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Research Article | Open Access

Volume 2013 |Article ID 186326 | https://doi.org/10.1155/2013/186326

Puyu Cui, Yufeng Lu, "Hyponormal Toeplitz Operators on the Dirichlet Spaces", Abstract and Applied Analysis, vol. 2013, Article ID 186326, 5 pages, 2013. https://doi.org/10.1155/2013/186326

# Hyponormal Toeplitz Operators on the Dirichlet Spaces

Revised02 Oct 2013
Accepted24 Oct 2013
Published17 Nov 2013

#### Abstract

We completely characterize the hyponormality of bounded Toeplitz operators with Sobolev symbols on the Dirichlet space and the harmonic Dirichlet space.

#### 1. Introduction

Let be the open unit disk in the complex plane and be the normalized Lebesgue area measure on . and denote the essential bounded measurable function space and the space of square integral functions on with respect to , respectively. The Bergman space consists of all analytic functions in . The Sobolev space is the space of functions with the following norm: is a Hilbert space with the inner product The Dirichlet space consists of all analytic functions in with . The Sobolev space is defined by with the norm Let be the orthogonal projection of onto . is an integral operator represented by where is the reproducing kernel of . For , the Toeplitz operator with symbol is defined by is a bounded operator for on .

Yu gave a decomposition of the Sobolev space in . Let be the set of all the following polynomials: where and run over a finite subset of and . Let denote the closure of in , and let denote . Since the set of all polynomials in and is dense in , there is the following decomposition:

Since and by the above decomposition, it follows that, if , then , where , , (the space of the analytic functions on ) with .

For the space , there is the following proposition.

Proposition 1 (see ). Let . Then .

A bounded linear operator on a Hilbert space is called hyponormal if is a positive operator. There is an extensive literature on hyponormal Toeplitz operators on (the Hardy space on ) . The corresponding problems for the Toeplitz operators on the Bergman space have been characterized in . In the case of the Dirichlet space and the harmonic Dirichlet space, Lu and Yu proved that there are no nonconstant hyponormal Toeplitz operators with certain symbols . In this paper, we completely characterize the Toeplitz operators with on Dirichlet space and harmonic Dirichlet space .

#### 2. Case on the Dirichlet Space

In this section, the hyponormality of with on will be discussed.

Theorem 2. Let with , , and . Then is hyponormal on if and only if .

Proof. By Proposition 1, we only need to prove the necessity with .
Let . Simple calculations imply that
Furthermore,
Therefore
Similarly, we have Denote for . Since is hyponormal, we have For , implies that Hence Letting , since and are convergent and is disconvergent, we get . Similarly, by choosing , we get for . Note that implies that . Thus for and the proof is finished.

The following corollary generalizes Theorems 1 and 2 in . Denote where is the space of the bounded analytic functions on .

Corollary 3. Let . Then is hyponormal on if and only if is a constant function.

#### 3. Case on the Harmonic Dirichlet Space

In this section, we will characterize the hyponormality of with on .

The harmonic Dirichlet space consists of all harmonic functions in . It is a closed subspace of , and hence it is a Hilbert space with the following reproducing kernel:

Let be the orthogonal projection of onto . is an integral operator represented by For , the Toeplitz operator with symbol is defined by is a bounded operator for on (see ).

Theorem 4. Let with , , and . Then is hyponormal on if and only if .

Proof. By Proposition 1, we only need to prove the necessity with .
Let . Since is hyponormal on , we have . Note that
Thus
Similarly, we have For , let and for . It follows that Therefore, For every , we have where . Letting , Since and are convergent and ( is fixed) is disconvergent, we get for . The proof is finished.

The following corollary generalizes Theorem 3 in .

Corollary 5. Suppose that with . Then is hyponormal on if and only if is a constant function.

#### Acknowledgments

This research is supported by NSFC (no. 11271059) and Research Fund for the Doctoral Program of Higher Education of China.

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