# Composition Operators from Certainμ-Bloch Spaces to Q

Authors: Chunyu Tan; Maofa Wang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Some necessary and sufficient conditions are established for composition operatorsC
φ to be bounded or compact from μ-Bloch type spaces B
μ to Q
p spaces. Moreover, the boundedness, compactness, and Fredholmness of composition operators on little spaces Q
p
,
0 are also characterized.


---

## Body

## 1. Introduction

LetD be the unit disc in the complex plane C and H
(
D
) the space of all analytic functions on D with the topology of uniform convergence on compact subsets of D. If f
∈
H
(
D
), we let f
r
(
z
)
=
f
(
r
z
), 0
<
r
<
1, be the dilation of f. The H
∞ space consists of all functions f
∈
H
(
D
) satisfying sup
⁡
z
∈
D
f
z
<
∞. The Bloch space B consists of all functions f
∈
H
(
D
) for which
(1)
f
B
∶
=
sup
⁡
z
∈
D
1
-
z
2
f
′
z
<
∞
.
B equipped with the norm f
∶
=
f
0
+
f
B becomes a Banach space (see [1, 2]). For α
>
0, the α-Bloch space B
α consists of all analytic functions f on D such that
(2)
f
B
α
∶
=
sup
⁡
z
∈
D
1
-
z
2
α
f
′
z
<
∞
.
Refer to [3] for more details on α-Bloch spaces. Recently, many authors have studied different classes of Bloch type spaces, where the typical weight function (
1
-
z
2
)
α is replaced by a continuous positive function μ defined on D. More precisely, let μ
:
D
→
(
0
,
∞
) be a radial weight function; that is, μ
(
z
)
=
μ
(
|
z
|
), z
∈
D, which is decreasing in a neighborhood of 1, continuous and lim
⁡
|
z
|
→
1
-
μ
(
|
z
|
)
=
0. The Bloch type space B
μ consists of all f
∈
H
(
D
) such that
(3)
f
B
μ
∶
=
sup
⁡
z
∈
D
μ
z
f
′
z
<
∞
.
It is easy to check that f
μ
∶
=
|
f
0
|
+
f
B
μ is a norm on B
μ, and B
μ is a Banach space equipped with this norm (see, e.g., [4]). Clearly, B
μ includes B
α as its special case. Indeed, if μ
(
z
)
=
(
1
-
|
z
|
2
)
α with α
>
0, B
μ becomes α-Bloch space B
α. When α
=
1, B
α is just the classical Bloch space B. For μ
(
z
)
=
(
1
-
|
z
|
2
)
ln
⁡
e
/
1
-
|
z
|
2, B
μ is logarithmic Bloch space, which first appeared in characterizing the multipliers of the Bloch spaces. The little Bloch-type space B
μ
,
0
=
B
μ
,
0
(
D
) consists of all f
∈
B
μ such that
(4)
lim
⁡
z
→
1
-
μ
z
f
′
z
=
0
.Fora
∈
D, let σ
a
(
z
)
=
a
-
z
/
1
-
a
-
z be the involutive automorphism of the unit disc which interchanges a and 0. We recall that in [5], for p
≥
0, f
∈
H
(
D
) belongs to Q
p if
(5)
sup
⁡
a
∈
D
∫
D
‍
f
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
∞
.
Q
p
,
0 is the subclass of Q
p consisting of all f
∈
Q
p such that
(6)
lim
⁡
a
→
1
-
∫
D
‍
f
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
=
0
.
With the norm,
(7)
f
Q
p
∶
=
f
0
+
sup
⁡
a
∈
D
∫
D
‍
f
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
1
/
2
,
Q
p is a Banach space, and Q
p
,
0 is the closure of all polynomials in Q
p. It is well known that Q
1
=
BMOA, the space of all analytic functions of the bounded mean oscillation on D. Q
0 is the classical Dirichlet space D. For all 1
<
p
<
∞, Q
p is the Bloch space B. Also, Q
1,0
=
VMOA, the subspace of BMOA consisting of all analytic functions with vanishing mean oscillation, and for p
>
1, Q
p
,
0
=
B
0; see [5, 6] for more details on those spaces.LetH
1 and H
2 be two linear subspaces of H
(
D
). If φ is an analytic self-map of D, then φ induces a composition operator C
φ
:
H
1
→
H
2 defined by
(8)
C
φ
(
f
)
∶
=
f
∘
φ
.
Composition operators have been studied by numerous authors in many subspaces of H
(
D
). Among others, Madigan and Matheson characterized the continuity and compactness of composition operators on the classical Bloch space B in [7]. Lou studied composition operators on Q
p spaces in [8]. Composition operators between the logarithmic Bloch-type space and Q
log
⁡
p are studied in [9–11].This paper studies composition operators fromμ-Bloch type spaces B
μ to Q
p spaces. After some necessary background materials, Section 2 gives some function-theoretic characterizations of bounded and compact composition operators C
φ
:
B
μ
→
Q
p by using the Hadamard gap series technique. Section 3 characterizes the continuity, compactness of C
φ
:
B
μ
→
Q
p
,
0, and the Fredholmness of C
φ on Q
p
,
0.Throughout the paper we use the same letterC to denote various positive constants which may change at each occurrence. Variables indicating the dependency of constants C will be often specified in the parenthesis. We use the notation X
≲
Y or Y
≳
X for nonnegative quantities X and Y to mean X
≤
C
Y for some inessential constant C
>
0. Similarly, we use the notation X
≈
Y if both X
≲
Y and Y
≲
X hold.


## 2. Composition Operators fromB
μ to Q
p

We recall that an analytic functionf on the unit disk D has Hadamard gaps if
(9)
f
(
z
)
=
∑
k
=
1
∞
‍
a
k
z
n
k
with n
k
+
1
/
n
k
≥
λ
>
1 for all k
∈
N. The following results are cited from [12].Theorem A.
Assume thatf
(
z
)
=
∑
n
=
0
∞
a
n
z
n
∈
B
μ. Then
(10)
limsup
⁡
n
→
∞
n
μ
1
-
1
n
a
n
<
∞
.Theorem B.
Assume thatμ is a nonincreasing radial weight satisfying
(11)
liminf
⁡
k
→
∞
μ
1
-
1
/
n
k
μ
1
-
1
/
n
k
+
1
=
q
>
1
,
μ
1
-
ln
⁡
1
z
≈
μ
z
,
a
s
z
⟶
1
-
,
and such that F
(
t
)
=
1
/
t
μ
(
1
-
1
/
t
) is a positive nonincreasing absolutely continuous function on the interval [
1
,
∞
) satisfying lim
⁡
t
→
∞
t
F
′
(
t
)
/
F
(
t
)
=
0 and lim
⁡
t
→
∞
t
2
F
(
t
)
=
∞, where {
n
k
} is a sequence such that n
k
+
1
/
n
k
=
p
>
1, k
∈
N. Let f
(
z
)
=
∑
k
=
1
∞
‍
a
k
z
n
k
∈
H
(
D
). If
(12)
limsup
⁡
k
→
∞
n
k
μ
1
-
1
n
k
a
k
<
∞
,
then f
∈
B
μ
(
D
).In the sequel, we always suppose thatμ is as in Theorem B. The next lemma will play a key role in our main results.Lemma 1.
There exist two functionsf
,
g
∈
B
μ such that
(13)
f
′
z
+
g
′
z
≳
1
μ
z
,
z
∈
D
.Proof.
We consider the function:(14)
f
(
z
)
=
ε
z
+
∑
j
=
1
∞
‍
q
j
-
1
μ
1
-
1
/
q
j
z
q
j
,
z
∈
D
,
where q is an appropriately large integer, and ε
>
0 is sufficiently small. It follows from Theorem B that f
∈
B
μ. We claim that
(15)
μ
z
f
′
z
≳
1
for 1
-
q
-
l
≤
|
z
|
≤
1
-
q
-
(
l
+
1
/
2
), l
∈
N. Indeed,
(16)
f
′
z
=
ε
+
∑
j
=
1
∞
‍
1
μ
1
-
1
/
q
j
z
q
j
-
1
>
1
μ
1
-
1
/
q
l
z
q
l
-
ε
+
∑
j
=
1
l
-
1
‍
1
μ
1
-
1
/
q
j
z
q
j
-
∑
j
=
l
+
1
∞
‍
1
μ
1
-
1
/
q
j
z
q
j
=
:
I
1
-
I
2
-
I
3
.
For q large enough, since
(17)
1
-
q
-
l
q
l
≤
z
q
l
≤
1
-
q
-
l
+
1
/
2
q
l
+
1
/
2
q
-
1
/
2
,
then
(18)
1
3
≤
z
q
l
≤
1
2
q
-
1
/
2
.
Hence
(19)
I
1
≥
1
3
1
μ
1
-
1
/
q
l
.
On the other hand, for q large enough,
(20)
I
2
≤
ε
+
∑
j
=
1
l
-
1
‍
1
μ
1
-
1
/
q
j
≤
1
μ
1
-
1
/
q
l
ε
+
1
q
-
1
,
I
3
≤
z
q
l
+
1
q
l
μ
(
1
-
1
/
q
l
)
∑
j
=
l
+
1
∞
‍
q
j
z
q
j
-
q
l
+
1
=
z
q
l
+
1
q
l
μ
1
-
1
/
q
l
∑
s
=
0
∞
‍
q
s
+
l
+
1
z
q
l
+
2
-
q
l
+
1
s
≤
z
q
l
q
μ
1
-
1
/
q
l
q
1
-
q
z
(
q
l
+
2
-
q
l
+
1
)
≤
q
2
-
q
1
/
2
μ
(
1
-
1
/
q
l
)
(
1
-
q
2
-
(
q
3
/
2
-
q
1
/
2
)
)
.
It follows from (19) and (20) that
(21)
f
′
z
≥
1
μ
1
-
1
/
q
l
×
1
3
-
(
ε
+
1
q
-
1
)
-
q
2
-
q
1
/
2
(
1
-
q
2
-
(
q
3
/
2
-
q
1
/
2
)
)
.
Since μ
(
1
-
1
/
q
l
+
1
/
2
)
≈
μ
(
1
-
1
/
q
l
) for sufficient large q,
(22)
f
′
z
≳
1
μ
(
1
-
1
/
q
l
)
≳
1
μ
(
1
-
1
/
q
l
+
1
/
2
)
≳
1
μ
(
z
)
for 1
/
q
l
+
1
/
2
≤
1
-
|
z
|
≤
1
/
q
l. That is (15).
Now with a similar argument for1
-
q
-
(
l
+
1
/
2
)
≤
|
z
|
≤
1
-
q
-
(
l
+
1
),  l
∈
N and q large enough, we have
(23)
μ
z
g
′
z
≳
1
,
where
(24)
g
(
z
)
=
∑
j
=
1
∞
‍
q
j
+
1
/
2
-
1
μ
1
-
1
/
q
j
+
1
/
2
z
q
j
+
1
/
2
.
Now inequality (13) follows immediately from (15) and (23) on the annulus 1
-
q
-
1
<
z
<
1.
On the other hand, in the disc|
z
|
≤
1
-
q
-
1, we have that g
′
(
0
)
=
0, f
′
(
0
)
≠
0, and f
′ and g
′ have a finite number of zeros in the disc. Hence if f
′ and g
′ have common zeros in the disc |
z
|
≤
1
-
q
-
1, then one can replace g by the function g
0
(
z
)
=
g
(
e
i
θ
z
) for an appropriate θ and obtain a pair of functions which satisfy inequality (13).We now characterize the boundedness of the composition operatorC
φ
:
B
μ
→
Q
p.Theorem 2.
Letp
>
0 and φ be an analytic self-map of the unit disc. Then C
φ
:
B
μ
→
Q
p is bounded if and only if
(25)
sup
⁡
a
∈
D
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
<
∞
.Proof.
Assume thatC
φ
:
B
μ
→
Q
p is bounded; then C
φ
f
∈
Q
p, for f
∈
B
μ. By Lemma 1, there exist f
,
g
∈
B
μ such that |
f
′
(
z
)
|
+
|
g
′
(
z
)
|
≳
1
/
μ
(
z
). So
(26)
∞
>
sup
⁡
a
∈
D
∫
D
‍
f
∘
φ
′
z
2
+
g
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
≳
sup
⁡
a
∈
D
∫
D
‍
f
∘
φ
′
z
+
g
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
=
sup
⁡
a
∈
D
∫
D
‍
f
′
φ
z
+
g
′
φ
z
2
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
≳
sup
⁡
a
∈
D
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
,
which implies (25).
Conversely, for anyf
∈
B
μ, it is clear to see that
(27)
sup
⁡
a
∈
D
∫
D
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
=
sup
⁡
a
∈
D
∫
D
‍
f
′
φ
z
2
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
sup
⁡
a
∈
D
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
·
f
B
μ
2
.
By (25), C
φ
f
∈
Q
p. Then C
φ
:
B
μ
→
Q
p is bounded by the closed graph theorem.Now, we are going to characterize the compactness of composition operatorsC
φ
:
B
μ
→
Q
p. In [13], Tjani showed the following result.Lemma 3.
LetX, Y be two Banach spaces of analytic functions on D. Suppose the following:(1)
The point evaluation functions onY are continuous.
(2)
The closed unit ball ofX is compact subset of X in the topology of uniform convergence on compact sets.
(3)
T
:
X
→
Y is continuous when X and Y are given the topology of uniform convergence on compact sets.Then, T is a compact operator if and only if, given a bounded sequence {
f
n
} in X such that f
n
→
0 uniformly on compact sets, the sequence {
T
f
n
} converges to zero in the norm of Y.Observe that for any fixedz
∈
D we have
(28)
f
z
≲
f
0
+
log
⁡
1
1
-
z
f
B
≲
f
0
+
log
⁡
1
1
-
z
f
Q
p
,
so the point evaluation functionals on Q
p are continuous. Thus, as a consequence of Lemma 3, we have the following result.Lemma 4.
The composition operatorC
φ
:
B
μ
→
Q
p is compact if and only if for every bounded sequence {
f
n
}
n
∈
N
⊆
B
μ, which converges uniformly to zero on any compact subset of the unit disk, C
φ
(
f
n
)
Q
p
→
0 as n
→
∞.We now use Lemma4 above to give a characterization of compact composition operator C
φ
:
B
μ
→
Q
p.Theorem 5.
Letp
>
0 and φ be an analytic self-map of the unit disc. Then C
φ
:
B
μ
→
Q
p is compact if and only if φ
∈
Q
p and
(29)
lim
⁡
t
→
1
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
=
0
.Proof.
We first assume thatC
φ
:
B
μ
→
Q
p is compact; then φ
∈
Q
p. Since z
n
/
n
B
μ
≲
1 and z
n
/
n
→
0 as n
→
∞, uniformly on any compact subsets of the unit disk, then by Lemma 4, C
φ
(
z
n
/
n
)
Q
p
→
0 as n
→
∞. So for each t
∈
(
0,1
) and each ε
>
0, there exists n
0
∈
N such that
(30)
t
2
(
n
0
-
1
)
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
ε
.
If we choose t
≥
2
-
1
/
2
(
n
0
-
1
), then
(31)
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
2
ε
.
We now consider the functions f
r
(
z
)
=
f
(
r
z
) and r
∈
(
0,1
) for f with f
B
μ
≤
1. Since f
r
B
μ
≲
1 and f
r uniformly to f on any compact subsets of the unit disk, for ε
>
0 there exists r
0
∈
(
0,1
) such that, for all r
≥
r
0,
(32)
sup
⁡
a
∈
D
∫
D
‍
f
∘
φ
′
z
-
f
r
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
z
<
ε
.
Note that, for t
≥
2
-
1
/
2
(
n
0
-
1
),
(33)
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
2
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
∘
φ
′
z
-
f
r
0
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
+
2
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
r
0
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
2
ε
+
2
f
r
0
′
H
∞
2
×
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
4
ε
1
+
f
r
0
′
H
∞
2
.
Namely, for each f
B
μ
≤
1 and ε
>
0, there is 0
<
δ
<
1 and some constant C
(
f
) depending only on f such that, for t
∈
[
δ
,
1
),
(34)
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
C
(
f
)
ε
.
Since C
φ is compact, it maps the unit ball of B
μ into a relative compact subset of Q
p. Thus for each ε
>
0, there exists a finite collection of functions f
1
,
f
2
,
…
,
f
N in the unit ball of B
μ, such that for each f
B
μ
≤
1 there is a k
∈
{
1,2
,
…
,
N
} with
(35)
sup
⁡
a
∈
D
∫
D
‍
f
∘
φ
′
z
-
f
k
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
ε
.
If we take C
=
max
⁡
1
≤
k
≤
N
C
(
f
k
), then for t
∈
[
δ
,
1
)
(36)
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
k
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
C
ε
.
Then
(37)
sup
⁡
f
B
μ
≤
1
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≲
C
ε
,
which implies the desired estimate (29) by using Lemma 1 in a similar way as in the proof of Theorem 2.
Conversely, we assume thatφ
∈
Q
p and (29) holds. Let {
f
n
}
n
∈
N be a sequence of functions in the unit ball of B
μ, such that f
n
→
0 uniformly on the compact subsets of the unit disc as n
→
∞. We notice that, for t
∈
(
0,1
),
(38)
f
n
∘
φ
Q
p
2
≲
f
n
φ
0
2
+
sup
⁡
a
∈
D
∫
φ
z
≤
t
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
+
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
z
=
:
J
1
+
J
2
+
J
3
.
Since f
n
→
0 uniformly on the compact subsets of the unit disc, as n
→
∞, then f
n
′
→
0 as n
→
∞ uniformly on the compact subsets of the unit disc. So for every ε
>
0, there is n
0
∈
N such that, for each n
>
n
0, J
1
≤
ε, and J
2
≤
ε
φ
Q
p
2. Also notice that
(39)
J
3
≤
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
.
By (29) there exists t
0
∈
(
0,1
) such that, for every t
>
t
0, J
3
≤
ε. Thus C
φ
(
f
n
)
Q
p
→
0 as n
→
∞, which completes the proof by Lemma 4.The following corollary is an immediate result of Theorems2 and 5.Corollary 6.
Letp
∈
(
0
,
∞
). Then(1)
B
μ is embedded boundedly into Q
p if and only if
(40)
sup
⁡
a
∈
D
∫
D
‍
1
-
σ
a
z
2
p
μ
z
2
d
m
(
z
)
<
∞
.
(2)
B
μ is embedded compactly into Q
p if and only if
(41)
lim
⁡
t
→
1
sup
⁡
a
∈
D
∫
φ
z
>
t
‍
1
-
σ
a
z
2
p
μ
z
2
d
m
(
z
)
<
0
.


## 3. Composition Operators fromB
μ to Q
p
,
0

In this section, we investigate composition operators fromB
μ to Q
p
,
0. Contrast with the case C
φ
:
B
μ
→
Q
p, here the boundedness and compactness of C
φ
:
B
μ
→
Q
p
,
0 are equivalent. Last, we also characterize the Fredholmness of composition operators on Q
p
,
0. We first begin with the following.Lemma 7.
Suppose that0
<
p
<
∞ and φ is an analytic self-map of D. Then C
φ
:
B
μ
→
Q
p
,
0 is compact if and only if
(42)
lim
⁡
a
→
1
sup
⁡
f
B
μ
≤
1
∫
D
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
=
0
.Proof.
Suppose thatC
φ
:
B
μ
→
Q
p
,
0 is compact; then C
φ
(
B
μ
) is relatively compact in Q
p
,
0, where B
μ is the unit ball of B
μ. Let ε
>
0; then there is an (
ε
/
4
)-net f
1
,
f
2
,
…
,
f
N of C
φ
(
B
μ
). Then for any fixed f
∈
B
μ, there exists i
0
∈
{
1,2
,
…
,
N
} such that
(43)
f
-
f
i
0
∘
(
φ
)
Q
p
<
ε
4
.
Clearly, there is δ
>
0 such that
(44)
∫
D
‍
f
i
0
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
ε
4
for |
a
|
>
δ. So
(45)
∫
D
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
2
∫
D
‍
f
∘
φ
′
z
-
f
i
0
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
+
2
∫
D
‍
f
i
0
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
2
ε
4
+
2
ε
4
=
ε
for |
a
|
>
δ. So (42) is proved.
Conversely, suppose that (42) holds and (
f
n
)
⊆
B
μ with f
n
B
μ
≤
1, converging uniformly to 0 on compact subsets of D; we now prove
(46)
lim
⁡
n
→
∞
C
φ
f
n
Q
p
=
0
.
For any given ε
>
0, by (42), there is δ
>
0 such that, for all f
n,
(47)
sup
⁡
δ
<
a
<
1
∫
D
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
z
<
ε
;
that is, f
n
∘
φ
∈
Q
p
,
0. For a
∈
D, r
∈
(
0,1
) and D
r
=
{
z
∈
D
:
|
φ
(
z
)
|
>
r
}, set
(48)
T
r
(
a
)
=
∫
D
r
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
,
and then lim
⁡
r
→
1
T
r
(
a
)
=
0, which means that, for each a
∈
D, there exists r
a such that T
r
(
a
)
<
ε for all r
>
r
a. The same as in the proof of Lemma 1.3 in [14], T
r
(
a
) is a continuous function of a, so there is a neighbourhood N
(
a
)
⊆
D of a such that T
r
a
(
z
)
<
ε for all z
∈
N
(
a
). Since {
a
:
|
a
|
≤
δ
}
⊆
⋃
a
∈
{
a
:
|
a
|
≤
δ
}
‍
N
(
a
) and {
a
:
|
a
|
≤
δ
} is compact, there exist N
(
a
1
)
,
…
,
N
(
a
M
) such that {
a
:
|
a
|
≤
δ
}
⊆
⋃
i
=
1
M
‍
N
(
a
i
). For a
i, i
=
1
,
…
,
M, there exists r
a
i such that T
r
a
i
(
z
)
<
ε, z
∈
N
(
a
i
), i
=
1
,
…
,
M. Setting r
0
=
max
⁡
{
r
a
1
,
…
,
r
a
M
}, T
r
0
(
a
)
<
ε for all |
a
|
≤
δ. That is,
(49)
sup
⁡
a
≤
δ
∫
D
r
0
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
ε
.
On the other hand, since f
n converge to 0 uniformly on compact subsets of D, there exists n
0, such that, for all n
≥
n
0, |
f
n
′
(
z
)
|
2
≤
ε for |
z
|
≤
r
0. It follows from (42) that φ
∈
Q
p
,
0. So
(50)
sup
⁡
a
≤
δ
∫
D
∖
D
r
0
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
sup
⁡
a
≤
δ
ε
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
ε
φ
Q
p
2
.
It follows from (49) and (50) that for n
≥
n
0
(51)
sup
⁡
a
≤
δ
∫
D
‍
f
n
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
(
1
+
φ
Q
p
2
)
ε
.
Combining (47) and (51) implies that C
φ
(
f
n
)
Q
p
→
0 as n
→
∞, which completes the proof.The following theorem characterizes the equivalence of boundedness and compactness of composition operators fromB
μ to Q
p
,
0.Theorem 8.
Let0
<
p
<
∞ and φ is an analytic self-map of D. Then the following are equivalent. (1)
C
φ
:
B
μ
→
Q
p
,
0 is bound.
(2)
C
φ
:
B
μ
→
Q
p
,
0 is compact.
(3)
(52)
lim
⁡
a
→
1
-
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
=
0
.Proof.
(
1
)
⇔
(
3
). Assume that C
φ
:
B
μ
→
Q
p
,
0 is bounded; then φ
(
z
)
∈
Q
p
,
0 by taking f
(
z
)
=
z. By Lemma 1, there exist f
,
g
∈
B
μ such that |
f
′
(
z
)
|
+
|
g
′
(
z
)
|
≳
1
/
μ
(
z
). So
(53)
0
⟵
lim
⁡
a
→
1
-
∫
D
‍
f
∘
φ
′
z
2
+
g
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
≳
lim
⁡
a
→
1
-
∫
D
‍
f
∘
φ
′
z
+
g
∘
φ
′
z
2
×
1
-
σ
a
z
2
p
d
m
(
z
)
=
lim
⁡
a
→
1
-
∫
D
‍
f
′
φ
z
+
g
′
φ
z
2
×
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≳
lim
⁡
a
→
1
-
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
,
which implies (52).
Conversely, for anyf
∈
B
μ, it is clear that
(54)
lim
⁡
a
→
1
-
∫
D
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
=
lim
⁡
a
→
1
-
∫
D
‍
f
′
φ
z
2
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
lim
⁡
a
→
1
-
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
·
f
B
μ
2
.
By (52), C
φ
f
∈
Q
p
,
0. Then C
φ
:
B
μ
→
Q
p
,
0 is bounded by the closed graph theorem.
(
2
)
⇔
(
3
). Let C
φ
:
B
μ
→
Q
p
,
0 be compact. By Lemma 7, for any given ε
>
0, there is δ
>
0 such that
(55)
sup
⁡
f
B
μ
<
1
∫
D
‍
f
∘
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
<
ε
for |
a
|
>
δ, which implies (52) by Lemma 1.
Conversely, suppose that (52) holds; then for any function f
∈
B
μ,
(56)
∫
D
‍
f
′
φ
z
2
φ
′
z
2
1
-
σ
a
z
2
p
d
m
(
z
)
≤
∫
D
‍
φ
′
z
2
1
-
σ
a
z
2
p
μ
φ
z
2
d
m
(
z
)
·
f
B
μ
⟶
0
as |
a
|
→
1
-. Hence, C
φ
:
B
μ
→
Q
p
,
0 is compact by Lemma 7, which completes the proof.Finally, we consider the Fredholmness of composition operators onQ
p
,
0 spaces. For a Banach space X, recall that a bounded linear operator T on X is said to be Fredholm if both the dimension of its kernel and the codimension of its image are finite. This occurs if and only if T is invertible modulo the compact operators; that is, there is a bounded linear operator S such that both T
S
-
I and S
T
-
I are compact. We also notice that an operator is Fredholm if and only if its dual is Fredholm (see, e.g., [15]).Before giving our result on Fredholmness, we need a useful result due to Wirths and Xiao [16].Lemma 9.
Letp
∈
(
0
,
∞
) and f
∈
Q
p with f
r
(
z
)
∶
=
f
(
r
z
) for r
∈
(
0,1
). Then the following are equivalent. (1)
f
∈
Q
p
,
0.
(2)
lim
⁡
r
→
1
f
r
-
f
Q
p
=
0.
(3)
f belongs to the closure of the class of the polynomials in the norm ·
Q
p.
(4)
For anyϵ
>
0 there is a g
∈
Q
p
,
0 such that g
-
f
Q
p
<
ϵ.Theorem 10.
Letφ be an analytic self-map of the unit disc D. Then the following are equivalent.(1)
φ is a Möbius transformation of D.
(2)
C
φ
:
Q
p
,
0
→
Q
p
,
0 is invertible.
(3)
C
φ
:
Q
p
,
0
→
Q
p
,
0 is Fredholm.Proof.
(
1
)
⇒
(
2
). If φ
(
z
)
=
φ
a
(
z
)
=
a
-
z
/
1
-
a
-
z, a
∈
D, then φ
a
∘
φ
a
(
z
)
=
z; that is, φ
a
=
φ
a
-
1. Since Q
p
,
0 is Möbius invariant by [15], we get C
φ
-
1
=
C
φ
-
1. If φ
(
z
)
=
λ
z with |
λ
|
=
1, we also have C
φ
-
1
=
C
φ
-
1. Since any Möbius transformation φ can be expressed that φ
(
z
)
=
λ
a
-
z
/
1
-
a
-
z
(
|
λ
|
=
1
,
a
∈
D
), C
φ is invertible.
(
2
)
⇒
(
3
) is obvious.
(
3
)
⇒
(
1
). Suppose C
φ
:
Q
p
,
0
→
Q
p
,
0 is Fredholm. Note that φ cannot be a constant mapping. Otherwise, if φ
(
z
)
≡
a, we have (
z
-
a
)
n
∈
ker
⁡
C
φ and dim
⁡
ker
⁡
C
φ
=
∞, which contradicts the Fredholmness of C
φ.
Assumeφ is not one to one. So there exist z
1
,
z
2
∈
D, z
1
≠
z
2 with φ
(
z
1
)
=
φ
(
z
2
). Select the neighborhoods U, V of z
1, z
2, respectively, such that U
∩
V
=
∅. φ
(
U
)
∩
φ
(
V
) is a nonempty and open set due to φ being open by the Open Mapping Theorem, so there exist infinite sequences {
z
n
1
}
⊆
U, {
z
n
2
}
⊆
V such that φ
(
z
n
1
)
=
φ
(
z
n
2
)
=
ω
n which are distinct. Hence C
φ
*
δ
z
n
1
=
δ
φ
(
z
n
1
)
=
δ
φ
(
z
n
2
)
=
C
φ
*
δ
z
n
2; namely, C
φ
*
(
δ
z
n
1
-
δ
z
n
2
)
=
0, where δ
z
:
f
→
f
(
z
) is evaluation function, which is a bounded linear functional on Q
p
,
0. Since Q
p
,
0 contains all polynomials by Lemma 9, we have that each evaluation function is not a linear combination of other evaluation functions, so the sequence {
δ
z
n
1
-
δ
z
n
2
} is linearly independent in the kernel of the adjoint operator C
φ
*. It is worth pointing out that C
φ
* is also Fredholm. It is a contradiction, so φ is injective.
We now showφ is surjective. Assume that φ is not surjective. Then we can find z
0
∈
D
∩
∂
φ
(
D
) and {
z
n
}
⊆
D such that φ
(
z
n
)
→
z
0 as n
→
∞. Further, we get, by the Open Mapping Theorem, that |
z
n
|
→
1 as n
→
∞. For arbitrary f
∈
Q
p
,
0,
(57)
C
φ
*
δ
z
n
f
=
δ
φ
(
z
n
)
f
=
f
∘
φ
z
n
⟶
f
z
0
=
δ
z
0
f
;
we get δ
φ
(
z
n
)
w
*
⟶
δ
z
0 and {
δ
φ
(
z
n
)
} is bounded uniformly. Again, it is obvious that δ
z
n
→
∞ as n
→
∞. Therefore, δ
φ
(
z
n
)
/
δ
z
n
=
C
φ
*
δ
z
n
/
δ
z
n
→
0. On the other hand, since C
φ
* is also Fredholm, there are operators K and S on Q
p
,
0
*, with K compact and S bounded, such that S
C
φ
*
=
I
+
K. Thus, δ
z
n
/
δ
z
n
+
K
δ
z
n
/
δ
z
n
→
0. Because K is compact and {
δ
z
n
/
δ
z
n
} is bounded, there exists subsequence {
δ
z
n
k
/
δ
z
n
k
} such that K
δ
z
n
k
/
δ
z
n
k
→
h, δ
z
n
k
/
δ
z
n
k
→
-
h, which means h
=
1. Moreover, Q
p
,
0 is the closure of all polynomials with respect to the norm ·
Q
p by Lemma 9, which gets (
δ
z
n
k
/
δ
z
n
k
)
w
*
⟶
0. This implies that (
δ
z
n
k
/
δ
z
n
k
)
w
*
⟶
-
h
=
0. This is a contradiction. So φ is surjective. Thus φ is a Möbius transformation, which completes the proof.



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*Source: 892146-2014-12-22.xml*