On Similarity and Reducing Subspaces of the -Shift plus Certain Weighted Volterra Operator
Let be an -degree polynomial . Inspired by Sarason’s result, we introduce the operator defined by the multiplication operator plus the weighted Volterra operator on the Bergman space. We show that the operator is similar to on some Hilbert space . Then for , by using matrix manipulations, the reducing subspaces of the corresponding operator on the Bergman space are characterized.
The invariant subspace and reducing subspace problems are interesting and important themes in operator theory. The conjecture is that every bounded linear operator on a separable Hilbert space has a nontrivial closed invariant subspace. A closed linear nontrivial subspace of is called an invariant subspace for if is different from and such that . If and are both invariant subspaces for , then is said to be a reducing subspace for . The invariant subspace and reducing subspace problems on the Hardy space and the Bergman space have been studied extensively in the literature. We mention here the papers [1–13] and the books [14–17] which include a lot of the information on the corresponding operator theory.
Let be the unit disk in the complex plane , and let denote the Bergman space of analytic functions which belongs to , where is the space of square integrable functions on . It is well known that is a Hilbert space. If , then where is the normalized area measure on , and For , let and then the inner product of and is defined by In this inner product, has an orthonormal basis , where Let denote the algebra of bounded analytic functions on . For , is an analytic multiplication operator on the Bergman space defined by is a bounded linear operator on with
Over the years it has been shown that many familiar classes of operators do have invariant subspaces. The lattice of shift operator acting on the Hardy space is completely described by Beurling’s Theorem . Sarason (see ) characterized all closed invariant subspaces of the Volterra operator In , Aleman characterized boundedness and compactness of the integral operator between Hardy space and for . In , using Beurling’s Theorem, Aleman and Korenblum studied the complex Volterra operator in the Hardy space defined by Then they characterized the lattice of closed invariant subspaces of . Sarason (see ) studied the lattice of closed invariant subspaces of acting on . Montes-Rodriguez et al. (see ) and Cowen et al. (see ) used the idea of Sarason to study the invariant subspaces of certain classes of composition operators on the Hardy space. Following Sarason’s work, Čučković and Paudyal (see ) characterized the lattice of closed invariant subspaces of the shift plus complex Volterra operator on the Hardy space. In their paper, the operator is defined by Ball (see ) and Nordgren (see ) studied the problem of determining the reducing subspaces for an analytic Toeplitz operator on the Hardy space. In , Stessin and Zhu gave a complete description of the weighted unilateral shift operator of finite multiplicity on some Hilbert spaces type I and type II. In , Zhu described the properties of the commutant of analytic Toeplitz operators with inner function symbols on the Hardy space and the Bergman space and characterized the reducing subspaces of a class of multiplication operators. In 2011, Douglas and Kim in  studied the reducing subspaces for an analytic multiplication operator on the Bergman space of the annulus .
Based on the above works, for an -degree polynomial , we introduce the operator defined by the multiplication operator plus the weighted Volterra operator on the Bergman space. We show that the operator is similar to on some Hilbert space . Then for , by using matrix manipulations, the reducing subspaces of corresponding operator on the Bergman space are characterized.
2. The Similarity of the Operator
For an -degree polynomial , the operator is defined by
To prove our result, we introduce the space defined by where is the space of holomorphic functions on the unit disk, and is the differentiation operator. From the definition of , for , we have So . It can be shown that, for any holomorphic function with and , then . So the norm of is defined by Corresponding inner product is given by
In the following, we suppose that has no zero points on . This condition guarantees that is closed under the given norm.
Theorem 1. Let be the weighted Volterra operator on . Then the following statements hold:(i)Range of is .(ii) is a bounded isomorphism from to , and its inverse is .(iii)The operator acting on is similar under to the multiplication operator on .
Proof. (i) Let be in the range of , and then there exists such that So , , and . Therefore, . Conversely, suppose that ; then Hence belongs to the range of .
(ii) First we want to show is a bounded operator on . For , we have Clearly is linear. To show is one to one, suppose that , satisfying that Differentiating both sides, we obtain that and hence is one to one. From the definition of we have that , for , and , for .
Therefore, is a bounded bijective linear operator from onto , and .
(iii) Suppose belongs to , and . Note that Now applying on both sides of the above equality, we obtain that So and . That is to say, transforms the operator into the multiplication operator on .
3. The Reducing Subspaces of the Operator
In this section, for fixed , we consider the case of . That is, the operator is defined by Since the -shift operator is a contraction operator, and the operator is a bounded operators, is a bounded operator on .
Let denote the commutant of , that is, , where represents the collection of all bounded linear operators on a Hilbert space . Then we have the following lemma.
Lemma 2. Let be the Bergman space. If is a bounded operator on , then if and only if admits the following matrix representation: with respect to the orthonormal basis of , where and . Moreover, if is a projection, then if and only if or .
Proof. Denote the orthonormal basis of by . Note that So the operator admits the following matrix representation with respect to the above basis: Suppose that has the following matrix representation with respect to the orthonormal basis of : From , we have Thus So we obtain Conversely, if admits the matrix representation (30) with respect to the above basis, simple computation shows that . So . Moreover, if is a projection, we deduce that if and only if has the following form: where or 0.
Remark 3. In fact, since is irreducible on , any projection in is or .
The following lemma will be used in the proof of main theorem.
Lemma 4. Let . Then (i) forms an orthonormal basis of .(ii).(iii) is a reducing subspace of .
Proof. (i) and (ii) are obvious. We only need to show (iii). Note that So we have and as desired.
Set . Then we have the following theorem.
Theorem 5. If is a projection, then if and only ifwhere is or .
Proof. If , note that , and then the operator can be decomposed in the following form: where . The condition yields that Suppose that has the following matrix representation with respect to the orthonormal basis of ;where .
From (32), we know the operator admits the following matrix representation with respect to the basis of ;Case 1 (). From (37), we get . Applying Lemma 2, we know has the form similar to (30). Note that is a projection, so . Hence .
Case 2 (). From (37), we have that . This equality is equivalent to So we obtain On the other hand, from (37) we also have . In the same way as for , we get is a projection that yields . Thus we have . Solving the system of equations we get . Therefore, . implies that where is or 0.
Conversely, if where is or 0, it is obvious that .
From , we know that determining the reducing subspaces of is equivalent to finding the projection in the commutant of . Thus we have the following conclusion.
Theorem 6 (main theorem). Let be the Bergman space. For , the operator has reducing subspaces with minimal reducing subspaces .
Proof. Suppose that for a projection . Note that . HenceBy Theorem 5, the projection operatorwhere is or 0. Applying Lemma 4, we have that the reducing subspaces of are and the minimal reducing subspaces are .
4. Some Consequences
In this section, we use the characterization of the reducing subspaces of to obtain a description of reducing subspaces of on , which is similar to .
The space is defined by If , it follows that , and then . Since , we have , and So . The norm of is defined by Corresponding inner product is given by
The proof of the following theorem is similar to the proof of Theorem 1, so we omit it.
Theorem 7. Let be the weighted Volterra operator on . Then the following statements hold:(i)Range of is .(ii) is a bounded isomorphism from to , and its inverse is . (Remark: Note acting on , and from the definition of , we know that is a removable singular point of , for . We can define by . Then can be viewed as an analytic point of ).(iii)The operator acting on is similar under to the multiplication operator on .
Note that ; from (52), we have Thus, the orthonormal basis of is given by
In order to characterize the reducing subspaces of on , we need the following lemma which is similar to Lemma 4.
Lemma 8. Let . Then (i) forms an orthonormal basis of .(ii).(iii) is a reducing subspace of .
Corollary 9. Let be described as the above. For , the multiplication operator has reducing subspaces with minimal reducing subspaces .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research is supported by NNSF of China (11371119).