Research Article | Open Access
Restriction of Toeplitz Operators on Their Reducing Subspaces
We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that is similar to the sum of copies of the Bergman shift.
Let be a bounded linear operator on a Hilbert space ; a subspace of is called an invariant subspace of if and a reducing subspace of if is an invariant subspace of and . A reducing subspace of is called minimal if for every reducing subspace of such that then either or . For a concrete bounded operator on a separable Hilbert space , it is important to determine invariant subspaces and reducing subspaces for .
Let be the unit open disc in the complex plane and be the normalized area measure on . The Bergman space consists of all analytic functions in the Lebesgue space . It is clear that is a closed subspace of , and let denote the projection from onto . The Toeplitz operator on with symbol is defined by ; it is called an analytic Toeplitz operator if .
An th-order Blaschke product is the analytic function on given bywhere is a real number and for . A Blaschke product is very important in the theory of Hardy space.
Characterization of reducing subspaces of an analytic Toeplitz operator on Bergman space has been of great interest for last two decades. Thomson [1, 2] showed that it suffices to study reducing subspace of for a finite Blaschke product in the case of Hardy space. It can be generalized to Bergman spaces easily.
Zhu studied the reducing subspaces of for a Blaschke product of order 2 firstly and showed that has exactly two distinct minimal reducing subspaces (cf. ). Motivated by this fact, Zhu conjectured that the number of minimal reducing subspaces of equals the order of (cf. ). Guo et al. showed that in general this is not true (cf. ), and they found that the number of minimal reducing subspaces of equals the number of connected components of the Riemann surface of when the order of is . Then they conjectured that the number of minimal reducing subspaces of equals the number of connected components of the Riemann surface of for any finite Blaschke product (called the refined Zhu’s conjecture, cf. ). Douglas et al. confirmed the conjecture in [5, 6] by using local inverses of Blaschke products . Tikaradze  generalized a part of results in  to bounded smooth pseudoconvex domains in . Douglas and Kim  studied reducing subspace on Bergman space of the annulus; the case of Hardy space was summarized in .
In , Douglas et al. generalized the bundle shift  to the case of Bergman spaces, constructed a vector bundle model for analytic Toeplitz operator on the Bergman space , and tried to build vector bundle models for restrictions of to its minimal reducing subspaces, but it is not completed. Douglas  studied unitary equivalence of the restrictions by computing their curvatures of corresponding geometric models. Hu et al.  showed that, for , there is a distinguished reducing subspace such that the restriction of on is the Bergman shift. In this paper, we analyze the concrete examples to see what are the possible models for these restrictions for further research.
2. Unitary Equivalence and Similarity of Weighted Shifts
Let be an operator on a separable infinite-dimensional Hilbert space . is called a one-side weighted shift if there exist an orthonormal basis for and a bounded sequence of complex numbers such that . Similarly, is called a two-side weighted shift if there exist an orthonormal basis for and a bounded sequence of complex numbers such that for all .
Lemma 1 (see ). Suppose that and are two injective one-side weighted shifts with weights and respectively; then is unitarily equivalent to if and only if for all .
Lemma 2 (see ). Suppose that and are two injective one-side weighted shifts with weights and , respectively; then is similar to if and only if there exist two constants and such thatfor all .
3. Models for Restriction of Toeplitz Operators on Their Minimal Reducing Subspaces
3.1. The Bergman Spaces on the Slit Disc
The domain is called the slit disk. Let denote the normalized area measure on . is the set of analytic functions in the Lebesgue space . For a nonnegative measurable function on , we can define the weighted Bergman space with respect to to be the set of all analytic functions in the Lebesgue space . Ross studied invariant subspaces of Bergman spaces on slit domains in . Aleman et al. defined and studied the Hardy space of a slit domain and in particular they studied the invariant subspace of the slit disk; one can consult  for details.
3.2. Slit Disc Models for
It is easy to check that is a measurable function on and that is a probability measure on .
Lemma 3. The Bergman space contains constants and power functions . Furthermore, and are orthonormal basis of and , respectively.
Proof. For the Bergman space and ,so .
For the Bergman space and ,so .
Lemma 4. The multiplication operator is a bounded operator on and .
Proof. For the Bergman space , it is easy to show thatSo is a weighted shift with weight and . Similarly, on the Bergman space , is a weighted shift with weight and so bounded.
We know that are the only two minimal reducing subspaces of on .
Theorem 5. , are unitarily equivalent to the multiplication operators on the following two spaces, respectively:
Proof. Define the following maps:Then are two isometries on , respectively. As a fact, by changing of variable , , we haveso is an isometry. Andit shows that is an isometry. The next is to show that on . For every function , we haveOne can show that on similarly.
Suppose that is a Blaschke product with two different zeroes and ; let be the normalized reproducing kernel at .
Lemma 6 (see ). Let with two different zeroes . Let be the geodesic midpoint between and . Then the operatorhasas its only two proper reducing subspaces, where .
Theorem 7. The restrictions of on and are unitarily equivalent to on and , respectively, where .
Proof. Now we define the following operators:Then are two isometries on , respectively. As a fact, by changing of variables two times, (, for the first time; , for the second time), we haveso is a isometry. Andit shows that is a isometry. Moreover, for any , we havewhere and is the midpoint between and . On the other hand, Note that is a function in . For , one can prove similarly that
3.3. Weighted Shift Models
Proposition 8. The restrictions of on its minimal reducing subspaces are one-side weighted shifts, and they are not unitarily equivalent to each other.
Proof. We know thatare the two minimal reducing subspaces of on the Bergman space . It is clear thatso and are weighted shifts with the weight sequences and , respectively. It is clear that is not unitarily equivalent to by Lemma 1.
Proposition 9. The restrictions of on its minimal reducing subspaces are one-side weighted shifts, and they are similar to each other.
Proof. Let and be the minimal reducing subspaces of ; we know that is the weighted shift with weight and is the weighted shift with weight . By a direct computation, we haveSo for any , we havesince is decreasing as increases to infinity. It implies that is similar to by Lemma 2.
Remark 10. is just the Bergman shift, so we know that is similar to the direct sum of two copies of the Bergman shift.
3.4. For Reducing Subspace of
Theorem 11. The slit disc models for restrictions of are the on the following spaces:
Proof. LetWe define the following operators, for :and then we can show that are isometries; as a fact, by changing of variable , , soMoreover, for any , we have
Remark 12. is a finite measure on for , so is a finite measure on .
Proposition 13. The restrictions of on its minimal reducing subspaces are one-side weighted shifts, and they are not unitarily equivalent to each other.
Proof. For the Bergman space and any , we havethat is, . It implies thatthat is, , and so is the weighted shift with weight , and so is bounded and is not unitarily equivalent to when .
Proposition 14. The restrictions of on its minimal reducing subspaces are one-side weighted shifts, and they are similar to each other.
Proof. and are the one-side weighted shift with weights and , where . So, for any ,Then,since is decreasing as increases to infinity. It implies that is similar to for any .
Remark 15. For any , is similar to which is the Bergman shift, and so is similar to the direct sum of copies of the Bergman shift.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors thank R. G. Douglas and D. K. Keshari for the helpful discussions. Anjian Xu is supported in part by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1500938), Chongqing Science and Technology Commission (Grant no. CSTC2015jcyjA00045), and NSF of China (11501068). Yang Zou is supported in part by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1501414).
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