AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 186326 10.1155/2013/186326 186326 Research Article Hyponormal Toeplitz Operators on the Dirichlet Spaces http://orcid.org/0000-0003-0796-5705 Cui Puyu Lu Yufeng Atakishiyev Natig M. School of Mathematical Sciences Dalian University of Technology Dalian 116024 China dlut.edu.cn 2013 17 11 2013 2013 28 06 2013 02 10 2013 24 10 2013 2013 Copyright © 2013 Puyu Cui and Yufeng Lu. 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.

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 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 La2 consists of all analytic functions in L2(𝔻,dA). The Sobolev space W1,2(𝔻) is the space of functions f:𝔻 with the following norm: (1)f=[|𝔻fdA|2+𝔻|fz|2+|fz¯|2dA  ]1/2<.W1,2(𝔻) is a Hilbert space with the inner product (2)f,g=𝔻fdA𝔻g¯dA+fz,gzL2(𝔻,dA)+fz¯,gz¯L2(𝔻,dA). 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 (3)W1,(𝔻)={uW1,2(𝔻):u,uz,uz¯L(𝔻,dA)}, with the norm (4)u1,=max{u,uz,uz¯}. Let P be the orthogonal projection of W1,2(𝔻) onto 𝒟. P is an integral operator represented by (5)P(f)(z)=𝔻fw(Kzw)-dA, where Kz(w)=k=1(z¯kwk/k) is the reproducing kernel of 𝒟. For uW1,(𝔻), the Toeplitz operator Tu with symbol u is defined by (6)Tuh=P(uh)h𝒟.Tu is a bounded operator for uW1,(𝔻) on 𝒟.

Yu gave a decomposition of the Sobolev space W1,2(𝔻) in . Let 𝒫0 be the set of all the following polynomials: (7)j-ll0al+j,lzl+jz¯l, where j and l run over a finite subset of and l0al+j,l=0. Let 𝒜0 denote the closure of 𝒫0 in W1,2(𝔻), and let 𝒜 denote 𝒜0+. Since the set of all polynomials in z and z¯ is dense in W1,2(𝔻), there is the following decomposition: (8)W1,2(𝔻)=𝒜𝒟𝒟¯.

Since W1,(𝔻)W1,2(𝔻) and by the above decomposition, it follows that, if uW1,(𝔻), then u=u0+f+g¯, where u0𝒜, f, gH(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 [<xref ref-type="bibr" rid="B10">1</xref>]).

Let ϕW1,(𝔻). Then ϕ𝒜0𝒜0.

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) . 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 Tu with uW1,(𝔻) on Dirichlet space 𝒟 and harmonic Dirichlet space 𝒟h.

2. Case on the Dirichlet Space

In this section, the hyponormality of Tu with uW1,(𝔻) on 𝒟 will be discussed.

Theorem 2.

Let u=u0+c+f+g¯W1,(𝔻) with u0+c𝒜, f,gH(𝔻), and f(0)=g(0)=0. Then Tu is hyponormal on 𝒟 if and only if u𝒜.

Proof.

By Proposition 1, we only need to prove the necessity with u(z)=f(z)+g(z)¯=k=1fkzk+k=1g¯kz¯k.

Let h(z)=k=1hkzk𝒟. Simple calculations imply that (9)Tf+g¯h(z)=Tfh(z)+Tg¯h(z)=f(z)h(z)+P(g¯h)(z)=f(z)h(z)+  g¯h,Kz=m=2(k=1m-1fkhm-k)zm+l=2hl(k=1l-1g¯kzl-k)=m=2(k=1m-1fkhm-k)zm+m=1k=1g¯khk+mzm.

Furthermore, (10)(Tf+g¯h)z=m=1(m+1)(k=1mfkhm-k+1)zm+m=0(m+1)(k=1g¯khk+m+1)zm=k=1g¯khk+1+m=1(m+1)  ×(k=1mfkhm-k+1+k=1g¯khk+m+1)zm.

Therefore (11)Tf+g¯h2=|k=1g¯khk+1|2+m=1|k=1mfkhm-k+1+k=1g¯khk+m+1|2.

Similarly, we have (12)Tf+g¯*h(z)=Tf+g¯*h,Kz=h,Tf+g¯Kz=m=21m(l=1m-1lgm-lhl)zm+m=1(k=1m+kmf¯khk+m)zm,(Tf+g¯*h)z=k=1(k+1)f¯khk+1+m=1[k=1(m+k+1)f¯khm+k+1+l=1mlgm-l+1hl]zm,Tf+g¯*h2=|k=1(k+1)f¯khk+1|2+m=11m+1|k=1(m+k+1)fk¯hm+k+1+l=1mlgm-l+1hl|2. Denote ei(z)=(1/i)zi for i1. Since Tu is hyponormal, we have (13)Tuei2-Tu*ei20fori1. For i2, Tuei2-Tu*ei20 implies that (14)1i2(k=1i-1|gk|2+l=1|fl|2)k=1i-11i-k|fk|2+l=11i+l|gl|21i|f1|2. Hence (15)(i=2N1i2)(l=1|gl|2+l=1|fl|2)(i=2N1i)|f1|2forN2. Letting N, since i=2N(1/i2) and (l=1|gl|2+l=1|fl|2) are convergent and i=2N(1/i) is disconvergent, we get f1=0. Similarly, by choosing i, we get fl=0 for l1. Note that Tue12-Tu*e120 implies that l=1|fl|2l=1(1/(l+1))|gl|2. Thus gl=0 for l1 and the proof is finished.

The following corollary generalizes Theorems 1 and 2 in . Denote (16)Ω={u:u=f+g¯,f,gH(𝔻),|f|2dAisa𝒟-Carleson measure}, where H(𝔻) is the space of the bounded analytic functions on 𝔻.

Corollary 3.

Let uΩ. 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 uW1,(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: (17)Rz(w)=Kz(w)¯+Kz(w)+1=ln11-zw¯+ln11-z¯w+1.

Let Q be the orthogonal projection of W1,2(𝔻) onto 𝒟h. Q is an integral operator represented by (18)Q(f)(z)=f,Rz=𝔻fw(Kzw)¯dA(w)+𝔻fw¯(Kz¯w¯)¯dA(w)+𝔻fdA. For uW1,(𝔻), the Toeplitz operator Tu~ with symbol u is defined by (19)Tu~h=Q(uh)h𝒟h.Tu~ is a bounded operator for uW1,(𝔻) on 𝒟h (see ).

Theorem 4.

Let u=u0+c+f+g¯W1,(𝔻) with u0+c𝒜, f,gH(𝔻), and f(0)=g(0)=0. Then Tu~ is hyponormal on 𝒟h if and only if u𝒜.

Proof.

By Proposition 1, we only need to prove the necessity with u(z)=f(z)+g(z)¯=k=1fkzk+k=1g¯kz¯k.

Let h(z)=a0+k=1akzk+k=1bkz¯k=k=0akzk+k=1bkz¯k𝒟h. Since Tu~ is hyponormal on 𝒟h, we have T~f+g¯h2-T~f+g¯*h20. Note that (20)T~f+g¯h(z)=Q[(f+g¯)h](z)=(f+g¯)h,Rz=𝔻[(f+g¯)h]wz1-zw¯dA+𝔻[(f+g¯)h]w¯z¯1-z¯wdA+𝔻(f+g¯)hdA=k=11k+1(bkfk+akg¯k)+[a0f1+k=1(bkfk+1+ak+1g¯k)]z+m=2[k=1m-1am-kfk+a0fm+k=1(bkfk+m+ak+mg¯k)]zm+[a0g¯1+k=1(akg¯k+1+bk+1fk)]z¯+m=2[k=1m-1bm-kg¯k+a0g¯m+k=1(akg¯k+m+bk+mfk)]z¯m.

Thus (21)T~f+g¯h2=|𝔻T~f+g¯hdA|2+𝔻|T~f+g¯hz|2dA+𝔻|T~f+g¯hz¯|2dA=|k=11k+1(bkfk+akg¯k)|2+  |a0f1+k=1(bkfk+1+ak+1g¯k)|2+m=2|k=1m-1am-kfk+a0fm  +k=1(bkfk+m+ak+mg¯k)|2+|a0g¯1+k=1(akg¯k+1+bk+1fk)|2+m=2|k=1m-1bm-kg¯k+a0g¯m  +k=1(akg¯k+m+bk+mfk)|2.

Similarly, we have (22)T~f+g¯*h(z)=T~f+g¯*h,Rz=h,T~f+g¯Rz=h,(f+g¯)Rz=k=1k(akf¯k+bkgk)+{a02g1+k=1[kbkgk+1+(k+1)ak+1fk¯]}z+m=21m{k=1m-1kakgm-k+a0m+1gm+k=1[(m+k)ak+mf¯k+kbkgk+m]}zm+{a02f¯1+k=1[kakf¯k+1+(k+1)bk+1gk]}z¯+m=21m{k=1m-1kbkf¯m-k+a0m+1f¯m+k=1[(m+k)bk+mgk+kakf¯k+m]}z¯mT~f+g¯*h2=|𝔻T~f+g¯*hdA|2+𝔻|(T~f+g¯*h)zdA|2+𝔻|(T~f+g¯*h)z¯|2dA=|k=1k(akf¯k+bkgk)|2+|a02g1+k=1[kbkgk+1+(k+1)ak+1fk¯]|2+  m=21m|k=1m-1kakgm-k+a0m+1gm  +k=1[(m+k)ak+mf¯k+kbkgk+m]|2+|a02f¯1+k=1[kakf¯k+1+(k+1)bk+1gk]|2+m=21m|k=1m-1kbkf¯m-k+a0m+1f¯m+k=1[(m+k)bk+mgk+kakf¯k+m]|2. For i2, let ai=1/i and aj=0 for ji. It follows that (23)1i2(k=1|fk|2+k=1i-1|gk|2+|1i+1gi|2+k=i+1|gk|2)k=11i+k|gk|2+|fi|2+k=1i-11k|fi-k|2+k=11k|fi+k|2. Therefore, (24)1i2(k=1|fk|2+k=1|gk|2)k=11i+k(|fk|2+|gk|2). For every k1, we have (25)(i=2N1i2)(l=1|fl|2+l=1|gl|2)(i=2N1i+k)(|fk|2+|gk|2), where N2. Letting N, Since i=2N(1/i2) and (l=1|fl|2+l=1|gl|2) are convergent and i=2N(1/(i+k)) (k1 is fixed) is disconvergent, we get |fk|=|gk|=0 for k1. The proof is finished.

The following corollary generalizes Theorem 3 in .

Corollary 5.

Suppose that u=f+g¯W1,(𝔻) with f,gH(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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