JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2017/5807909 5807909 Research Article A Note on Reducing Subspaces of Toeplitz Operator on the Weighted Analytic Function Spaces of the Bidisk Hw2D2 http://orcid.org/0000-0002-2180-7837 Lin Hongzhao 1 Curto Raúl E. College of Computer and Information Sciences Fujian Agriculture and Forestry University Fujian 350002 China fjau.edu.cn 2017 752017 2017 22 02 2017 28 03 2017 752017 2017 Copyright © 2017 Hongzhao Lin. 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 discussed the reducing subspaces of Toeplitz operators Tz1Nz2MNM on the weighted analytic function spaces of bidisk Hw2(D2). The result shows that if the weight w is of type-I, the structure of reducing subspaces of Tz1Nz2MNM on Hw2D2 is very simple.

National Natural Science Foundation of China 11601081
1. Introduction

Let D denote the open unit disk of complex plane C and w=(wi,j) be an infinite matrix with wi,j>0; we consider the Hilbert space Hw2(D2) consisting of analytic functions: (1)fz1,z2=i,j=0ai,jz1iz2jon bidisk D2 such that(2)f2=i,j=0ai,j2wi,j<.

There are many examples for Hw2D2. Recall the definition of Dirichlet space and Bergman space; Hw2D2 is Dirichlet space of the bidisk DD2 if wi,j=i(i+1)+j(j+1)/i+1j+1 and Hw2(D2) is Bergman space of the bidisk A2D2 if wi,j=1/(i+1)(j+1). In particular, if we take α=(α1,α2) with α1,α2>-1 and wi,j=i!j!Γ(α1+1)Γ(α2+1)/Γ(i+α1+2)Γ(j+α2+2), Hw2(D2) is Bergman space of the bidisk with weight α, which is usually denoted by Aα2D2.

Given two positive integers N and M with NM, note that z1iz2j/wi,j is the orthonormal basis of Hw2(D2), and it is easy to see that the operator Tz1Nz2M is bounded on Hw2(D2) if and only if(3)M0=supwi+N,j+Mwi,j;i0,j0<.Throughout the paper, we fix a weight matrix w=(wi,j) and two distinct positive integers N,M satisfying (3). We will study the reducing subspace lattice of the operator: (4)S=Tz1Nz2M=Tz1NTz2M.It is easy to check that S has proper reducing subspaces as (5)Xi,j=spanz1i+hNz2j+hM;h0,where 0iN-1 and 0jM-1. It is natural to ask when all the reducing subspaces of S are Xi,j, in other words, when Xi,j is the minimal reducing subspace of S.

Recall that if M is a closed subspace of Hilbert space H, M is called a reducing subspace of the operator T if T(M)M and T(M)M. A reducing subspace M is said to be minimal if there are none nontrivial reducing subspaces of T contained in M.

Stessin and Zhu  completely characterize the reducing subspaces of weighted unilateral shift operators of finite multiplicity. As a consequence, they give the description of the reducing subspaces of TzN on the Bergman space and Dirichlet space of the unit disk. For more general symbols, the reducing subspaces of the Toeplitz operators with finite Blaschke product are well studied (see, e.g., ). Recently, Lu, Shi, and Zhou extend the result in  to Bergman space with several variables. They completely characterize the reducing subspaces of Tz1N and Tz1Nz2N in  on the weighted Bergman space of the bidisk and on the weighted Bergman space over polydisk in , respectively. Moreover, they  solve the problems of Tz1Nz2M with NM on both settings. Motivated by the above work, we have investigated the reducing subspaces of Toeplitz operators Tz1N (or Tz2N) and Tz1Nz2N on the weighted Dirichlet space of the bidisk in .

In this paper, we will consider the problem for Toeplitz operators Tz1Nz2M (NM) on the weighted analytic function spaces of the bidisk Hw2D2. We say that w=(wi,j) is of type-I if for nonnegative integers k,  m,  i,  j,  N:(6)wi+hN,j+hMwi,j=wk+hN,m+hMwk,m,hN,if and only if (i,j)=(k,m).

Theorem 1.

If M is a reducing subspace of Tz1Nz2M on Hw2(D2) with type-1 weight, then there exist integers i and j with 0i<N or 0j<M such that MijM is the minimal reducing subspace of Tz1Nz2M, where (7)Mij=spanz1iz2jz1lNz2lN:  lN.In particular, M is minimal if and only if M=Mij for some i,  j.

2. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

At first, we will give some useful lemmas. The following lemma describes the projection of monomial on reducing subspace M.

Lemma 2.

If M is a reducing subspace of Tz1Nz2M on Hw2(D2) with type-I weight, then, for each multiindex (k,m),  z1kz2mM or z1kz2mM.

Proof.

Let PM be the projection onto M and z1kz2m=fz1,z2+gz1,z2 be the orthogonal decomposition on M, where f(z1,z2)=i,j=0fi,jz1iz2jM and g(z1,z2)M.

Writing Sh=Tz1hNz2hM and (S)h=Tz1hNz2hM, we calculate(8)ShShPMz1kz2m=ShShPMf+g=ShShf=Shi,j=0fi,jz1i+hNz2j+hM=i,j=0fi,jwi+hN,j+hMwi,jz1iz2j.On the other hand, direct computation shows that (9)PMShShz1kz2m=PMShz1k+hNz2m+hN=PMwk+hN,m+hMwk,mz1kz2m=i,j=0fi,jwk+hN,m+hMwk,mz1iz2j.Note that ShShPMz1kz2m=PMShShz1kz2m; if there exists some fi,j0 (otherwise, PMz1kz1z2m=0), it follows that(10)wi+hN,j+hMwi,j=wk+hN,m+hMwk,m,for each positive integer h. Since the weight is of type-I, it reaches that (i,j)=(k,m), which means that (11)PMz1kz2m=z1kz2m.

Lemma 3.

Let Λ be an index set; Hilbert space X is the direct sum of its closed subspace Xi  iΛ, that is, X=iΛXi, M is a reducing subspace of bounded linear operator T on X, and PMXiXi. If f=iΛfiM with fiXi, then fiM for each iΛ.

Proof.

Note that(12)iΛfi=f=PMf=iΛPMfi.The result follows from fi=PMfi since fiXi and PMfiXi.

The following result is immediately achieved by Lemmas 2 and 3.

Lemma 4.

If M is a nontrivial reducing subspace of Tz1Nz2M on Hw2(D2) and f=i,j=0fi,jz1iz2jM, then z1iz2jM for fi,j0.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Suppose f=i,j0fi,jz1iz2jM. If fi,j0, by Lemma 4, z1iz2jM. Since M is a reducing subspace of S, SlMM and SlMM for any lN. Thus, (13)z1i+lNz2j+lMM,where i=i-hN and j=j-hM with h=mini/N,j/M. It follows that MijM.

Observe that each reducing subspace M contains a reducing subspace such as Mij, which means that Mij consist of all the minimal reducing subspaces. If M is a minimal reducing subspace, then M=Mij for some Mij. This completes the proof.

3. Some Examples Example 1.

Dirichlet space of bidisk DD2 is Hw2D2 with type-1 weight. By Theorem 1, Mij is the minimal reducing subspace of Tz1Nz2M.

Recall that D(D2)=Hw2(D2) with wi,j=i(i+1)+j(j+1)/(i+1)(j+1). The following lemma shows that the weight w is of type-I; then Theorem 1 holds.

Lemma 5.

Supposing that k,  m,  i,  j,  N are nonnegative integers, then (14)wi+hN,j+hMwi,j=wk+hN,m+hMwk,m,hN,if and only if (i,j)=(k,m).

Proof.

We only need to prove the necessity. By the assumption, (15)wi+hN,j+hMwk+hN,m+hM=wi,jwk,m,hN.Taking h in the left side, it follows that (16)limhwi+hN,j+hMwk+hN,m+hM=1.Thus, for any positive integer h, (17)wi+hN,j+hMwk+hN,m+hM=1,since the right side is constant. By definition of wi,j, it is equivalent to(18)I1=I2,where (19)I1=m+hM+1k+hN+1i+hNi+hN+1+j+hMj+hM+1,I2=j+hM+1i+hN+1k+hNk+hN+1+m+hMm+hM+1.By combining like terms, I1=I2 converts to(20)i+hN+1k+hN+1C1h+C2=j+hM+1m+hM+1C1h+C3,where C1=m-jN+i-kM,  C2=m+1i-j+1k, and C3=mi+1-jk+1. Comparing the coefficient of h3, we have (21)N2C1=M2C1,which implies that C1=0 since NM. Thus, (20) turns to(22)i+hN+1k+hN+1C2=j+hM+1m+hM+1C3.We claim that C2=0 and C3=0. Otherwise, we may assume C20.

By comparing the coefficient of h2,  h,  1 in (22), we have (23)N2M2=i+k+2Nj+m+2M=i+1k+1j+1m+1=C3C2.Note that C1=0, the first equality implies that i+1/j+1=N/M. Then the second equality gives that k+1/m+1=N/M. Thus(24)i-kj-m=NM.However, recalling the definition of C2 and C3, we have (25)C3C2=mi+1-jk+1m+1i-j+1k=mN/Mj+1-jN/Mm+1M/Nk+1i-M/Ni+1k=N2m-jM2i-k.Note that C3/C2=N2/M2, it follows that (26)m-ji-k=1,which contradicts (24).

Thus the claim that C2=0 and C3=0 holds. By the definition of C2 and C3, simple computation shows that i-k=m-j. Since C1=0, it follows that i=k and m=j. This completes the proof.

Example 2.

The weighted Bergman space of bidisk Aα2(D2) such that weight α=(a,a) with a>-1 is Hw2(D2) with type-1 weight. By Theorem 1, Mij is the minimal reducing subspace of Tz1Nz2M.

Recall that Aα2(D2)=Hw2(D2) with wi,j=i!j!Γ2(a+1)/Γ(i+a+2)Γ(j+a+2). The proof of Theorem  3.2 in  indicated that the weight w=(wi,j) is of type-I; then Theorem 1 holds.

However, the case of the unweighted Bergman space of the bidisk A2(D2) is different since A2(D2)=Hw2(D2) with w=(wi,j) where wi,j=ii+1+jj+1/i+1j+1, which is not of type-I by Lemma  2.3 in . The structure of reducing subspaces of Tz1Nz2MNM on this case is more complicated. In fact, Theorem  2.4 in  showed that if M is a reducing subspace, then there exist nonnegative integers n,  m with 0mN-1 and a,bC such that M contains a reducing subspace as follows:(27)Mn,m,a,b=spanaz1hN+nz2hM+m+bz1ρ2hM+nz2ρ1hN+m:hN,where ρ1(hN+m)=(hN+m+1)M/N-1, ρ2(hM+n)=hM+n+1N/M-1. In particular, if ρ1hN+m (or ρ2hM+n) is not a positive integer, then b=0. Moreover, M is minimal if and only if M=Mn,m,a,b.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11601081).

Stessin M. Zhu K. Reducing subspaces of weighted shift operators Proceedings of the American Mathematical Society 2002 130 9 2631 2639 10.1090/S0002-9939-02-06382-7 MR1900871 Zbl1035.47015 2-s2.0-0036720483 Zhu K. Reducing subspaces for a class of multiplication operators Journal of the London Mathematical Society. Second Series 2000 62 2 553 568 10.1112/S0024610700001198 MR1783644 Guo K. Sun S. Zheng D. Zhong C. Multiplication operators on the Bergman space via the Hardy space of the bidisk Journal für die Reine und Angewandte Mathematik. [Crelle's Journal] 2009 628 129 168 10.1515/CRELLE.2009.021 MR2503238 Guo K. Huang H. On multiplication operators on the Bergman space: similarity, unitary equivalence and reducing subspaces Journal of Operator Theory 2011 65 2 355 378 MR2785849 Zbl1222.47040 2-s2.0-79955664954 Lu Y. Zhou X. Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk Journal of the Mathematical Society of Japan 2010 62 3 745 765 MR2648061 10.2969/jmsj/06230745 Zbl1202.47008 2-s2.0-77957931479 Zhou X. Shi Y. Lu Y. Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc Science China Mathematics 2011 41 5 427 438 10.1360/012010-627 Shi Y. Lu Y. Reducing subspaces for Toeplitz operators on the polydisk Bulletin of the Korean Mathematical Society 2013 50 2 687 696 10.4134/BKMS.2013.50.2.687 MR3137713 Zbl1280.47039 2-s2.0-84887063641 Lin H. Z. Hu Y. Y. Lu Y. F. Reducing subspaces of Toeplitz operator on the weighted dirichlet space of the bidisk Chinese Annals of Mathematics. Series A 2016 37 3 311 328 MR3587338