Discrete Dynamics in Nature and Society

Volume 2008 (2008), Article ID 154263, 14 pages

http://dx.doi.org/10.1155/2008/154263

## On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia

Received 1 May 2008; Accepted 20 July 2008

Academic Editor: Leonid Berezansky

Copyright © 2008 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.

#### Abstract

We introduce an integral-type operator, denoted by , on the space of holomorphic functions on the unit ball , which is an extension of the product of composition and integral operators on the unit disk. The operator norm of from the weighted Bergman space to the Bloch-type space or the little Bloch-type space is calculated. The compactness of the operator is characterized in terms of inducing functions and . Upper and lower bounds for the essential norm of the operator , when , are also given.

#### 1. Introduction

Let be the open unit ball in the complex vector space its boundary, the open unit disk in the complex plane the Lebesgue measure on where and where the constant is chosen such that the normalized rotation invariant measure on (that is, ), the class of all holomorphic functions on the unit ball and the space of all bounded holomorphic functions on with the norm

Let and be points in ,and .

For with the Taylor expansion , letbe the radial derivative of where is a multi-index, and . It is well known (see, e.g., [1]) that

For the Hardy space consists of all such thatIt is well known that for every the radial limitexists almost everywhere on

The weighted Bergman space , consists of all such thatWhen , the weighted Bergman space with the norm becomes a Banach space. If , it is a Frechet space with the translation invariant metric

Since for every we will also use the notation for the Hardy space .

A positive continuous function on is called normal (see [2]) if there is and and such thatFrom now on if we say that a function is normal, we will also assume that it is radial, that is,

The weighted space consists of all such thatwhere is normal. For , , we obtain the weighted space (see, e.g., [3–5]).

The little weighted space is a subspace of consisting of all such that

The class of all such thatwhere is normal, is called the Bloch-type space, and is denoted by . With the normthe Bloch-type space becomes a Banach space.

The little Bloch-type space is a subspace of consisting of those such thatThe -Bloch space is obtained for (see, e.g., [6–11]). For the space becomes the classical Bloch space.

Let be a holomorphic self-map of For any the composition operator is defined byIt is of interest to provide function theoretic characterizations when induces bounded or compact composition operators on spaces of holomorphic functions. For some classical results in the topic (see, e.g., [12]). For some recent results see, for example, [3–5, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references therein.

Let and be a holomorphic self-map of For products of integral-type and composition operator are defined as follows:

When , operators in (1.17) are reduced to the integral operator introduced in [24]. For some other results on the operator; see, for example, [25, 26], and related references therein. Some results on related integral-type operators on spaces of holomorphic functions in can be found, for example, in [27–41] (see also the references therein).

In [42], among other results, we proved the following theorem regarding the boundedness of the operator .

Theorem 1.1. *Assume that is normal, and is a holomorphic self-map of .
Then is bounded if and only if*

One of the interesting questions is to extend operators in (1.17) in the unit ball settings and to study their function theoretic properties on spaces of holomorphic functions on the unit ball in terms of inducing functions.

Assume that , and is a holomorphic self-map of . We introduce the following important integral-type operator on the space of holomorphic functions on :

First note that when , the operator is reduced to an operator of the form as the second operator in (1.17). Indeed, since and , it follows that for some By using this fact and the change of variables , we obtainHence operator (1.19) is a natural extension of the second operator in (1.17).

Now we formulate the following big research project related to the operator .

*Research Project 1. *Let and be two Banach spaces of holomorphic functions
on the unit ball in (e.g., the weighted Bergman space ,
the Bloch-type space ,
the Hardy space space, the weighted space ,
the Besov space ,
BMOA etc.) Characterize the boundedness, compactness, essential norms, and
other operator theoretic properties of the operator in terms of function theoretic properties of
inducing functions and .

Another interesting question is to find the exact value of the norm of operators on spaces of holomorphic functions. Majority of papers in the area only find asymptotics of the operator norm of certain linear operators on some spaces of holomorphic functions. There are a few papers which calculate the operator norm of these operators. Recently in [4] we calculated operator norm of the weighted composition operator mapping the Bloch space to the weighted space which motivates us to find the norms of weighted composition and other closely related operators between various spaces of holomorphic functions.

*Research Project 2. *Let and be two Banach spaces of holomorphic functions
as in Research project 1. Calculate the operator norm of in terms of inducing functions and .

In this paper, among other results, we will calculate the operator norm of . We will also characterize the boundedness, compactness, and the essential norm of the operator. These results partially solve problems posed in the above research projects.

Throughout the paper, denotes a positive constant not necessarily the same at each occurrence. The notation means that there is a positive constant such that

#### 2. Auxiliary Results

In this section, we give several auxiliary results, which are used in the proofs of the main results.

Lemma 2.1 (see [43, Corollary 3.5]). *Suppose that and .
Then for all and ,
the following inequality holds:*

The following criterion for the compactness follows by standard arguments (see, e.g., [12, 20, 34–36]). Hence, we omit its proof.

Lemma 2.2. *Suppose
that is normal, and is a holomorphic self-map of .
Then the operator is compact if and only if is bounded and for every bounded sequence in converging to zero uniformly on compacts of , one has as .*

The following result can be found in [44]. For closely related results, see also [11, 45–52] and the references therein.

Lemma 2.3. *Suppose
that ,
then**for every (here ). *

The following lemma can be proved similar to [53, Lemma 1].

Lemma 2.4. *Suppose
that is normal. A closed set in is compact if and only if it is bounded and*

The following lemma is related to [32, Lemma 1] and [34, Lemma 2].

Lemma 2.5. *Assume that and .
Then*

*Proof. *Since the function is holomorphic and ,
it has the Taylor expansion in the following form Thenas claimed.

#### 3. The Norm of the Operator

In this section, we calculate the norm .

Theorem 3.1. *Assume that is normal, is a holomorphic self-map of ,
and is bounded. Then*

*Proof. *If then by Lemmas 2.5 and 2.1 we obtainfrom which it follows that

Now we prove the reverse inequality. For fixed, setWe have that for each For this fact is well known. The proof for the
case could be less known, and we give a proof of it
for the lack of a specific reference and for the benefit of the reader. Let , ,
then we have where we have used the following
formula (see, e.g., [1])

From this and the boundedness of ,
we haveTaking the supremum in (3.7) over ,
we obtainFrom (3.3) and (3.8), it follows
that

Sinceand the proof of (3.8) does not
depend on the space (we may replace it by ) the second equality in (3.1) also holds.

Corollary 3.2. *Assume that is normal, and is a holomorphic self-map of .
Then is bounded if and only if*

*Proof. *If is bounded, then (3.10) follows from Theorem 3.1.
If (3.10) holds, then the boundedness of follows from (3.3).

#### 4. The Boundedness of the Operator

Here we characterize the boundedness of the operator .

Theorem 4.1. *Assume that is normal, and is a holomorphic self-map of Then is bounded if and only if is bounded and *

*Proof. *Assume that is bounded and Then, for each polynomial ,
we havefrom which it follows that .

Since the set of all polynomials is dense in we have that for every there is a sequence of polynomials such thatFrom this and since the operator is bounded, it follows thatas Hence Since is a closed subset of ,
the boundedness of follows.

Now assume that is bounded. Then clearly is bounded. Taking the test function ,
we obtain

#### 5. Compactness of the Operator

This section is devoted to studying of the compactness of the operator . We prove the following result.

Theorem 5.1. *Assume that is normal, is a holomorphic self-map of ,
and the operator is bounded. Then the operator is compact if and only if*

*Proof. *First assume that the operator is compact. If then condition (5.1) is vacuously satisfied.
Hence, assume that and assume to the contrary that (5.1) does not
hold. Then there is a sequence satisfying the condition as and such thatFor fixed, setwhere is defined in (3.4). Recall that for each Then and it is easy to see that uniformly on compacts of as .
Hence, by Lemma 2.2, it follows that

On the other hand, by Lemma 2.5 and (5.2), we obtainfor every which contradicts with (5.4).

Now assume that (5.1) holds. Then for every there is an such that when ,

On the other hand, since the operator is bounded, for ,
we obtain

Assume that is a bounded sequence in converging to zero uniformly on compacts of as Let Then by Lemma 2.1 and (5.6), for ,
we obtainIf we haveFrom (5.7) and (5.8), it follows
that as ,
from which the compactness of the operator follows.

#### 6. Compactness of the Operator

This section characterizes the compactness of the operator .

Theorem 6.1. *Assume is normal, is a holomorphic self-map of ,
and the operator is bounded. Then the operator is compact if and only if*

*Proof. *Assume is compact. Then clearly is bounded and as in Theorem 4.1 we have that .

Hence if ,
thenfrom which the result follows in
this case.

Now assume .
By using the test functions , ,
defined in (5.3) we obtain that condition (5.1) holds, which implies that for
every ,
there is an such that for ,
condition (5.6) holds.

Since there is such that for

Hence, if and ,
we haveFrom (5.6) and (6.4), condition
(6.1) follows.

Now assume that condition (6.1) holds. Then the
quantity in Theorem 3.1 is finite. Using this fact and
the following inequalityit follows that the set is bounded in moreover, in view of (6.1), it is bounded in Taking the supremum in the last inequality
over the unit ball in ,
then letting ,
using condition (6.1) and employing Lemma 2.4, we obtain the compactness of the
operator as desired.

#### 7. Essential Norm of

Let and be Banach spaces, and let be a bounded linear operator. The essential norm of the operator , denoted by , is defined as follows:where denote the operator norm.

From this definition and since the set of all compact operators is a closed subset of the set of bounded operators, it follows that operator is compact if and only if

In this section, we study the essential norm of the operator for the case

Theorem 7.1. *Assume that is a holomorphic self-map of ,
and is bounded. Then the following inequalities
hold:*

*Proof. *Assume that is a sequence in such that as .
Note that the sequence (where is defined in (3.4)) is such that for each and it converges to zero uniformly on compacts
of From this and by
[11, Theorems 2.12 and 4.50], it follows that weakly in ,
as (here we use the condition ). Hence, for every compact operator ,
we have that as Thus, for every such sequence and for every
compact operator ,
we have that

Taking the infimum in (7.3) over the set of all compact
operators ,
we obtainfrom which the first inequality
follows.

In the sequel we prove the second inequality. Assume
that is a sequence which increasingly converges to
1. Consider the operators defined byWe prove that these operators
are compact. Indeed, since ,
it follows that condition (5.1) in Theorem 5.1 is vacuously satisfied, from which
the claim follows.

Recall that Let be fixed for a moment. Employing Lemma 2.1, and
using the factwhich follows by using the
triangle inequality for the norm, the monotonicity of the integral meansand the polar coordinates, we
have

LetIf ,
then by using the mean value theorem, the subharmonicity of the partial
derivatives of and Lemma 2.3, we have

If ,
then applying in (7.10) the known fact that for each compact for some independent of (see [11]), we obtain that (7.11) also holds in this case.

Letting in (7.8), using (7.11), and then letting ,
the second inequality in (7.2) follows, finishing the proof of the theorem.

Motivated by Theorem 7.1, we leave the following open problem.

*Open Problem 1. *Find the
exact value of the essential norm of the operator .

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