinica,
Chinese Series, 50(4) (2007), 927-942. 1
[13] R. Yoneda. The composition operator on weighted Bloch space, Arch. Math., 78 (2002), 310-317. 1
[14] K. Zhu. Spaces of Holomorphic functions in the unit ball,Springer, New York, 2005. 1
[15] X. Zhu. Volterra composition operators on logarithmic Bloch spaces. Banach J. Math. Anal. 3(1) (2009), 122-130.
1