Abstract and Applied Analysis
Volume 2013, Issue 1 186326
Research Article
Open Access

Hyponormal Toeplitz Operators on the Dirichlet Spaces

Puyu Cui, 

Corresponding Author

Puyu Cui

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China dlut.edu.cn

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Yufeng Lu, 

Yufeng Lu

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China dlut.edu.cn

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First published: 17 November 2013
https://doi.org/10.1155/2013/186326
Academic Editor: Natig M. Atakishiyev

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 𝔻 be the open unit disk in the complex plane ℂ and dA be the normalized Lebesgue area measure on 𝔻. L∞(𝔻,  dA) and L2(𝔻,  dA) denote the essential bounded measurable function space and the space of square integral functions on 𝔻 with respect to dA, respectively. The Bergman space consists of all analytic functions in L2(𝔻,  dA). The Sobolev space W1,2(𝔻) is the space of functions f : 𝔻 → ℂ with the following norm:
()
W1,2(𝔻) is a Hilbert space with the inner product
()
The Dirichlet space 𝒟 consists of all analytic functions h in W1,2(𝔻) with h(0) = 0. The Sobolev space W1,∞(𝔻) is defined by
()
with the norm
()
Let P be the orthogonal projection of W1,2(𝔻) onto 𝒟. P is an integral operator represented by
()
where is the reproducing kernel of 𝒟. For u ∈ W1,∞(𝔻), the Toeplitz operator Tu with symbol u is defined by
()
Tu is a bounded operator for u ∈ W1,∞(𝔻) on 𝒟.
Yu gave a decomposition of the Sobolev space W1,2(𝔻) in [1]. Let 𝒫0 be the set of all the following polynomials:
()
where j and l run over a finite subset of ℤ and ∑l≥0 al+j,l = 0. Let 𝒜0 denote the closure of 𝒫0 in W1,2(𝔻), and let 𝒜 denote 𝒜0 + ℂ. Since the set of all polynomials in z and is dense in W1,2(𝔻), there is the following decomposition:
()

Since W1,∞(𝔻)⊆W1,2(𝔻) and by the above decomposition, it follows that, if u ∈ W1,∞(𝔻), then , where u0 ∈ 𝒜, f, g ∈ H(𝔻) (the space of the analytic functions on 𝔻) with f(0) = g(0) = 0.

For the space 𝒜0, there is the following proposition.

Proposition 1 (see [1].)Let ϕ ∈ W1,∞(𝔻). Then ϕ𝒜0 ⊂ 𝒜0.

A bounded linear operator A on a Hilbert space is called hyponormal if A*A − AA* is a positive operator. There is an extensive literature on hyponormal Toeplitz operators on H2(𝕋) (the Hardy space on 𝕋) [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 Tu with u ∈ W1,∞(𝔻) on Dirichlet space 𝒟 and harmonic Dirichlet space 𝒟h.

2. Case on the Dirichlet Space

In this section, the hyponormality of Tu with u ∈ W1,∞(𝔻) on 𝒟 will be discussed.

Theorem 2. Let with u0 + c ∈ 𝒜, f, g ∈ H(𝔻), and f(0) = g(0) = 0. Then Tu is hyponormal on 𝒟 if and only if u ∈ 𝒜.

Proof. By Proposition 1, we only need to prove the necessity with .

Let . Simple calculations imply that

()

Furthermore,

()

Therefore

()

Similarly, we have

()
Denote ei(z) = (1/i)zi for i ≥ 1. Since Tu is hyponormal, we have
()
For i ≥ 2, implies that
()
Hence
()
Letting N → ∞, since and are convergent and is disconvergent, we get f1 = 0. Similarly, by choosing i, we get fl = 0 for l ≥ 1. Note that implies that . Thus gl = 0 for l ≥ 1 and the proof is finished.

The following corollary generalizes Theorems 1 and 2 in [10]. Denote
()
where H∞(𝔻) is the space of the bounded analytic functions on 𝔻.

Corollary 3. Let u ∈ Ω. Then Tu is hyponormal on 𝒟 if and only if u is a constant function.

3. Case on the Harmonic Dirichlet Space

In this section, we will characterize the hyponormality of Tu with u ∈ W1,∞(𝔻) on 𝒟h.

The harmonic Dirichlet space 𝒟h consists of all harmonic functions in W1,2(𝔻). It is a closed subspace of W1,2(𝔻), and hence it is a Hilbert space with the following reproducing kernel:
()
Let Q be the orthogonal projection of W1,2(𝔻) onto 𝒟h. Q is an integral operator represented by
()
For u ∈ W1,∞(𝔻), the Toeplitz operator with symbol u is defined by
()
is a bounded operator for u ∈ W1,∞(𝔻) on 𝒟h (see [11]).

Theorem 4. Let with u0 + c ∈ 𝒜, f, g ∈ H(𝔻), and f(0) = g(0) = 0. Then is hyponormal on 𝒟h if and only if u ∈ 𝒜.

Proof. By Proposition 1, we only need to prove the necessity with .

Let . Since is hyponormal on 𝒟h, we have . Note that

()

Thus

()

Similarly, we have

()
For i ≥ 2, let ai = 1/i and aj = 0 for j ≠ i. It follows that
()
Therefore,
()
For every k ≥ 1, we have
()
where N ≥ 2. Letting N → ∞, Since and are convergent and (k ≥ 1 is fixed) is disconvergent, we get |fk | = |gk | = 0 for k ≥ 1. The proof is finished.

The following corollary generalizes Theorem 3 in [10].

Corollary 5. Suppose that with f, g ∈ H(𝔻). Then Tu is hyponormal on 𝒟h if and only if u 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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