#### 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 [1]. 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 [1]). *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 ) [2–4]. The corresponding problems for the Toeplitz operators on the Bergman space have been characterized in [5–9]. 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 [10]. 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 [10]. 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 [11]).

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 [10].

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.