A Note on Reducing Subspaces of Toeplitz Operator on the Weighted Analytic Function Spaces of the Bidisk
We discussed the reducing subspaces of Toeplitz operators on the weighted analytic function spaces of bidisk . The result shows that if the weight is of type-I, the structure of reducing subspaces of on is very simple.
Let denote the open unit disk of complex plane and be an infinite matrix with ; we consider the Hilbert space consisting of analytic functions: on bidisk such that
There are many examples for . Recall the definition of Dirichlet space and Bergman space; is Dirichlet space of the bidisk if and is Bergman space of the bidisk if . In particular, if we take with and , is Bergman space of the bidisk with weight , which is usually denoted by .
Given two positive integers and with , note that is the orthonormal basis of , and it is easy to see that the operator is bounded on if and only ifThroughout the paper, we fix a weight matrix and two distinct positive integers satisfying (3). We will study the reducing subspace lattice of the operator: It is easy to check that has proper reducing subspaces as where and . It is natural to ask when all the reducing subspaces of are , in other words, when is the minimal reducing subspace of .
Recall that if is a closed subspace of Hilbert space , is called a reducing subspace of the operator if and . A reducing subspace is said to be minimal if there are none nontrivial reducing subspaces of contained in .
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 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., [2–4]). Recently, Lu, Shi, and Zhou extend the result in  to Bergman space with several variables. They completely characterize the reducing subspaces of and 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 with on both settings. Motivated by the above work, we have investigated the reducing subspaces of Toeplitz operators (or ) and on the weighted Dirichlet space of the bidisk in .
In this paper, we will consider the problem for Toeplitz operators () on the weighted analytic function spaces of the bidisk . We say that is of type-I if for nonnegative integers , , , , :if and only if .
Theorem 1. If is a reducing subspace of on with type-1 weight, then there exist integers and with or such that is the minimal reducing subspace of , where In particular, is minimal if and only if for some , .
2. Proof of Theorem 1
At first, we will give some useful lemmas. The following lemma describes the projection of monomial on reducing subspace .
Lemma 2. If is a reducing subspace of on with type-I weight, then, for each multiindex , or .
Proof. Let be the projection onto and be the orthogonal decomposition on , where and .
Writing and , we calculateOn the other hand, direct computation shows that Note that ; if there exists some (otherwise, ), it follows thatfor each positive integer . Since the weight is of type-I, it reaches that , which means that
Lemma 3. Let be an index set; Hilbert space is the direct sum of its closed subspace , that is, , is a reducing subspace of bounded linear operator on , and . If with , then for each .
Proof. Note thatThe result follows from since and .
Lemma 4. If is a nontrivial reducing subspace of on and , then for .
Proof of Theorem 1. Suppose . If , by Lemma 4, Since is a reducing subspace of , and for any . Thus, where and with . It follows that .
Observe that each reducing subspace contains a reducing subspace such as , which means that consist of all the minimal reducing subspaces. If is a minimal reducing subspace, then for some . This completes the proof.
3. Some Examples
Example 1. Dirichlet space of bidisk is with type-1 weight. By Theorem 1, is the minimal reducing subspace of .
Recall that with . The following lemma shows that the weight is of type-I; then Theorem 1 holds.
Lemma 5. Supposing that , , , , are nonnegative integers, then if and only if .
Proof. We only need to prove the necessity. By the assumption, Taking in the left side, it follows that Thus, for any positive integer , since the right side is constant. By definition of , it is equivalent towhere By combining like terms, converts towhere , , and . Comparing the coefficient of , we have which implies that since . Thus, (20) turns toWe claim that and . Otherwise, we may assume .
By comparing the coefficient of , , in (22), we have Note that , the first equality implies that . Then the second equality gives that . ThusHowever, recalling the definition of and , we have Note that , it follows that which contradicts (24).
Thus the claim that and holds. By the definition of and , simple computation shows that . Since , it follows that and . This completes the proof.
Example 2. The weighted Bergman space of bidisk such that weight with is with type-1 weight. By Theorem 1, is the minimal reducing subspace of .
However, the case of the unweighted Bergman space of the bidisk is different since with where , which is not of type-I by Lemma 2.3 in . The structure of reducing subspaces of on this case is more complicated. In fact, Theorem 2.4 in  showed that if is a reducing subspace, then there exist nonnegative integers , with and such that contains a reducing subspace as follows:where , . In particular, if (or ) is not a positive integer, then . Moreover, is minimal if and only if .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work is supported by National Natural Science Foundation of China (Grant no. 11601081).