Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 517689, 18 pages

doi:10.1155/2012/517689

Research Article

Properties of Toeplitz Operators on Some Holomorphic Banach Function Spaces

Ahmed El-Sayed Ahmed1,2 and Mahmoud Ali Bakhit3,4

¹Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

²Department of Mathematics, Faculty of Science, Taif University , P.O. Box 888, El-Hawiyah, El-Taif 5700, Saudi Arabia

³Department of Mathematics, Faculty of Science, Assiut Branch, Al-Azhar University, Assiut 32861, Egypt

⁴Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia

Received 5 March 2012; Accepted 18 April 2012

Academic Editor: Dashan Fan

Copyright © 2012 Ahmed El-Sayed Ahmed and Mahmoud Ali Bakhit. 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 characterize complex measures on the unit ball of , for which the general Toeplitz operator is bounded or compact on the analytic Besov spaces , also on the minimal Möbius invariant Banach spaces in the unit ball .

1. Introduction

Let be the unit ball of the -dimensional complex Euclidean space . We denote the class of all holomorphic functions on the unit ball by . The ball centered at with radius will be denoted by . For , let , where is the normalized Lebesgue volume measure on and (where denotes the Gamma function) so that .

For any , , the inner product is defined by . For , we write

For and , set

where is the Bergman metric on , that is,

For , the Besov spaces consists of all functions for which (see [1])

From [1], we know that for , the Besov space is nontrivial if and only if .

The analytic Besov space is the minimal Möbius invariant Banach space (see [2]) defined by

For , a function is said to belong to the -Bloch spaces if (see [3])

The little Bloch space consists of all such that

With the norm , we know that becomes a Banach space. For , the spaces and become the Bloch and the little Bloch space (see, e.g., [2]).

For every point , the Möbius transformation is defined by

where and (see, e.g., [2] or [4]). The map has the following properties that , , and

where and are arbitrary points in . In particular,

The following result can be found in [3].

Proposition 1.1. Let . Then if and only if

For and , the weighted Bergman spaces consists of all functions for which

It is clear that and is a linear subspace of . When , we simply write for . In the special case when is a Hilbert space. It is well known that for the Bergman kernel of is given by

For , a complex measure such that

define a Toeplitz operator as follows:

where and .

For , define the function , for by:

In case we write instead of and we have that , where 1 stands for the constant function. For the function is equivalent to the fractional derivative of . The Bergman projection is the orthogonal form onto defined by:

The Bergman projection naturally extends to an integral operator on .

Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see [5, 6]. Boundedness and compactness of general Toeplitz operators on the -Bloch spaces have been investigated in [7] on the unit disk for . Also in [8], the authors extend the Toeplitz operator to in the unit ball of and completely characterize the positive Borel measure such that is bounded or compact on with . Recently, in [9], general Toeplitz operators on the analytic Besov spaces with have been investigated. Under a prerequisite condition, the authors characterized complex measure on the unit disk for which is bounded or compact on Besov space . For more details on several studies of different classes of Toeplitz operators we refer to [6, 10–16] and others.

In the present paper, we will extend the general Toeplitz operators to in the unit ball o f and completely characterize the positive Borel measure such that is bounded or compact on the spaces with . The extension requires some different techniques from those used in [9].

Let be the Bergman metric on . Denote the Bergman metric ball at by , where and .

Lemma 1.2 (see [2, Theorem  2.23]). For fixed , there is a sequence such that

(i); (ii)there is a positive integer such that each is contained in at most of the sets .

A positive Borel measure on the unit ball is said to be a Carleson measure for if there exists such that

The following characterization of Carleson measures can be found in [2] or in [5]. A positive Borel measure on the unit ball is said to be a Carleson measure for the Bergman space if

It is well known that a positive Borel measure is a -Carleson measure if and only if

where is the sequence in Lemma 1.2. If satisfies that

then is called vanishing Carleson measure for .

These two are special cases of a more general notion of Carleson measures on normed spaces of analytic functions.

In general, let be a positive measure on and a Möbius invariant space. For ; then is an -Carleson measure if there is a constant so that (see [2])

Also, define

We say that is vanishing -Carleson measure if for any sequence with and such that uniformly on compact subset of , we have that

Throughout the paper, we will say that the expressions and are equivalent, and write , whenever there exist positive constants and such that . As usual, the letter will denote a positive constant, possibly different on each occurrence.

2. Bounded Toeplitz Operators on Spaces

We are going to work with Toeplitz operators acting on Besov spaces in the unit ball of .

We start with the following lemma.

Lemma 2.1. Let , , . If

then is a bounded operator from into if and only if .

Proof. Let

By Theorem  2.10 in [2], we know that is bounded on if and only if . However, it is obvious that is bounded from into if and only if is bounded on .

Theorem 2.2. Let and let be a positive Borel measure on . If is a -Carleson measure, then the Toeplitz operator is bounded on spaces if and only if is a -Carleson measure.

Proof. Let where and let . We know that the dual spaces of are under the paring

To prove the boundedness of , it suffices to show that

for all and , where is a positive constant that does not depend on or .

Now we define by the following:

Then

Since

we have

This implies

By Proposition 1.1, we have

Since is a -Carleson measure, taking , then as in [8] (see also Proposition  1.4.10 of [4]), we get

Then,

Therefore

By Fubini’s Theorem we have

Using the operator , divide the integral

we have

where is the identity operator, and

By Proposition 1.1, we have

By Lemma 2.1, the operator is bounded from into whenever . Since if and only if , and we have from above, maps boundedly into , whenever , or , which is always true if . Thus . It can easily seen that and that . Thus

By (2.11), we get

Next consider , we have

Therefore, is bounded on if and only if

if and only if the measure is a -Carleson measure.

Now, we will characterize boundedness of Toeplitz operators on the minimal Möbius invariant Banach spaces of holomorphic functions in the unit ball of .

Theorem 2.3. Let be a positive Borel measure on . If is a -Carleson measure, then the Toeplitz operator is bounded on spaces if and only if

is a -Carleson measure.

Proof. We will use the fact that the dual spaces of are the Bloch space under the paring

Similarly, as in the proof of Theorem 2.2, by duality, we have that is bounded on spaces if and only if

for all and , where

Using the fact that

for , we have that , which means that . Now using the operator , we have

By Proposition 1.1, we have

Next consider , notice first that

Thus,

It is known that under the paring

Since for by the above duality we get

Therefore, is bounded on if and only if

if and only if the measure is a -Carleson measure.

3. Compact Toeplitz Operators on Spaces

In this section we will characterize compact Toeplitz operators on spaces in the unit ball of . We need the following lemma.

Lemma 3.1. Let , and be bounded linear operator from into in the unit ball. Then is compact on spaces if and only if as wheneve r is a bounded sequence in that converges to 0 uniformly on .

Proof. This lemma can be proved by Montel’s Theorem.

Theorem 3.2. Let and let be a positive Borel measure on . If is a vanishing -Carleson measure, then the Toeplitz operator is compact on spaces if and only if is a vanishing -Carleson measure.

Proof. Let where and let be a sequence in satisfying and such that converges to 0 uniformly as on compact subsets of , and let . By duality, we have that is compact on if and only if

As in the proof of Theorem 2.2,

where

Also as in the proof of Theorem 2.2,

Since and is a vanishing -Carleson measure, and converges to 0 uniformly as on compact subsets of , we get that

Thus is compact on if and only if

Using the operator , we have that

For and , we have

For a fixed , since is a vanishing -Carleson measure, let sufficiently close to 1 so that

Similarly, as in the proof of Theorem 2.2, by Proposition 1.1,

Since as on compact subsets of , we cane choose big enough so that

Therefore,

Hence , where does not depend on , and so

Thus, is compact on if and only if

Again, as in the proof of Theorem 2.2, we have

Therefore, is compact on if and only if

which is equivalent to say that is a vanishing -Carleson measure.

Theorem 3.3. Let be a positive Borel measure on . If is a -Carleson measure, then the Toeplitz operator is compact on spaces if and only if

is a vanishing -Carleson measure.

Proof. Let be a sequence in satisfying and such that converges to 0 uniformly as on compact subsets of , and let . By duality, we have that is compact on if and only if

Thus, is compact on if and only if

Using the operator , we have that

As in the proof of Theorem 2.3, we have

Notice that implies that . Since converges to 0 uniformly as on compact subsets of , and is a -Carleson measure, we get that and . Thus

Therefore, is compact on if and only if

We have shown in the proof of Theorem 2.3

Therefore, is compact on if and only if

which is equivalent to saying that the measure is a vanishing -Carleson measure.

References

K. T. Hahn and E. H. Youssfi, “Möbius invariant Besov -spaces and Hankel operators in the Bergman space on the ball in ,” Complex Variables, vol. 17, no. 1-2, pp. 89–104, 1991.

K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, NY, USA, 2004.

S. Li and S. Stević, “Some characterizations of the Besov space and the α-Bloch space,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 262–273, 2008.

W. Rudin, Function Theory in the Unit Ball of Cn, Springer, New York, NY, USA, 1980.

K. Zhu, “Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains,” Journal of Operator Theory, vol. 20, no. 2, pp. 329–357, 1988.

K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, NY, USA, 1990.

Z. Wu, R. Zhao, and N. Zorboska, “Toeplitz operators on Bloch-type spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 12, pp. 3531–3542, 2006.

X. Wang and T. Liu, “Toeplitz operators on Bloch-type spaces in the unit ball of ,” Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 727–735, 2010.

Z. Wu, R. Zhao, and N. Zorboska, “Toeplitz operators on analytic Besov spaces,” Integral Equations and Operator Theory, vol. 60, no. 3, pp. 435–449, 2008.

S. C. Arora and J. Bhola, “Essentially slant Toeplitz operators,” Banach Journal of Mathematical Analysis, vol. 3, no. 2, pp. 1–8, 2009.

A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2nd edition, 2006.

C. C. Cowen and E. A. Gallardo-Gutiérrez, “Unitary equivalence of one-parameter groups of Toeplitz and composition operators,” Journal of Functional Analysis, vol. 261, no. 9, pp. 2641–2655, 2011.

A. V. Harutyunyan and G. Harutyunyan, “Holomorphic Besov spaces in the polydisc and bounded Toeplitz operators,” Analysis, vol. 30, no. 4, pp. 365–381, 2010.

T. Nakazi, “Invariant subspaces of Toeplitz operators and uniform algebras,” Bulletin of the Belgian Mathematical Society, vol. 15, no. 1, pp. 1–8, 2008.

Y. J. Lee and N. Q. Dieu, “Toeplitz operators on the Dirichlet spaces of planar domains,” Proceedings of the American Mathematical Society, vol. 139, no. 2, pp. 547–558, 2011.

D. Vukotić, “Analytic Toeplitz operators on the Hardy space : a survey,” Bulletin of the Belgian Mathematical Society, vol. 10, no. 1, pp. 101–113, 2003.