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Abstract and Applied Analysis
Volume 2010, Article ID 134969, 9 pages
http://dx.doi.org/10.1155/2010/134969
Research Article

Norm and Essential Norm of an Integral-Type Operator from the Dirichlet Space to the Bloch-Type Space on the Unit Ball

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

Received 1 August 2010; Accepted 1 October 2010

Academic Editor: Irena Lasiecka

Copyright © 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.

Abstract

Operator norm and essential norm of an integral-type operator, recently introduced by this author, from the Dirichlet space to the Bloch-type space on the unit ball in are calculated here.

1. Introduction

Let be the open unit ball in , the open unit disk in , the class of all holomorphic functions on , and , the space consisting of all such that

For an with the Taylor expansion , let be the radial derivative of where is a multi-index, and . Let

The Dirichlet space contains all , such that The quantity is a norm on which for is equal to usual norm where is the normalized area measure on .

The inner product, between two functions on is defined by

For , let and , then it is easy to see that the family is an orthonormal basis for , and hence the reproducing kernel for is given by ([1]) as follows: where is the inner product in .

Clearly for each and , the next reproducing formula holds: Note that for from (1.8), we obtain

Also, by the Cauchy-Schwarz inequality and (1.9), we have that, for each and , Note that inequality (1.10) is exact since it is attained for

The weighted-type space ([2, 3]) consists of all such that where is a positive continuous function on (weight).

The Bloch-type space consists of all such that where is a weight.

Let , , and be a holomorphic self-map of , then the following integral-type operator: has been recently introduced in [4] and considerably studied (see, e.g., [58] and the related references therein). For some related operators, see also [916] and the references therein.

Usual problem in this research area is to provide function-theoretic characterizations for when and induce bounded or compact integral-type operator on spaces of holomorphic functions. Majority of papers only find asymptotics of operator norm of linear operators. Somewhat concrete but perhaps more interesting problem is to calculate operator norm of these operators between spaces of holomorphic functions on various domains. Some results on this problem can be found, for example, in [3, 1726] (for related results see also [7, 2732]). Having published paper [3], we started with systematic investigation of methods for calculating operator norms of concrete operators between spaces of holomorphic function.

Here, we calculate the operator norm as well as the essential norm of the operator , considerably extending our recent result in note [33].

2. Auxiliary Results

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

Lemma 2.1 (see [4]). Let , , and be a holomorphic self-map of , then

The next Schwartz-type lemma ([34]) can be proved in a standard way. Hence, we omit its proof.

Lemma 2.2. Assume that , , is a weight, and is an analytic self-map of , then is compact if and only if is bounded and for any bounded sequence in converging to zero uniformly on compacts of as , one has

Lemma 2.3. Assume that , , is a weight, is an analytic self-map of such that , and the operator is bounded, then is compact.

Proof. First note that since is bounded and by Lemma 2.1, it follows that . Now, assume that is a bounded sequence in converging to zero on compacts of as , then we have as , since is contained in the ball which is a compact subset of , according to the assumption . Hence, by Lemma 2.2, the operator is compact.

3. Operator Norm of

In this section, we calculate the operator norm of .

Theorem 3.1. Assume that , , and is a holomorphic self-map of , then

Proof. Using Lemma 2.1, reproducing formula (1.8), the Cauchy-Schwarz inequality, and finally (1.9), we get that, for each and , Taking the supremum in (3.2) over as well as the supremum over the unit ball in and using the fact , for each , which follows from the assumption , we get
Now assume that the operator is bounded. From (1.9), we obtain that, for each , From (1.9) and (3.4), it follows that Hence, if is bounded, then from (3.3) and (3.5) we obtain (3.1).
In the case is unbounded, the result follows from inequality (3.3).

4. Essential Norm of

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

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

Here, we calculate the essential norm of the operator

Theorem 4.1. Assume that , is a weight, is a holomorphic self-map of , and is bounded. If , then , and if , then

Proof. Since is bounded, for the test function , we get . If , then from Lemma 2.3 it follows that is compact which is equivalent with . On the other hand, it is clear that in this case the condition is vacuous, so that (4.3) is vacuously satisfied.
Now assume that and that is a sequence in such that as . For fixed, set By (1.9), we have that for each Hence, the sequence is such that , for each , and clearly it converges to zero uniformly on compacts of From this and by [35, Theorems ], it easily follows that weakly in , as . Hence, for every compact operator , we have that Thus, for every such sequence and for every compact operator , we have that
Taking the infimum in (4.6) over the set of all compact operators , we obtain from which an inequality in (4.3) follows.
In the sequel, we prove the reverse inequality. Assume that is a sequence of positive numbers which increasingly converges to 1. Consider the operators defined by Since , by Lemma 2.3, we have that these operators are compact.
Since is bounded, then Let be fixed for a moment. By Lemma 2.1, we get
Further, we have
From (1.10), (4.11), and the fact , we obtain in particular
Let The mean value theorem along with the subharmonicity of the moduli of partial derivatives of , well-known estimates among the partial derivatives of analytic functions, Theorem , and Proposition in [35], yield where and is the Lebesgue volume measure on .
Using (4.13) in (4.10), letting in (4.9), using (4.15), and then letting , the reverse inequality follows, finishing the proof of the theorem.

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