Hyponormal Toeplitz Operators on the Dirichlet Spaces

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 𝐿²_𝑎 consists of all analytic functions in 𝐿²(𝔻,𝑑𝐴). The Sobolev space 𝑊^1,2(𝔻) is the space of functions 𝑓∶𝔻→ℂ with the following norm: 𝑓=_𝔻𝑓𝑑𝐴²+_𝔻𝜕𝑓𝜕𝑧²+𝜕𝑓𝜕𝑧²𝑑𝐴^1/2<∞. (1)𝑊^1,2(𝔻) is a Hilbert space with the inner product 𝑓,𝑔=_𝔻𝑓𝑑𝐴_𝔻𝑔𝑑𝐴+𝜕𝑓,𝜕𝑧𝜕𝑔𝜕𝑧_𝐿²_{(𝔻,𝑑𝐴)}+𝜕𝑓𝜕𝑧,𝜕𝑔𝜕𝑧_𝐿²_{(𝔻,𝑑𝐴)}. (2) The Dirichlet space 𝒟 consists of all analytic functions ℎ in 𝑊^1,2(𝔻) with ℎ(0)=0. The Sobolev space 𝑊^1,∞(𝔻) is defined by 𝑊^1,∞(𝔻)=𝑢∈𝑊^1,2(𝔻)∶𝑢,𝜕𝑢,𝜕𝑧𝜕𝑢𝜕𝑧∈𝐿^∞(𝔻,𝑑𝐴), (3) with the norm ‖𝑢‖_1,∞=max‖𝑢‖_∞,𝜕𝑢𝜕𝑧_∞,𝜕𝑢𝜕𝑧_∞. (4) Let 𝑃 be the orthogonal projection of 𝑊^1,2(𝔻) onto 𝒟. 𝑃 is an integral operator represented by 𝑃𝑓(𝑧)=_𝔻𝜕𝑓𝜕𝑤𝜕𝐾_𝑧𝜕𝑤𝑑𝐴, (5) where 𝐾_𝑧(𝑤)=∑^∞_𝑘=1(𝑧^𝑘𝑤^𝑘/𝑘) is the reproducing kernel of 𝒟. For 𝑢∈𝑊^1,∞(𝔻), the Toeplitz operator 𝑇_𝑢 with symbol 𝑢 is defined by 𝑇_𝑢ℎ=𝑃(𝑢ℎ)ℎ∈𝒟. (6)𝑇_𝑢 is a bounded operator for 𝑢∈𝑊^1,∞(𝔻) on 𝒟.

Yu gave a decomposition of the Sobolev space 𝑊^1,2(𝔻) in [1]. Let 𝒫₀ be the set of all the following polynomials: _{𝑗≥−𝑙𝑙≥0}𝑎_{𝑙+𝑗,𝑙}𝑧^𝑙+𝑗𝑧^𝑙, (7) where 𝑗 and 𝑙 run over a finite subset of ℤ and ∑_𝑙≥0𝑎_{𝑙+𝑗,𝑙}=0. Let 𝒜₀ denote the closure of 𝒫₀ in 𝑊^1,2(𝔻), and let 𝒜 denote 𝒜₀+ℂ. Since the set of all polynomials in 𝑧 and 𝑧 is dense in 𝑊^1,2(𝔻), there is the following decomposition: 𝑊^1,2(𝔻)=𝒜⊕𝒟⊕𝒟. (8)

Since 𝑊^1,∞(𝔻)⊆𝑊^1,2(𝔻) and by the above decomposition, it follows that, if 𝑢∈𝑊^1,∞(𝔻), then 𝑢=𝑢₀+𝑓+𝑔, 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∈𝑊^1,∞(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 𝑇_𝑢 with 𝑢∈𝑊^1,∞(𝔻) on Dirichlet space 𝒟 and harmonic Dirichlet space 𝒟_ℎ.

2. Case on the Dirichlet Space

In this section, the hyponormality of 𝑇_𝑢 with 𝑢∈𝑊^1,∞(𝔻) on 𝒟 will be discussed.

Theorem 2. Let 𝑢=𝑢₀+𝑐+𝑓+𝑔∈𝑊^1,∞(𝔻) with 𝑢₀+𝑐∈𝒜, 𝑓,𝑔∈𝐻(𝔻), and 𝑓(0)=𝑔(0)=0. Then 𝑇_𝑢 is hyponormal on 𝒟 if and only if 𝑢∈𝒜.

Proof. By Proposition 1, we only need to prove the necessity with 𝑢(𝑧)=𝑓(𝑧)+𝑔(𝑧)=∑^∞_𝑘=1𝑓_𝑘𝑧^𝑘+∑^∞_𝑘=1𝑔_𝑘𝑧^𝑘.
Let ℎ(𝑧)=∑^∞_𝑘=1ℎ_𝑘𝑧^𝑘∈𝒟. Simple calculations imply that 𝑇_𝑓+𝑔ℎ(𝑧)=𝑇_𝑓ℎ(𝑧)+𝑇_𝑔ℎ(𝑧)=𝑓(𝑧)ℎ(𝑧)+𝑃𝑧)(𝑧)ℎ(𝑧)+𝑔ℎ(=𝑓𝑔ℎ,𝐾_𝑧=^∞_𝑚=2^𝑚−1_𝑘=1𝑓_𝑘ℎ_𝑚−𝑘𝑧^𝑚+^∞_𝑙=2ℎ_𝑙^𝑙−1_𝑘=1𝑔_𝑘𝑧^𝑙−𝑘=^∞_𝑚=2^𝑚−1_𝑘=1𝑓_𝑘ℎ_𝑚−𝑘𝑧^𝑚+^∞∞_𝑚=1𝑘=1𝑔_𝑘ℎ_𝑘+𝑚𝑧^𝑚. (9)
Furthermore, 𝜕𝑇_𝑓+𝑔ℎ=𝜕𝑧^∞_𝑚=1(𝑚+1)^𝑚_𝑘=1𝑓_𝑘ℎ_{𝑚−𝑘+1}𝑧^𝑚+^∞_𝑚=0(𝑚+1)^∞_𝑘=1𝑔_𝑘ℎ_𝑘+𝑚+1𝑧^𝑚=^∞_𝑘=1𝑔_𝑘ℎ_𝑘+1+^∞_𝑚=1()×𝑚+1_𝑚𝑘=1𝑓_𝑘ℎ_{𝑚−𝑘+1}+^∞_𝑘=1𝑔_𝑘ℎ_𝑘+𝑚+1𝑧^𝑚. (10)
Therefore 𝑇_𝑓+𝑔ℎ²=^∞_𝑘=1𝑔_𝑘ℎ_𝑘+1²+^∞_𝑚=1^𝑚_𝑘=1𝑓_𝑘ℎ_{𝑚−𝑘+1}+^∞_𝑘=1𝑔_𝑘ℎ_𝑘+𝑚+1². (11)
Similarly, we have 𝑇^∗_𝑓+𝑔ℎ(𝑧)=𝑇^∗_𝑓+𝑔ℎ,𝐾_𝑧=ℎ,𝑇_𝑓+𝑔𝐾_𝑧=^∞_𝑚=21𝑚^𝑚−1_𝑙=1𝑙𝑔_𝑚−𝑙ℎ_𝑙𝑧^𝑚+^∞_𝑚=1^∞_𝑘=1𝑚+𝑘𝑚𝑓_𝑘ℎ_𝑘+𝑚𝑧^𝑚,𝜕𝑇^∗_𝑓+𝑔ℎ=𝜕𝑧^∞_𝑘=1(𝑘+1)𝑓_𝑘ℎ_𝑘+1+^∞_𝑚=1^∞_𝑘=1(𝑚+𝑘+1)𝑓_𝑘ℎ_𝑚+𝑘+1+^𝑚_𝑙=1𝑙𝑔_{𝑚−𝑙+1}ℎ_𝑙𝑧^𝑚,𝑇^∗_𝑓+𝑔ℎ²=^∞_𝑘=1(𝑘+1)𝑓_𝑘ℎ_𝑘+1²+^∞_𝑚=11𝑚+1^∞_𝑘=1(𝑚+𝑘+1)𝑓_𝑘ℎ_𝑚+𝑘+1+^𝑚_𝑙=1𝑙𝑔_{𝑚−𝑙+1}ℎ_𝑙². (12) Denote 𝑒_𝑖(𝑧)=(1/𝑖)𝑧^𝑖 for 𝑖≥1. Since 𝑇_𝑢 is hyponormal, we have 𝑇_𝑢𝑒_𝑖²−𝑇^∗_𝑢𝑒_𝑖²≥0for𝑖≥1. (13) For 𝑖≥2, ‖𝑇_𝑢𝑒_𝑖‖²−‖𝑇^∗_𝑢𝑒_𝑖‖²≥0 implies that 1𝑖^2𝑖−1_𝑘=1𝑔_𝑘²+^∞_𝑙=1𝑓_𝑙²≥^𝑖−1_𝑘=11𝑓𝑖−𝑘^𝑘2+^∞_𝑙=11𝑔𝑖+𝑙^𝑙2≥1𝑖𝑓₁². (14) Hence ^𝑁_𝑖=21𝑖^2∞_𝑙=1𝑔_𝑙²+^∞_𝑙=1𝑓_𝑙²≥^𝑁_𝑖=21𝑖𝑓₁²for𝑁≥2. (15) Letting 𝑁→∞, since ∑^𝑁_𝑖=2(1/𝑖²) and (∑^∞_𝑙=1|𝑔_𝑙|²+∑^∞_𝑙=1|𝑓_𝑙|²) are convergent and ∑^𝑁_𝑖=2(1/𝑖) is disconvergent, we get 𝑓₁=0. Similarly, by choosing 𝑖, we get 𝑓_𝑙=0 for 𝑙≥1. Note that ‖𝑇_𝑢𝑒₁‖²−‖𝑇^∗_𝑢𝑒₁‖²≥0 implies that ∑^∞_𝑙=1|𝑓_𝑙|²≥∑^∞_𝑙=1(1/(𝑙+1))|𝑔_𝑙|². Thus 𝑔_𝑙=0 for 𝑙≥1 and the proof is finished.

The following corollary generalizes Theorems 1 and 2 in [10]. Denote Ω=𝑢∶𝑢=𝑓+𝑔,𝑓,𝑔∈𝐻^∞(𝔻),𝑓²𝑑𝐴isa𝒟-Carlesonmeasure, (16) where 𝐻^∞(𝔻) is the space of the bounded analytic functions on 𝔻.

Corollary 3. Let 𝑢∈Ω. Then 𝑇_𝑢 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 𝑇_𝑢 with 𝑢∈𝑊^1,∞(𝔻) on 𝒟_ℎ.

The harmonic Dirichlet space 𝒟_ℎ consists of all harmonic functions in 𝑊^1,2(𝔻). It is a closed subspace of 𝑊^1,2(𝔻), and hence it is a Hilbert space with the following reproducing kernel: 𝑅_𝑧(𝑤)=𝐾_𝑧(𝑤)+𝐾_𝑧(𝑤)1+1=ln1−𝑧𝑤1+ln1−𝑧𝑤+1. (17)

Let 𝑄 be the orthogonal projection of 𝑊^1,2(𝔻) onto 𝒟_ℎ. 𝑄 is an integral operator represented by 𝑄𝑓(𝑧)=𝑓,𝑅_𝑧=_𝔻𝜕𝑓𝜕𝑤𝜕𝐾_𝑧(𝑤)+𝜕𝑤𝑑𝐴_𝔻𝜕𝑓𝜕𝑤𝜕𝐾_𝑧𝜕𝑤𝑑𝐴(𝑤)+_𝔻𝑓𝑑𝐴. (18) For 𝑢∈𝑊^1,∞(𝔻), the Toeplitz operator 𝑇_𝑢 with symbol 𝑢 is defined by 𝑇_𝑢ℎ=𝑄(𝑢ℎ)ℎ∈𝒟_ℎ. (19)𝑇_𝑢 is a bounded operator for 𝑢∈𝑊^1,∞(𝔻) on 𝒟_ℎ (see [11]).

Theorem 4. Let 𝑢=𝑢₀+𝑐+𝑓+𝑔∈𝑊^1,∞(𝔻) with 𝑢₀+𝑐∈𝒜, 𝑓,𝑔∈𝐻(𝔻), and 𝑓(0)=𝑔(0)=0. Then 𝑇_𝑢 is hyponormal on 𝒟_ℎ if and only if 𝑢∈𝒜.

Proof. By Proposition 1, we only need to prove the necessity with 𝑢(𝑧)=𝑓(𝑧)+𝑔(𝑧)=∑^∞_𝑘=1𝑓_𝑘𝑧^𝑘+∑^∞_𝑘=1𝑔_𝑘𝑧^𝑘.
Let ℎ(𝑧)=𝑎₀+∑^∞_𝑘=1𝑎_𝑘𝑧^𝑘+∑^∞_𝑘=1𝑏_𝑘𝑧^𝑘=∑^∞_𝑘=0𝑎_𝑘𝑧^𝑘+∑^∞_𝑘=1𝑏_𝑘𝑧^𝑘∈𝒟_ℎ. Since 𝑇_𝑢 is hyponormal on 𝒟_ℎ, we have ‖𝑇_𝑓+𝑔ℎ‖²−‖𝑇^∗_𝑓+𝑔ℎ‖²≥0. Note that 𝑇_𝑓+𝑔ℎ(𝑧)=𝑄𝑓+𝑔ℎ(𝑧)=𝑓+𝑔ℎ,𝑅_𝑧=_𝔻𝜕𝑓+𝑔ℎ𝑧𝜕𝑤1−𝑧𝑤+𝑑𝐴_𝔻𝜕𝑓+𝑔ℎ𝜕𝑤𝑧1−𝑧𝑤𝑑𝐴+_𝔻𝑓+𝑔=ℎ𝑑𝐴_∞𝑘=11𝑏𝑘+1^𝑘𝑓_𝑘+𝑎_𝑘𝑔_𝑘+𝑎₀𝑓₁+^∞_𝑘=1𝑏_𝑘𝑓_𝑘+1+𝑎_𝑘+1𝑔_𝑘𝑧+^∞_𝑚=2^𝑚−1_𝑘=1𝑎_𝑚−𝑘𝑓_𝑘+𝑎₀𝑓_𝑚+^∞_𝑘=1𝑏_𝑘𝑓_𝑘+𝑚+𝑎_𝑘+𝑚𝑔_𝑘𝑧^𝑚+𝑎₀𝑔₁+^∞_𝑘=1𝑎_𝑘𝑔_𝑘+1+𝑏_𝑘+1𝑓_𝑘𝑧+^∞_𝑚=2^𝑚−1_𝑘=1𝑏_𝑚−𝑘𝑔_𝑘+𝑎₀𝑔_𝑚+^∞_𝑘=1𝑎_𝑘𝑔_𝑘+𝑚+𝑏_𝑘+𝑚𝑓_𝑘𝑧^𝑚. (20)
Thus 𝑇_𝑓+𝑔ℎ²=_𝔻𝑇_𝑓+𝑔ℎ𝑑𝐴²+_𝔻𝜕𝑇_𝑓+𝑔ℎ𝜕𝑧²𝑑𝐴+_𝔻𝜕𝑇_𝑓+𝑔ℎ𝜕𝑧²=𝑑𝐴_∞𝑘=11𝑏𝑘+1^𝑘𝑓_𝑘+𝑎_𝑘𝑔_𝑘²+𝑎₀𝑓₁+^∞_𝑘=1𝑏_𝑘𝑓_𝑘+1+𝑎_𝑘+1𝑔_𝑘²+^∞_𝑚=2^𝑚−1_𝑘=1𝑎_𝑚−𝑘𝑓_𝑘+𝑎₀𝑓_𝑚+^∞_𝑘=1𝑏_𝑘𝑓_𝑘+𝑚+𝑎_𝑘+𝑚𝑔_𝑘²+𝑎₀𝑔₁+^∞_𝑘=1𝑎_𝑘𝑔_𝑘+1+𝑏_𝑘+1𝑓_𝑘²+^∞_𝑚=2^𝑚−1_𝑘=1𝑏_𝑚−𝑘𝑔_𝑘+𝑎₀𝑔_𝑚+^∞_𝑘=1𝑎_𝑘𝑔_𝑘+𝑚+𝑏_𝑘+𝑚𝑓_𝑘². (21)
Similarly, we have 𝑇^∗_𝑓+𝑔ℎ(𝑧)=𝑇^∗_𝑓+𝑔ℎ,𝑅_𝑧=ℎ,𝑇_𝑓+𝑔𝑅_𝑧=ℎ,𝑓+𝑔𝑅_𝑧=^∞_𝑘=1𝑘𝑎_𝑘𝑓_𝑘+𝑏_𝑘𝑔_𝑘+𝑎₀2𝑔₁+^∞_𝑘=1𝑘𝑏_𝑘𝑔_𝑘+1+(𝑘+1)𝑎_𝑘+1𝑓_𝑘𝑧+^∞_𝑚=21𝑚^𝑚−1_𝑘=1𝑘𝑎_𝑘𝑔_𝑚−𝑘+𝑎₀𝑔𝑚+1^𝑚+^∞_𝑘=1(𝑚+𝑘)𝑎_𝑘+𝑚𝑓_𝑘+𝑘𝑏_𝑘𝑔_𝑘+𝑚𝑧^𝑚+𝑎₀2𝑓₁+^∞_𝑘=1𝑘𝑎_𝑘𝑓_𝑘+1+(𝑘+1)𝑏_𝑘+1𝑔_𝑘𝑧+^∞_𝑚=21𝑚^𝑚−1_𝑘=1𝑘𝑏_𝑘𝑓_𝑚−𝑘+𝑎₀𝑚+1𝑓_𝑚+^∞_𝑘=1(𝑚+𝑘)𝑏_𝑘+𝑚𝑔_𝑘+𝑘𝑎_𝑘𝑓_𝑘+𝑚^𝑧𝑚𝑇^∗_𝑓+𝑔ℎ²=_𝔻𝑇^∗_𝑓+𝑔ℎ𝑑𝐴²+_𝔻𝜕𝑇^∗_𝑓+𝑔ℎ𝜕𝑧𝑑𝐴²+_𝔻𝜕𝑇^∗_𝑓+𝑔ℎ𝜕𝑧²𝑑𝐴=^∞_𝑘=1𝑘𝑎_𝑘𝑓_𝑘+𝑏_𝑘𝑔_𝑘²+𝑎₀2𝑔₁+^∞_𝑘=1𝑘𝑏_𝑘𝑔_𝑘+1+(𝑘+1)𝑎_𝑘+1𝑓_𝑘²+^∞_𝑚=21𝑚^𝑚−1_𝑘=1𝑘𝑎_𝑘𝑔_𝑚−𝑘+𝑎₀𝑔𝑚+1^𝑚+^∞_𝑘=1(𝑚+𝑘)𝑎_𝑘+𝑚𝑓_𝑘+𝑘𝑏_𝑘𝑔_𝑘+𝑚²+𝑎₀2𝑓₁+^∞_𝑘=1𝑘𝑎_𝑘𝑓_𝑘+1+(𝑘+1)𝑏_𝑘+1𝑔_𝑘²+^∞_𝑚=21𝑚^𝑚−1_𝑘=1𝑘𝑏_𝑘𝑓_𝑚−𝑘+𝑎₀𝑚+1𝑓_𝑚+^∞_𝑘=1(𝑚+𝑘)𝑏_𝑘+𝑚𝑔_𝑘+𝑘𝑎_𝑘𝑓_𝑘+𝑚². (22) For 𝑖≥2, let 𝑎_𝑖=1/𝑖 and 𝑎_𝑗=0 for 𝑗≠𝑖. It follows that 1𝑖^2∞_𝑘=1𝑓_𝑘²+^𝑖−1_𝑘=1𝑔_𝑘²+1𝑔𝑖+1^𝑖2+^∞_𝑘=𝑖+1𝑔_𝑘²≥^∞_𝑘=11𝑔𝑖+𝑘^𝑘2+𝑓_𝑖²+^𝑖−1_𝑘=11𝑘𝑓_𝑖−𝑘²+^∞_𝑘=11𝑘𝑓_𝑖+𝑘². (23) Therefore, 1𝑖^2∞_𝑘=1𝑓_𝑘²+^∞_𝑘=1𝑔_𝑘²≥^∞_𝑘=11𝑓𝑖+𝑘^𝑘2+𝑔_𝑘². (24) For every 𝑘≥1, we have ^𝑁_𝑖=21𝑖^2∞_𝑙=1𝑓_𝑙²+^∞_𝑙=1𝑔_𝑙²≥^𝑁_𝑖=21𝑓𝑖+𝑘^𝑘2+𝑔_𝑘², (25) where 𝑁≥2. Letting 𝑁→∞, Since ∑^𝑁_𝑖=2(1/𝑖²) and (∑^∞_𝑙=1|𝑓_𝑙|²+∑^∞_𝑙=1|𝑔_𝑙|²) are convergent and ∑^𝑁_𝑖=2(1/(𝑖+𝑘)) (𝑘≥1 is fixed) is disconvergent, we get |𝑓_𝑘|=|𝑔_𝑘|=0 for 𝑘≥1. The proof is finished.

The following corollary generalizes Theorem 3 in [10].

Corollary 5. Suppose that 𝑢=𝑓+𝑔∈𝑊^1,∞(𝔻) with 𝑓,𝑔∈𝐻(𝔻). Then 𝑇_𝑢 is hyponormal on 𝒟_ℎ 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.