- Research Article
- Open access
- Published:
Weighted Composition Operators from Logarithmic Bloch-Type Spaces to Bloch-Type Spaces
Journal of Inequalities and Applications volume 2009, Article number: 964814 (2009)
Abstract
The boundedness and compactness of the weighted composition operators from logarithmic Bloch-type spaces to Bloch-type spaces are studied here.
1. Introduction
Let be the unit disc in the complex plane
,
the normalized Lebesgue area measure on
,
the class of all holomorphic functions on
, and
the space of bounded holomorphic functions on
with the norm
The logarithmic Bloch-type space ,
was recently introduced in [1]. The space consists of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ1_HTML.gif)
The norm on is introduced as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ2_HTML.gif)
When ,
becomes the
-Bloch space
. For
-Bloch and other Bloch-type spaces, see, for example, [1–9], as well as the related references therein. For
,
is the logarithmic Bloch space, which appeared in characterizing the multipliers of the Bloch space (see [3, 9]).
The little logarithmic Bloch-type space ,
consists of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ3_HTML.gif)
The following theorem summarizes the basic properties of the logarithmic Bloch-type spaces. Here, as usual, for fixed
Theorem 1 A (see [1]).
The following statements are true.
(a)The logarithmic Bloch-type space is Banach with the norm given in (1.2).
(b) is a closed subset of
(c)Assume Then
if and only if
(d)The set of all polynomials is dense in .
(e)Assume then for each
,
. Moreover
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ4_HTML.gif)
A positive continuous function on
is called weight.
The Bloch-type space consists of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ5_HTML.gif)
where is a weight. With the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ6_HTML.gif)
the Bloch-type space becomes a Banach space.
The little Bloch-type space is a subspace of
consisting of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ7_HTML.gif)
Let be a holomorphic self-map of
and
. For
the corresponding weighted composition operator is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ8_HTML.gif)
It is of interest to provide function-theoretic characterizations for when and
induce bounded or compact weighted composition operators on spaces of holomorphic functions. For some classical results mostly on composition operators, see, for example, [10]. For some recent related results, mostly in
or related to Bloch-type or weighted-type spaces, see, for example, [4, 10–46] and the references therein.
Here we study the boundedness and compactness of the weighted composition operator from the logarithmic Bloch-type space and the little logarithmic Bloch-type space to the Bloch-type or the little Bloch-type space.
In this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation
means that there is a positive constant
such that
. We say that
, if both
and
hold.
2. Auxiliary Results
In this section we quote several auxiliary results which will be used in the proofs of the main results.
Lemma 2.1.
Assume ,
then the following statements are true.
(a)Assume then the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ9_HTML.gif)
is increasing on the interval
(b)The function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ10_HTML.gif)
is increasing on the interval
Proof.
-
(a)
We have
(2.3)
Since when
and
, and the function
is decreasing on the interval
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ12_HTML.gif)
from which this statement follows.
The proof of (b) is similar, hence it is omitted.
The next lemma regarding the point evaluation functional on follows from [1, Lemma  3] and some elementary asymptotic relationship, such as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ13_HTML.gif)
Lemma 2.2.
Let Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ14_HTML.gif)
for some independent of
The proof of the following lemma is similar to [25, Lemma  2.1], so we omit it.
Lemma 2.3.
Assume is a weight. A closed set
in
is compact if and only if it is bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ15_HTML.gif)
Remark 2.4.
If in Lemma 2.3 we assume that is not closed, then the word compact can be replaced by relatively compact.
The next characterization of compactness is proved in a standard way (see, e.g., the proofs of the corresponding lemmas in [10, 30, 47–49]). Hence we omit it.
Lemma 2.5.
Assume that ,
is a holomorphic self-map of
and
is a weight. Let
be one of the following spaces
,
and
one of the spaces
,
. Then the operator
is compact if and only if
is bounded and for every bounded sequence
converging to
uniformly on compacts of
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ16_HTML.gif)
Some concrete examples of the functions belonging to logarithmic Bloch-type spaces can be found in the next lemma.
Lemma 2.6.
The following statements are true.
(a)Assume that and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ17_HTML.gif)
where and
is a nonconstant function belonging to
(b)Assume that and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ18_HTML.gif)
where and
is a nonconstant function belonging to
(c)Assume that then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ19_HTML.gif)
where and
is a nonconstant function belonging to
Moreover, for each , it holds that
belong to the corresponding
space, and for fixed
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ20_HTML.gif)
Proof. (a) Let be fixed. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ22_HTML.gif)
where in (2.13) we have used that and in (2.14) we have used the fact that the function in (2.1) is increasing on the interval
.
From (2.13), since , and by Lemma 2.1(a), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ23_HTML.gif)
as , from which it follows that
as desired.
-
(b)
For fixed
, we have
(2.16)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ25_HTML.gif)
where in (2.16) we have used the assumption , while in (2.17), as in (a), we have used the fact that the function in (2.1) is increasing on the interval
.
From (2.16), and by Lemma 2.1(a), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ26_HTML.gif)
as . Hence
finishing the proof of this statement.
-
(c)
We have
(2.19)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ28_HTML.gif)
where we have used the assumption and the fact that function (2.1) is increasing on
.
From (2.19), Lemma 2.1(a), and since we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ29_HTML.gif)
as that is,
Estimations (2.12) follow from (2.14), (2.17), (2.20) and by using the following facts
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ30_HTML.gif)
we finish the proof of the lemma.
Remark 2.7.
Note that from Lemmas 2.2 and 2.6 the functions defined in (2.9)–(2.11) have maximal growths in the corresponding logarithmic Bloch-type spaces.
3. Boundedness and Compactness of the Operator![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_IEq126_HTML.gif)
This section studies the boundedness and compactness of the weighted composition operator .
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_IEq128_HTML.gif)
, .
Theorem 3.1.
Assume ,
,
is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is bounded if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ32_HTML.gif)
Proof.
First assume that (3.1) and (3.2) hold. Then, by Lemma 2.2 and the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ34_HTML.gif)
Applying (3.1) and (3.2) in (3.4), the boundedness of follows.
Now assume the operator is bounded. By taking the test functions
and
(which obviously belong to
), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ36_HTML.gif)
From (3.5) and (3.6), and since the function is bounded, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ37_HTML.gif)
For set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ38_HTML.gif)
We have that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ39_HTML.gif)
and as an easy consequence of Lemma 2.6(a), and
for each
Using these facts and the boundedness of , for the test functions
, where
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ40_HTML.gif)
From (3.10) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ41_HTML.gif)
On the other hand, by using (3.7) and Lemma 2.1(b), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ42_HTML.gif)
Hence, (3.11) and (3.12) imply (3.2).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ43_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ44_HTML.gif)
and by Lemma 2.6(a) we get , and
for every
. Using the boundedness of
, the test functions
, and equalities (3.14) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ45_HTML.gif)
for each ,
.
From (3.2), (3.5), (3.15), and using the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ46_HTML.gif)
condition (3.1) follows.
Theorem 3.2.
Assume ,
,
is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is compact if and only if
is bounded
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ47_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ48_HTML.gif)
Proof.
Suppose that is compact. Then it is clear that
is bounded. If
, then (3.17) and (3.18) are vacuously satisfied. Hence assume that
. Let
be a sequence in
such that
as
, and
where
is defined in (3.8). Then
,
uniformly on compacts of
as
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ49_HTML.gif)
Hence from (3.10) and Lemma 2.5 we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ50_HTML.gif)
from which (3.18) follows.
Let ,
where
is defined in (3.13). Then
and
uniformly on compact subsets of
as
. Since
is compact, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ51_HTML.gif)
From (3.15) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ52_HTML.gif)
which along with (3.16), (3.18), and (3.21) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ53_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ54_HTML.gif)
for some positive . From (3.23) and (3.24), equality (3.17) follows.
Conversely, assume that is bounded and (3.17) and (3.18) hold. From the proof of Theorem 3.1 we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ55_HTML.gif)
On the other hand, from (3.17) and (3.18) we have that, for every , there is a
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ56_HTML.gif)
whenever .
Assume is a sequence in
(or
) such that
and
converges to
uniformly on compact subsets of
as
Let
. Then from (3.25), (3.26), and by Lemma 2.2, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ57_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ58_HTML.gif)
Since converges to zero on compact subsets of
as
, by the Weierstrass theorem it follows that the sequence
also converges to zero on compact subsets of
as
, in particular
and
. Using these facts and letting
in the last inequality, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ59_HTML.gif)
Since is an arbitrary positive number it follows that the last limit is equal to zero. Applying Lemma 2.5, the implication follows.
Theorem 3.3.
Assume ,
,
is an analytic self-map of the unit disk,
, and
is a weight. Then
is bounded if and only if
is bounded
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ60_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ61_HTML.gif)
Proof.
First assume that is bounded. Then, it is clear that
is bounded, and as usual by taking the test functions
and
and using the fact
, we obtain (3.30) and (3.31).
Conversely, assume that the operator is bounded,
and condition (3.31) holds.
Then, for each polynomial , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ62_HTML.gif)
from which along with conditions (3.30) and (3.31) it follows that . Since according to Theorem A the set of all polynomials is dense in
we see that for every
there is a sequence of polynomials
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ63_HTML.gif)
From this and by the boundedness of the operator we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ64_HTML.gif)
as Hence
and consequently
is bounded.
Remark 3.4.
Note that Theorem 3.3 holds for all and
Theorem 3.5.
Assume ,
,
is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is compact if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ65_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ66_HTML.gif)
Proof.
If is compact, then it is bounded so that conditions (3.30) and (3.31) hold. On the other hand,
is compact, which implies that (3.17) and (3.18) hold.
By (3.18) we have that, for every , there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ67_HTML.gif)
when . From (3.31), there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ68_HTML.gif)
when , and where
is the function in Lemma 2.1(b).
Therefore, when and
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ69_HTML.gif)
On the other hand, if and
, from (3.38) and Lemma 2.1(b) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ70_HTML.gif)
Combining (3.39) and (3.40), we obtain (3.36). Similarly, from (3.17) and (3.30) is obtained (3.35), as claimed.
Conversely, assume that (3.35) and (3.36) hold. First note that (3.35) implies (3.30). Indeed if (3.30) did not hold then there would be a sequence and a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ71_HTML.gif)
and From this and the continuity of the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ72_HTML.gif)
we would have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ73_HTML.gif)
which is a contradiction with (3.35).
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ74_HTML.gif)
Using conditions (3.30), (3.35), and (3.36) in (3.44), it follows that for each
, moreover the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ75_HTML.gif)
is bounded in .
Taking the supremum in (3.44) over the unit ball of the space then letting
and using conditions (3.30), (3.35), and (3.36), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ76_HTML.gif)
from which along with Lemma 2.3 the compactness of the operator follows.
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_IEq266_HTML.gif)
, .
Theorem 3.6.
Assume that is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is bounded if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ77_HTML.gif)
Proof.
The proof of the theorem is similar to the proof of Theorem 3.1. The sufficiency follows by using the triangle inequality in (3.3) and then the third inequality in Lemma 2.2 and the definition of the space .
For the necessity it is enough to follow the lines of the corresponding part of the proof of Theorem 3.1 and use the test functions ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ78_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ79_HTML.gif)
which belong to (for the functions in (3.48) and (3.49) it easily follows by Lemma 2.6(b), where
is the function in (2.10). We omit the details.
The proofs of the following two theorems are similar to the proofs of Theorems 3.2 and 3.5, where the test functions in (3.48) and (3.49) are used as well as the lemmas in Section 2. Hence their proofs are omitted.
Theorem 3.7.
Assume that is an analytic self-map of the unit disk,
and
is a weight. Then the operator
is compact if and only if
is bounded
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ80_HTML.gif)
Theorem 3.8.
Assume that is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is compact if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ81_HTML.gif)
Case 3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_IEq285_HTML.gif)
.
The following results were proved in [15]. Hence we quote them for the benefit of the reader, and without any proof.
Theorem 3.9.
Assume that is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is bounded if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ82_HTML.gif)
Theorem 3.10.
Assume that is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is compact if and only if
is bounded
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ83_HTML.gif)
Theorem 3.11.
Assume that is an analytic self-map of the unit disk,
, and
is a weight. Then the operator
is compact if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ84_HTML.gif)
Case 4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_IEq299_HTML.gif)
, or and
Here we consider the cases , or
and
.
Theorem 3.12.
Assume that , or
and
,
,
is a weight, and
is a holomorphic self-map of
Then
is bounded if and only if
and condition (3.2) holds.
Proof.
The sufficiency follows by using the first inequality in Lemma 2.2 and the definition of the space in (3.3).
For the necessity, by using the test functions we first get conditions (3.5) and (3.7). To get (3.2) for the case
and
we use the test functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ85_HTML.gif)
Note that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ86_HTML.gif)
and similar to Lemma 2.6(b), and
for each
Hence for the family we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ87_HTML.gif)
from which along with (3.5) and the assumption , easily follows (3.2) in this case.
When , condition (3.2) follows as in Theorem 3.1, by using the test functions in (3.8).
Theorem 3.13.
Assume that , or
and
,
,
is a weight, and
is a holomorphic self-map of
, and
is bounded. Then
is compact if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F964814/MediaObjects/13660_2009_Article_2043_Equ88_HTML.gif)
and condition (3.18) holds.
Proof.
The proof is similar to the corresponding parts of the proofs of Theorems 3.2 and 3.7, so is omitted.
Remark 3.14.
Note that if , or
and
and
is compact, then condition (3.18) is proved as in Theorems 3.2 and 3.7, by using the test functions in (3.8) and (3.48). If
then condition (3.58) is vacuously satisfied. At the moment, we are not sure if the compactness implies condition (3.58) in the case
. Hence for the interested readers we leave this as an open problem.
The following theorem is proved as the corresponding part of Theorem 3.5.
Theorem 3.15.
Assume that , or
and
,
,
is a weight, and
is a holomorphic self-map of
. Then the operator
is compact if
and condition (3.36) holds.
Remark 3.16.
Note that if is compact, then clearly
.
References
Stević S: On new Bloch-type spaces. Applied Mathematics and Computation 2009,215(2):841–849. 10.1016/j.amc.2009.06.009
Anderson JM, Clunie J, Pommerenke Ch: On Bloch functions and normal functions. Journal für die reine und angewandte Mathematik 1974, 270: 12–37.
Brown L, Shields AL: Multipliers and cyclic vectors in the Bloch space. The Michigan Mathematical Journal 1991,38(1):141–146.
Clahane DD, Stević S: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball. Journal of Inequalities and Applications 2006, 2006:-11.
Li S, Stević S: Weighted-Hardy functions with Hadamard gaps on the unit ball. Applied Mathematics and Computation 2009,212(1):229–233. 10.1016/j.amc.2009.02.019
Stević S: On an integral operator on the unit ball in
. Journal of Inequalities and Applications 2005,2005(1):81–88. 10.1155/JIA.2005.81
Stević S: On Bloch-type functions with Hadamard gaps. Abstract and Applied Analysis 2007, 2007:-8.
Yamashita S: Gap series and
-Bloch functions. Yokohama Mathematical Journal 1980,28(1–2):31–36.
Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York, NY, USA; 2005:x+271.
Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.
Fang Z-S, Zhou Z-H: Differences of composition operators on the space of bounded analytic functions in the polydisc. Abstract and Applied Analysis 2008, 2008:-10.
Fu X, Zhu X: Weighted composition operators on some weighted spaces in the unit ball. Abstract and Applied Analysis 2008, 2008:-8.
Galanopoulos P: On
to
pullbacks. Journal of Mathematical Analysis and Applications 2008,337(1):712–725. 10.1016/j.jmaa.2007.02.049
Gu D: Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces. Journal of Inequalities and Applications 2008, 2008:-14.
Krantz SG, Stević S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Analysis: Theory, Methods & Applications 2009,71(5–6):1772–1795. 10.1016/j.na.2009.01.013
Li S, Stević S: Composition followed by differentiation between Bloch type spaces. Journal of Computational Analysis and Applications 2007,9(2):195–205.
Li S, Stević S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proceedings of Indian Academy of Sciences: Mathematical Sciences 2007,117(3):371–385. 10.1007/s12044-007-0032-y
Li S, Stević S: Weighted composition operators from
-Bloch space to
on the polydisc. Numerical Functional Analysis and Optimization 2007,28(7–8):911–925. 10.1080/01630560701493222
Li S, Stević S: Weighted composition operators from
to the Bloch space on the polydisc. Abstract and Applied Analysis 2007, 2007:-13.
Li S, Stević S: Generalized composition operators on Zygmund spaces and Bloch type spaces. Journal of Mathematical Analysis and Applications 2008,338(2):1282–1295. 10.1016/j.jmaa.2007.06.013
Li S, Stević S: Weighted composition operators between
and
-Bloch spaces in the unit ball. Taiwanese Journal of Mathematics 2008,12(7):1625–1639.
Li S, Stević S: Products of integral-type operators and composition operators between Bloch-type spaces. Journal of Mathematical Analysis and Applications 2009,349(2):596–610. 10.1016/j.jmaa.2008.09.014
Lindström M, Wolf E: Essential norm of the difference of weighted composition operators. Monatshefte für Mathematik 2008,153(2):133–143. 10.1007/s00605-007-0493-1
MacCluer BD, Zhao R: Essential norms of weighted composition operators between Bloch-type spaces. The Rocky Mountain Journal of Mathematics 2003,33(4):1437–1458. 10.1216/rmjm/1181075473
Madigan K, Matheson A: Compact composition operators on the Bloch space. Transactions of the American Mathematical Society 1995,347(7):2679–2687. 10.2307/2154848
Montes-RodrÃguez A: Weighted composition operators on weighted Banach spaces of analytic functions. Journal of the London Mathematical Society 2000,61(3):872–884. 10.1112/S0024610700008875
Ohno S: Weighted composition operators between
and the Bloch space. Taiwanese Journal of Mathematics 2001,5(3):555–563.
Ohno S, Zhao R: Weighted composition operators on the Bloch space. Bulletin of the Australian Mathematical Society 2001,63(2):177–185. 10.1017/S0004972700019250
Shi J, Luo L: Composition operators on the Bloch space of several complex variables. Acta Mathematica Sinica 2000,16(1):85–98. 10.1007/s101149900028
Stević S: Composition operators between
and
-Bloch spaces on the polydisc. Zeitschrift für Analysis und ihre Anwendungen 2006,25(4):457–466.
Stević S: Weighted composition operators between mixed norm spaces and
spaces in the unit ball. Journal of Inequalities and Applications 2007, 2007:-9.
Stević S: Essential norms of weighted composition operators from
-Bloch space to a weighted-type space on the unit ball. Abstract and Applied Analysis 2008, 2008:-11.
Stević S: Norm of weighted composition operators from Bloch space to
on the unit ball. Ars Combinatoria 2008, 88: 125–127.
Stević S: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball. Discrete Dynamics in Nature and Society 2008, 2008:-14.
Stević S: Essential norms of weighted composition operators from the Bergman space to weighted-type spaces on the unit ball. Ars Combinatoria 2009, 91: 391–400.
Stević S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. Journal of Mathematical Analysis and Applications 2009,354(2):426–434. 10.1016/j.jmaa.2008.12.059
Stević S: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball. Applied Mathematics and Computation 2009,212(2):499–504. 10.1016/j.amc.2009.02.057
Ueki S-I: Composition operators on the Privalov spaces of the unit ball of
. Journal of the Korean Mathematical Society 2005,42(1):111–127.
Ueki S-I, Luo L: Compact weighted composition operators and multiplication operators between Hardy spaces. Abstract and Applied Analysis 2008, 2008:-12.
Ueki S-I, Luo L: Essential norms of weighted composition operators between weighted Bergman spaces of the ball. Acta Scientiarum Mathematicarum 2008,74(3–4):829–843.
Wolf E: Compact differences of composition operators. Bulletin of the Australian Mathematical Society 2008,77(1):161–165.
Wolf E: Weighted composition operators between weighted Bergman spaces and weighted Bloch type spaces. Journal of Computational Analysis and Applications 2009,11(2):317–321.
Yang W: Weighted composition operators from Bloch-type spaces to weighted-type spaces. to appear in Ars Combinatoria to appear in Ars Combinatoria
Ye S: Weighted composition operator between the little
-Bloch spaces and the logarithmic Bloch. Journal of Computational Analysis and Applications 2008,10(2):243–252.
Zhu X: Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces. Indian Journal of Mathematics 2007,49(2):139–150.
Zhu X: Weighted composition operators from
spaces to
spaces to spaces. Abstract and Applied Analysis 2009, 2009:-14.
Li S, Stević S: Riemann-Stieltjes-type integral operators on the unit ball in
. Complex Variables and Elliptic Equations 2007,52(6):495–517. 10.1080/17476930701235225
Stević S: Boundedness and compactness of an integral operator on a weighted space on the polydisc. Indian Journal of Pure and Applied Mathematics 2006,37(6):343–355.
Stevich S: Boundedness and compactness of an integral operator in a mixed norm space on the polydisc. SibirskiÄ MatematicheskiÄ Zhurnal 2007,48(3):559–569.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Stević, S., Agarwal, R.P. Weighted Composition Operators from Logarithmic Bloch-Type Spaces to Bloch-Type Spaces. J Inequal Appl 2009, 964814 (2009). https://doi.org/10.1155/2009/964814
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/964814