Compactness of Composition Operators from the p-Bloch Space to the q-Bloch Space on the Classical Bounded Symmetric Domains ()
1. Introduction
Let
be a bounded homogeneous domain in
. The class of all holomorphic functions on
will be denoted by
. For
a holomorphic self-map of
and
, the composition
is denoted by
, and
is called the composition operator with symbol
.
The composition operators as well as related operators known as the weighted composition operators between the weighted Bloch spaces were investigated in [1] [2] in the case of the unit disk, and in [3] - [7] for the case of the unit ball. The study of the weighted composition operators from the Bloch space to the Hardy space
was carried out in [8] [9] for the unit ball. Characterizations of the boundedness and the compactness of the composition operators and the weighted ones between the Bloch spaces were given in [10] - [12] for the polydisc case, and in [13] - [18] for the case of the bounded symmetric domains. Furthermore, we will give some results about the composition operators for the case of the weighted Bloch space on the bounded symmetric domains.
In 1930s all irreducible bounded symmetric domains were divided into six types by E. Cartan. The first four types of irreducible domains are called the classical bounded symmetric domains, the other two types, called exceptional domains, consist of one domain each (a 16 and 27 dimensional domain).
The first three types of classical bounded symmetric domains can be expressed as follows [19] :
,
where
and
is the
identity matrix,
is the transpose of
;
![]()
![]()
Let
and
. The Kronecker product
of
and
is defined as the ![]()
matrix
such that the element at the
-th row and
-th column
[19] . Then the Berg- man metric of
is as follows (see [19] ):
(1.1)
where
is a complex vector,
is the conjugate transpose of
, and
.
Following Timoney’s approach (see [18] ), a holomorphic function
is in the Bloch space
, if
![]()
Now we define a holomorphic function
to be in the p-Bloch space
, if
(1.2)
where
![]()
![]()
We can prove that
is a Banach space with norm
which is similar
with the case on ![]()
Let
be a holomorphic self-map of
. We are concerned here with the question of when
will be a compact operator.
Let
denote a diagonal matrix with diagonal elements
. In this work,we shall de- note by
a positive constant, not necessarily the same on each occurrence.
In Section 2, we prove the equivalence of the norms defined in this paper and in [20] .
In Section 3, we state several auxiliary results most of which will be used in the proofs of the main results.
Finally, in Section 4, we establish the main result of the paper. We give a sufficient and necessary condition for the composition operator Cf from the p-Bloch space
to the q-Bloch space
to be compact, where
and
. Specifically,we prove the following result:
Theorem 1.1. Let
be a holomorphic self-map of
. Then
is compact if and only if, for every
, there exists a
such that
(1.3)
for all
whenever
,
.
The compactness of the composition operators for the weighted Bloch space on the bounded symmetric domains of
is similar with the case of
; we omit the details.
2. The Equivalence of the Norms
Denote [20]
.
Lemma 2.1. (Bloomfield-Watson) [21] Let
be an
Hermitian matrix. Then
(2.1)
where
is any
matrix and satisfies
.
Theorem 2.1.
and
are equivalent.
Proof. The metric matrix of
is
![]()
For any
, let
with
. Then
![]()
Denote
then
,
and
![]()
Thus
![]()
Hence
![]()
Furthermore,
![]()
Since
![]()
Thus
(2.2)
For
![]()
then we have
(2.3)
Combining (2.2) and (2.3),
![]()
Next,
![]()
and
![]()
Therefore, the proof is completed. □
3. Some Lemmas
Here we state several auxiliary results most of which will be used in the proof of the main result.
Lemma 3.1. [18] Let
be a bounded homogeneous domain. Then there exists a constant
, depending only on
, such that
(3.1)
for each
whenever f holomorphically maps
into itself. Here
denotes the Bergman metric
on
,
denotes the Jacobian matrix of
.
Lemma 3.2. Let
be a holomorphic self-map of
and
a compact subset of
.Then there exists a constant
such that
(3.2)
for all
whenever
.
Proof. For
, let ![]()
For any compact
, there exists a constant
such that
. Then there exists
such that
, whenever
.
Thus
(3.3)
Combining Lemma 3.1 with (3.3) shows that (3.2) holds. □
Lemma 3.3. (Hadamard) [21] Let
be an
Hermitian matrix. Then
(3.4)
and equality holds if and only if
is a diagonal matrix.
Lemma 3.4. Let
. Then
(3.5)
Proof. For any
, we have ![]()
Thus we have
, ![]()
It follows from Lemma 3.3 that
□
Lemma 3.5. Let
be a classical bounded symmetric domain, and
denote its metric matrix. Then a holomorphic function
on
is in
if and only if
(3.6)
If (3.6) holds, then
(3.7)
Proof. We can get the conclusion by the process of the proof on Theorem 2.1. □
Lemma 3.6. [18] Let
![]()
![]()
![]()
where
and
are unitary matrices and ![]()
Denote
,
. Then
(1) ![]()
(2) ![]()
(3)
;
(4)
for
;
(5)
for
;
(6)
for
.
Lemma 3.7.
is compact if and only if
as
for
any bounded sequence
in
that converges to 0 uniformly on compact subsets of
.
Proof. The proof is trial by using the normal methods. □
4. Proof of Theorem 1.1
Proof. Let
be a bounded sequence in
with
, and
uniformly on compact subsets of
.
Suppose (1.3) holds. Then for any
, there exists a
, such that
(4.1)
for all
whenever
and
.
By the chain rule, we have
.
If
and
, then we get
. If
and
, then
(4.2)
It follows from (4.1) and (4.2) that
![]()
(4.3)
(4.4)
whenever
and
.
On the other hand, there exists a constant
such that
![]()
So if
, then
![]()
We assume that
converges to 0 uniformly on compact subsets of
. By Weierstrass Theorem, it is easy to see that
converges to 0 uniformly on compact subsets of
. Thus, for given
, there exists
large enough such that
(4.5)
for any
,
whenever
and
. Then by in- equalities (4.3) and (4.5) and Lemma 3.2, it follows that, for
large enough,
(4.6)
whenever
and
.
Combining (4.4) and (4.6) shows that
as
large enough. So
is compact.
For the converse, arguing by contradiction, suppose
is compact and
the condition (1.3) fails. Then there exist an
, a sequence
in
with
as
and a sequence
in
, such that
(4.7)
for all
.
Now we will construct a sequence of functions
satisfying the following three conditions :
(I)
is a bounded sequence in
;
(II)
tends to 0 uniformly on any compact subset of
;
(III) ![]()
The existence of this sequence will contradict the compactness of
.
We will construct the sequence of functions
according to the following four parts: A - D.
Part A: Suppose that ![]()
where
is the
matrix whose element at the
row and
column is 1 and the other elements are 0. Since
maps
into itself,
and ![]()
Denote
by
Using formula (1.1), we have
![]()
Denote
![]()
then
(4.8)
We construct the sequence of functions
according to the following three different cases.
Case 1. If for some
,
(4.9)
then set
(4.10)
where
is any positive number.
Case 2. If for some
,
(4.11)
then set
(4.12)
where
, if for some
,
or for some
,
, replace the corresponding term
by 0 (the same below).
Case 3. If for some
,
(4.13)
then set
(4.14)
Next, we will prove that the sequences of functions
defined by (4.10), (4.12) and (4.14) all satisfy the conditions (I), (II) and (III).
To begin with, we will prove the sequence of functions
defined by (4.10) satisfies the three con- ditions. We can get that
![]()
It follows from Lemma 3.5 that
.
This proves that the sequence of functions
defined by (4.10) satisfies condition (I).
Let
be any compact subset of
. Then there exists a
such that
(4.15)
for any
. By (4.10), we have
![]()
Since
![]()
But
as
. Thus,
converges to 0 uniformly on
. Therefore,
converges to 0 uniformly on
as
. Thus, the sequence of functions
defined by (4.10) satisfies the condition (II).
Now (4.8) and (4.9) mean that
(4.16)
Combining (4.7) and (4.16), we have
![]()
Since
![]()
This proves that
as
, which means that the sequence of functions
defined by (4.10) satisfies condition (III).
We can prove that the sequence of functions
defined by (4.12) or (4.14) satisfies the conditions (I) - (III) by using the analogous method as above.
Part B: Now we assume that
![]()
It is clear that
and for
we can assume that
and
as
, where
.
If
, we can use the same methods as in Part A to construct a sequence of functions
satisfy- ing conditions (I)-(III).
Using formula (1.1), we have
![]()
Denote
![]()
Then,
(4.17)
We construct the sequence of functions
according to the following six different cases.
Case 1. If for some
,
![]()
then set
(4.18)
Case 2. If for some
,
![]()
then set
(4.19)
Case 3. If for some
,
![]()
then set
(4.20)
Case 4. If for some
,
![]()
then set
(4.21)
Case 5. If for some
,
![]()
then set
(4.22)
Case 6. If for some
,
![]()
then set
(4.23)
By using the same methods as in Part A, we can prove the sequences of functions
defined by (4.18)-(4.23) satisfying conditions (I) - (III).
Now, as an example,we will prove that the sequence of functions
defined by (4.19) satisfying the conditions (I) - (III).
For any
, we have
(4.24)
Thus
![]()
By Lemma
, we have
(4.25)
It follows from Lemma 3.5 and (4.25) that
. This proves that the sequence of functions
defined by (4.19) satisfy the condition (I).
Let
be any compact subset of
. Since there exists a
such that
,
Thus
![]()
Since
![]()
So
as
. Thus,
converges to 0 uni-
formly on E. Therefore,the sequence of
converges to 0 uniformly on
as
. Thus, the sequence of functions
defined by (4.19) satisfies the condition (II).
For case 2,
(4.26)
Combining (4.7) and (4.26), we have
![]()
Since
![]()
This proves that
as
, which means that the sequence of functions
defined by (4.19) satisfies condition (III).
If
, then by Lemma 3.6, there exist
and
in
such that
![]()
If we denote
, then
and
, where.
Denote
, where the sequence of functions
is the sequence obtained in Part A. We have
(4.27)
where
and
. Now (4.27) implies that
![]()
and
![]()
It is clear that
, and combining the discussion in Part A,we can get that
as
; that means the sequence of functions
satisfies condition (III).
We prove that the sequence of functions
is a bounded sequence in
.
Since
,
![]()
So
is bounded.
Next we prove that
converges to 0 uniformly on any compact subset
of
. Let
then by the definition of
and Lemma 3.6, we can get a calculation directly that
![]()
It is clear that
converges uniformly to
in
.
Since
and
, there similarly exist
in
such that
, and the first component of
is
. It is clear that
is holo-
morphic on
. Let
for
. For
, we know
. We may choose
such that
. Thus, for
large enough,
and from this it follows that
![]()
by the definition of
,
converges to 0 uniformly on
.
Hence
satisfies conditions (I)--(III), and this contradicts the compactness of
.
Part C: Assume that
![]()
where
. For
we may assume that
,
, ![]()
as
, where
,
.
Just as in Part B, we can use the same methods to prove the conclusion. And for
,
, we may only show the sequence of functions
which satisfy the conditions (I) - (III) here.
Using formula (1.1), we have
![]()
Denote
![]()
then,
(4.28)
We construct the sequence of functions
according to the following three different cases.
Case 1. If for some
,
![]()
then set
(4.29)
Case 2. If for some
,
![]()
then set
(4.30)
Case 3. If for some
,
![]()
then set
(4.31)
Using the same methods as in Part A and Part B, we can prove the sequences of functions
defined by (4.29)-(4.31) satisfying conditions (I) - (III).
Part D: In the general situation. For
, there exist an
unitary matrix
and an
unitary matrix
such that
![]()
We may assume that
and
as
. Let
and
;
means
that
as
for any
,
. Let
and
for
. Of course, P is an
unitary matrix,
is an
unitary matrix, and
con-
verges uniformly to
on
.
Let
,
where the sequence of
are the functions obtained in Part C.
From the same discussion as that in Part B, we know that
satisfies conditions (I) and (III). For the compact subset
,
is also a compact subset of
, so we can choose an open sub-
set
of
such that
. Since
converges uniformly to
on
, it follows that
as
. Since
tends to 0 uniformly on
, we know
tends to 0 uniformly on
. Thus,
satisfies condition (II). □
Acknowledgements
We thank the Editor and the referee for their comments. Research is funded by the National Natural Science Foundation of China (Grant No. 11171285) and the Postgraduate Innovation Project of Jiangsu Province of China (CXLX12-0980).
NOTES
*Corresponding author.