Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 801264, 14 pages
doi:10.1155/2010/801264
Research Article
Weighted Iterated Radial Composition Operators
between Some Spaces of Holomorphic Functions on
the Unit Ball
Stevo Stević
Mathematical Institute of the Serbian Academy of Science and Arts, Knez Mihailova 36/III, 11000 Beograd,
Serbia
Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs
Received 26 September 2010; Accepted 26 October 2010
Academic Editor: Chaitan Gupta
Copyright q 2010 Stevo Stević. 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.
The boundedness and compactness of weighted iterated radial composition operators from the
mixed-norm space to the weighted-type space and the little weighted-type space on the unit ball
are characterized here. We also calculate the Hilbert-Schmidt norm of the operator on the weighted
Bergman-Hilbert space as well as on the Hardy H 2 space.
1. Introduction
n
n
Let
z z1 , . . . , zn andn w w1 , . . . , wn be points in C n, z, w k1 zk wk , and |z| z, z. Let B {z ∈ C : |z| < 1} be the open unit ball in C , ∂B its boundary, and HB the
class of all holomorphic functions on B.
For an f ∈ HB with the Taylor expansion fz |β|≥0 aβ zβ , let
Rfz βaβ zβ
|β|≥0
1.1
be the radial derivative of f, where β β1 , β2 , . . . , βn is a multi-index, |β| β1 · · · βn and
β
β
zβ z11 · · · znn 1. It is easy to see that
Rfz ∇fz, z ,
where ∇f is the complex gradient of function f.
1.2
2
Abstract and Applied Analysis
The iterated radial derivative operator Rm f is defined inductively by
Rm f R Rm−1 f ,
m ∈ N \ {1}.
1.3
A positive continuous function ν on the interval 0, 1 is called normal 2 if there are
δ ∈ 0, 1 and τ and t, 0 < τ < t such that
νr
is decreasing on δ, 1,
1 − rτ
νr
1 − rt
is increasing on δ, 1,
νr
0,
− rτ
lim
r → 1 1
lim
νr
r → 1 1
− rt
1.4
∞.
If we say that a function ν : B → 0, ∞ is normal, we also assume that it is radial, that is,
νz ν|z|, z ∈ B.
Strictly positive continuous functions on B are called weights.
The weighted-type space Hμ∞ B Hμ∞ consists of all f ∈ HB such that
f ∞ : sup μzfz < ∞,
Hμ
1.5
z∈B
where μ is a weight see, e.g., 3, 4 as well as 5 for a related class of spaces.
∞
∞
The little weighted-type space Hμ,0
B Hμ,0
is a subspace of Hμ∞ consisting of all
f ∈ HB such that
lim μzfz 0.
1.6
|z| → 1
For 0 < p, q < ∞, and φ normal, the mixed-norm space Hp, q, φB Hp, q, φ
consists of all functions f ∈ HB such that
f Hp,q,φ
1
r
dr
1−r
φ
p
Mq f, r
0
p
1/p
< ∞,
1.7
where
1/q
frζq dσζ
,
Mq f, r 1.8
∂B
and dσ is the normalized surface measure on ∂B. For p q, φr 1 − r 2 α1/p , and α > −1,
p
p
the space is equivalent with the weighted Bergman space Aα B Aα , which is defined as
the class of all f ∈ HB such that
p
f p :
A
α
B
α
fzp 1 − |z|2 dV z < ∞,
1.9
Abstract and Applied Analysis
3
where dV z is the Lebesgue volume measure on B. Some facts on mixed-norm spaces in
various domains in Cn can be found, for example, in 6–8 see also the references therein.
For 0 < p < ∞ the Hardy space H p B H p consists of all f ∈ HB such that
f Hp
frζp dσζ
: sup
0<r<1
1/p
< ∞.
1.10
∂B
For p 2 the Hardy and the weighted Bergman space are Hilbert.
Let ϕ be a holomorphic self-map of B, u ∈ HB, and m ∈ N0 . For f ∈ HB, the
weighted iterated radial composition operator is defined by
m
Rm
u,ϕ f z uzR f ϕz ,
z ∈ B.
1.11
Note that the operator is the composition of the multiplication, composition and the iterated
radial operator, that is
m
Rm
u,ϕ Mu ◦ Cϕ ◦ R .
1.12
This is one of the product operators suggested by this author to be investigated at numerous
talks e.g., in 9. Note that for m 0 the operator Rm
u,ϕ becomes the weighted composition
operator see, e.g., 4, 8, 10. It is of interest to provide function-theoretic characterizations
for when ϕ and u induce bounded or compact weighted iterated radial composition operators
on spaces of holomorphic functions. Studying products of some concrete linear operators on
spaces of analytic functions attracted recently some attention see, for example, 11–32 as well
as the related references therein. Some operators on mixed-norm spaces have been studied,
for example, in 8, 10, 11, 16, 25, 26, 29, 33 see also the references therein.
Here we study the boundedness and compactness of weighted iterated radial
composition operators from mixed-norm spaces to weighted-type spaces on the unit ball for
the case m ∈ N. We also calculate the Hilbert-Schmidt norm of the operator on the weighted
Bergman-Hilbert space A2α B as well as on the Hardy H 2 B space.
In this paper, constants are denoted by C, they are positive and may differ from one
occurrence to the other. The notation a b means that there is a positive constant C such that
a ≤ Cb. If both a b and b a hold, then one says that a b.
2. Auxiliary Results
In this section we quote several lemmas which are used in the proofs of the main results.
The next characterization of compactness is proved in a standard way, hence we omit
its proof see, e.g., 34.
Lemma 2.1. Assume p, q > 0, ϕ is a holomorphic self-map of B, u ∈ HB, φ is normal and μ is
∞
a weight. Then the operator Rm
u,ϕ : Hp, q, φ → Hμ is compact if and only if for every bounded
sequence fk k∈N ⊂ Hp, q, φ converging to 0 uniformly on compacts of B as k → ∞, one has
lim Rm
u,ϕ fk k→∞
Hμ∞
0.
2.1
4
Abstract and Applied Analysis
The following lemma is a slight modification of Lemma 2.5 in 8 and is proved similar
to Lemma 1 in 35.
∞
is compact if and only if it is
Lemma 2.2. Assume μ is a normal weight. Then a closed set K in Hμ,0
bounded and
lim sup μzfz 0.
2.2
|z| → 1 f∈K
The following lemma is folklore and in the next form it can be found in 36.
Lemma 2.3. Assume that 0 < p, q < ∞, φ is normal, and m ∈ N. Then for every f ∈ HB the
following asymptotic relationship holds:
1
p
r
dr f0 1−r
φ
p
Mq f, r
0
p
1
0
φp r
p
dr.
Mq Rm f, r 1 − rmp
1−r
2.3
Lemma 2.4. Assume that m ∈ N, 0 < p, q < ∞, φ is normal and f ∈ Hp, q, φ. Then, there is a
positive constant C independent of f such that
m
R fz ≤ C
f Hp,q,φ
|z|
φ|z| 1 − |z|2
n/qm .
2.4
Proof. Let g Rm−1 f and z ∈ B. By the definition of the radial derivative, the Cauchy-Schwarz
inequality and the Chauchy inequality, we have that
supBz,1−|z|/4 gw
Rgz ≤ |z|∇gz ≤ C|z|
.
1 − |z|
2.5
From 2.3 with m → m − 1 we easily obtain the following inequality see, e.g., 8,
Lemma 2.1:
gz ≤ C
f Hp,q,φ
n/qm−1 .
φ|z| 1 − |z|2
2.6
From 2.5 and 2.6 and the asymptotic relations
1 − |w| 1 − |z|,
inequality 2.4 follows.
φ|z| φ|w|,
1 − |z|
,
for w ∈ B z,
2
2.7
Abstract and Applied Analysis
5
Lemma 2.5. Let
fa,s z 1
,
1 − z, as
z ∈ B.
2.8
Then,
Rm fa,s z s
Pm z, a
,
1 − z, asm
2.9
where
m
m
Pm w sm−1 wm pm−1 swm−1 · · · p2 sw2 w,
2.10
m
and where pj s, j 2, . . . , m − 1 are nonnegative polynomials for s > 0.
Proof. We prove the lemma by induction. For m 1,
Rfa,s z s
z, a
1 − z, as1
,
z ∈ B,
2.11
which is formula 2.9 with P1 w w.
Assume 2.9 is true for every m ∈ {1, . . . , l}. Taking the radial derivative operator on
equality 2.9 with m l, we obtain
l
Rl1 fa,s z sR
s
1 − z, asl
l
l
s l sl−1 z, al1 pl−1 sz, al · · · p2 sz, a3 z, a2
s
1 − z, asl1
l
l
lsl−1 z, al l−1pl−1 sz, al−1 · · ·2p2 sz, a2 z, a 1−z, a
s
l
sl−1 z, al pl−1 sz, al−1 · · · p2 sz, a2 z, a
1 − z, asl1
l
l
sl z, al1 · · · s l − 2p2 s 3p3 s z, a3
,
1 − z, asl1
l
s l − 1 2p2 s z, a2 z, a
1 − z, asl1
from which the inductive proof easily follows.
2.12
6
Abstract and Applied Analysis
∞
∞
3. Boundedness and Compactness of Rm
u,ϕ : Hp, q, φ → Hμ (or Hμ,0 )
This section characterizes the boundedness and compactness of Rm
u,ϕ : Hp, q, φ
∞
.
Hμ∞ or Hμ,0
→
Theorem 3.1. Assume m ∈ N, 0 < p, q < ∞, φ is normal, μ is a weight, ϕ is a holomorphic self-map
∞
of B, and u ∈ HB. Then Rm
u,ϕ : Hp, q, φ → Hμ is bounded if and only if
μz|uz|ϕz
M1 : sup 2 n/qm < ∞.
z∈B
φ ϕz 1 − ϕz
3.1
∞
Moreover, if Rm
u,ϕ : Hp, q, φ → Hμ is bounded, then the following asymptotic relationship holds
m Ru,ϕ Hp,q,φ → Hμ∞
M1 .
3.2
Proof. Assume 3.1 holds. Then by Lemma 2.4 for each f ∈ Hp, q, φ, we have that
μz|uz|ϕz
m
μzRu,ϕ fz ≤ C f Hp,q,φ 2 n/qm .
φ ϕz 1 − ϕz
3.3
Taking the supremum over the unit ball in 3.3 and using 3.1 the boundedness of operator
∞
Rm
u,ϕ : Hp, q, φ → Hμ follows and
m Ru,ϕ Hp,q,φ → Hμ∞
≤ CM1 .
3.4
∞
Now assume that operator Rm
u,ϕ : Hp, q, φ → Hμ is bounded. By using the test
functions
fj z zj ∈ H p, q, φ ,
j 1, . . . , n,
3.5
we obtain
∞
Rm
u,ϕ fj ∈ Hμ ,
j 1, . . . , n,
3.6
that is, for each j 1, . . . , n, holds
m Ru,ϕ fj Hμ∞
sup μz|uz|ϕj z ≤ zj Hp,q,φ Rm
u,ϕ z∈B
Hp,q,φ → Hμ∞
< ∞,
3.7
Abstract and Applied Analysis
7
which implies that
n
sup μz|uz|ϕz ≤
sup μz|uz|ϕj z
z∈B
j1 z∈B
≤ Rm
u,ϕ n zj < ∞.
∞
Hp,q,φ
Hp,q,φ → Hμ
3.8
j1
Let
fw z 1 − |w|2
t1
n/qt1
φw1 − z, w
.
3.9
It is known that L1 : supw∈B fw Hp,q,φ < ∞ see 8, Theorem 3.3. From this, using the
∞
boundedness of Rm
u,ϕ : Hp, q, φ → Hμ and by Lemma 2.5, we have that for each a ∈ B
L1 Rm
u,ϕ Hp,q,φ → Hμ∞
≥ Rm
f
u,ϕ ϕa Hμ∞
≥
≥
sup μz|uz|Rm fϕa ϕz z∈B
μa|ua|Pm ϕa2
n
t1
2 n/qm
q
φ ϕa 1 − ϕa
3.10
2
μa|ua|ϕa
n
t1
2 n/qm .
q
φ ϕa 1 − ϕa
From 3.10, we have that
∞>
2
μz|uz|ϕz
2 n/qm
|ϕz|>1/2 φ ϕz
1 − ϕz
sup
μz|uz|ϕz
≥ sup
2 n/qm .
|ϕz|>1/2 2φ ϕz
1 − ϕz
3.11
On the other hand, from 3.8 and since φ is normal, we obtain
2
μ ϕz |uz|ϕz
sup
≤ C sup μ ϕz |uz|ϕz < ∞.
n/qm
2
z∈B
|ϕz|≤1/2 φ ϕz
1 − ϕz
3.12
8
Abstract and Applied Analysis
From 3.8, 3.10, 3.11, and 3.12 condition 3.1 follows, and moreover
M1 ≤ C
Rm
u,ϕ Hp,q,φ → Hμ∞
.
3.13
From 3.4 and 3.13 asymptotic relationship 3.2 follows, finishing the proof of the theorem.
Theorem 3.2. Assume m ∈ N, 0 < p, q < ∞, φ is normal, μ is a weight, ϕ is a holomorphic self-map of
∞
m
∞
B and u ∈ HB. Then Rm
u,ϕ : Hp, q, φ → Hμ is compact if and only if Ru,ϕ : Hp, q, φ → Hμ
is bounded and
μz|uz|ϕz
lim
2 n/qm 0.
|ϕz| → 1 φ ϕz 1 − ϕz
3.14
∞
m
Proof. Suppose that Rm
u,ϕ : Hp, q, φ → Hμ is compact. Then it is clear that Ru,ϕ :
∞
Hp, q, φ → Hμ is bounded. If ϕ∞ < 1, then 3.14 is vacuously satisfied. Hence assume
that ϕ∞ 1. Let zk k∈N be a sequence in B such that |ϕzk | → 1 as k → ∞, and
fk z fϕzk z,k ∈ N, where fw is defined in 3.9. Then supk∈N fk Hp,q,φ < ∞, fk → 0
uniformly on compacts of B as k → ∞ since
2 t1
1 − ϕzk 0,
lim
k→∞
φ ϕzk 3.15
so that
lim Rm
u,ϕ fk k→∞
Hμ∞
0.
3.16
On the other hand, by 3.10, we have
2
μzk |uzk |ϕzk m ≤
C
f
∞.
R
k
u,ϕ
2 n/qm
Hμ
φ ϕzk 1 − ϕzk 3.17
From 3.16 and 3.17, equality 3.14 easily follows.
∞
Conversely, assume that Rm
u,ϕ : Hp, q, φ → Hμ is bounded and 3.14 holds. From
the proof of Theorem 3.1 we know that 3.1 holds. On the other hand, from 3.14, we have
that, for every ε > 0, there is a δ ∈ 0, 1, such that
μz|uz|ϕz
2 n/qm < ε
φ ϕz 1 − ϕz
whenever δ < |ϕz| < 1.
3.18
Abstract and Applied Analysis
9
Assume fk k∈N is a sequence in Hp, q, φ such that supk∈N fk Hp,q,φ ≤ L and fk
converges to 0 uniformly on compact subsets of B as k → ∞. Let K {z ∈ B : |ϕz| ≤ δ}.
Then from 3.18, and by Lemma 2.4, it follows that
m Ru,ϕ fk Hμ∞
sup μz|uz|Rm fk ϕz z∈B
≤ sup μz|uz|Rm fk ϕz sup μz|uz|Rm fk ϕz z∈K
z∈B\K
≤ sup μz|uz|ϕz∇Rm−1 fk ϕz z∈K
μz|uz|ϕz
C
fk Hp,q,φ sup
2 n/qm
z∈B\K φ ϕz
1 − ϕz
3.19
≤ K2 Csup∇Rm−1 fk ζ Cε
fk Hp,q,φ ,
|ζ|≤δ
where K2 : supz∈B μz|uz||ϕz| see 3.8. Therefore
m Ru,ϕ fk Hμ∞
≤ K2 sup∇Rm−1 fk ζ CLε.
|ζ|≤δ
3.20
Since fk k∈N converges to zero on compact subsets of B as k → ∞, by Cauchy’s estimates it
follows that the sequence |∇Rm−1 fk |k∈N also converges to zero on compact subsets of B as
k → ∞, in particular
lim sup∇Rm−1 fk ζ 0.
k → ∞ |ζ|≤δ
3.21
Using these facts and letting k → ∞ in 3.20, we obtain that
lim sup
Rm
f
k
u,ϕ k→∞
Hμ∞
≤ CLε .
3.22
Since ε is an arbitrary positive number it follows that the last limit is equal to zero. Applying
Lemma 2.1, the implication follows.
Theorem 3.3. Assume m ∈ N, 0 < p, q < ∞, φ, μ are normal, ϕ is a holomorphic self-map of B and
∞
m
∞
u ∈ HB. Then Rm
u,ϕ : Hp, q, φ → Hμ,0 is bounded if and only if Ru,ϕ : Hp, q, φ → Hμ is
bounded and
lim μz|uz|ϕz 0.
|z| → 1
3.23
∞
m
Proof. First assume that Rm
u,ϕ : Hp, q, φ → Hμ,0 is bounded. Then, it is clear that Ru,ϕ :
∞
Hp, q, φ → Hμ is bounded, and as in the proof of Theorem 3.1, by taking the test functions
fj z zj , j 1, . . . , n, we obtain 3.23.
10
Abstract and Applied Analysis
∞
Conversely, assume that the operator Rm
u,ϕ : Hp, q, φ → Hμ is bounded and
condition 3.23 holds. Then, for each polynomial p, we have
m ≤ μz|uz|ϕz
μz Rm
∇Rm−1 p
,
u,ϕ p z ≤ μz|uz| R p ϕz
∞
3.24
∞
from which along with condition 3.23 it follows that Rm
u,ϕ p ∈ Hμ,0 . Since the set of all
polynomials is dense in Hp, q, φ, we see that for every f ∈ Hp, q, φ there is a sequence of
polynomials pk k∈N such that
lim f − pk Hp,q,φ 0.
k→∞
3.25
∞
From this and by the boundedness of the operator Rm
u,ϕ : Hp, q, φ → Hμ , we have that
m
Ru,ϕ f − Rm
u,ϕ pk Hμ∞
≤ Rm
u,ϕ Hp,q,φ → Hμ∞
f − pk −→ 0
Hp,q,φ
3.26
∞
∞
m
as k → ∞. Hence Rm
u,ϕ Hp, q, φ ⊆ Hμ,0 , and consequently Ru,ϕ : Hp, q, φ → Hμ,0 is
bounded.
Theorem 3.4. Assume m ∈ N, 0 < p, q < ∞, φ, μ are normal, ϕ is a holomorphic self-map of B and
∞
u ∈ HB. Then Rm
u,ϕ : Hp, q, φ → Hμ,0 is compact if and only if
μz|uz|ϕz
lim
2 n/qm 0.
|z| → 1 φ ϕz 1 − ϕz
3.27
Proof. From 3.27, we see that 3.1 hold. This fact along with 3.3 implies that the set
∞
∞
Rm
u,ϕ {f | fHp,q,φ ≤ 1} is bounded in Hμ , moreover in Hμ,0 . By taking the supremum in
3.3 over the unit ball in Hp, q, φ, using 3.27 and applying Lemma 2.2 we obtain that the
∞
operator Rm
u,ϕ : Hp, q, φ → Hμ,0 is compact.
∞
m
is compact, then by Theorem 3.2, we have that condition
If Ru,ϕ : Hp, q, φ → Hμ,0
3.14 holds, which implies that for every ε > 0 there is an r ∈ 0, 1 such that
μz|uz|ϕz
2 n/qm < ε,
φ ϕz 1 − ϕz
3.28
for r < |ϕz| < 1.
As in Theorem 3.3, we have that 3.23 holds. Thus there is a σ ∈ 0, 1 such that
n/qm
μz|uz|ϕz < ε 1 − r 2
inf φ ϕz ,
|ϕz|≤r
for σ < |z| < 1.
3.29
Abstract and Applied Analysis
11
If |ϕz| ≤ r and σ < |z| < 1, then from 3.29, we obtain
μz|uz|ϕz
μz|uz|ϕz
< ε.
2 n/qm ≤ 1 − r 2 n/qm inf
|ϕz|≤r φ ϕz
1 − ϕz
φ ϕz
3.30
Using 3.30 and the fact that from 3.28, we have
μz|uz|ϕz
2 n/qm < ε,
φ ϕz 1 − ϕz
3.31
for σ < |z| < 1 and r < |ϕz| < 1, we get 3.27.
2
2
m
2
2
4. Hilbert-Schmidt Norm of Rm
u,ϕ : Aα → Aα and Ru,ϕ : H → H
2
2
In this section we calculate Hilbert-Schmidt norm of the operators Rm
u,ϕ : Aα → Aα and
m
2
2
Ru,ϕ : H → H . For some related results see 37, 38.
If H is a separable Hilbert space, then the Hilbert-Schmidt norm T HS of an operator
T : H → H is defined by
T HS ∞
1/2
T en 2
4.1
,
n1
where {en }n∈N is an orthonormal basis on H. The right-hand side in 4.1 does not depend on
the choice of basis. Hence, it is larger than the operator norm T op of T .
Let · , ·α , α ≥ −1, be the usual scalar product on A2α , where we regard that A2−1 H2 .
Since for each multi-index β β1 , . . . , βn ∈ Nn0
β! Γn α 1
β 2
,
z dvα z Γ
n β α 1
B
4.2
where β! β1 ! · · · βn !, and
B
zβ zγ dvα z 0,
β/
γ
4.3
see, e.g., 1, we have that the vectors
Γ n β α 1 β
z ,
eβ z β! Γn α 1
form an orthonormal basis in A2α .
β ∈ Nn0 ,
4.4
12
Abstract and Applied Analysis
2
Theorem 4.1. Let m ∈ N0 . Then Hilbert-Schmidt norm of the operator Rm
u,ϕ on Aα , α > −1 is
⎛
m Ru,ϕ HS
⎞1/2
⎞
⎜
1
⎜
⎟
2
m
⎜
⎠
⎝ |uz| ⎝R nα1
B
1 − nj1 wj
⎛
⎟
dvα z⎟
⎠
.
4.5
wj |ϕj z|2
Proof. By using the definition of the Hilbert-Schmidt norm and the monotone convergence
theorem, we have
2
Γ n β α 1 m β 2
m
m 2
Ru,ϕ z Ru,ϕ eβ Ru,ϕ HS
2,α
2,α
β! Γn α 1
β
β
n Γ n β α 1 m β
ϕj z2βj dvα z
|uz|2
β! Γn α 1
B
j1
β
4.6
n
m Γ n β α 1 β
ϕj z2βj dvα z.
|uz|
β! Γn α 1 j1
B
β
2
We also have that
⎞−nα1
⎛
⎞k
n
∞
n
⎝ wj ⎠ Γn α k 1
⎝1 − w j ⎠
k! Γn α 1
j1
j1
k0
⎛
n
∞ k! lj Γn α k 1
wj
l!
k! Γn α 1
j1
k0 |l|k
n
Γn α |l| 1 l! Γn α 1
l
4.7
lj
wj ,
j1
from which by taking the radial derivatives it follows that
Rm 1−
n
1
j1
wj
nα1 l
|l|m
n
Γn α |l| 1 lj
w .
l! Γn α 1 j1 j
4.8
From 4.6 and 4.8 the result easily follows.
Similar to Theorem 4.1 the following result regarding the case of the Hardy space is
proved. We omit the proof.
Abstract and Applied Analysis
13
2
Theorem 4.2. Let m ∈ N0 . Then, Hilbert-Schmidt norm of the operator Rm
u,ϕ on H , is
⎛
m Ru,ϕ HS
⎞
⎜
1
⎟
2⎜ m
⎜
sup ⎝ |urζ| ⎝R ⎠
n
S
0<r<1
1 − nj1 wj
⎞1/2
⎛
⎟
dσζ⎟
⎠
.
4.9
wj |ϕj rζ|2
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Wissenschaften, Springer, New York, NY, USA, 1980.
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