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Restriction of Toeplitz Operators on Their Reducing Subspaces.

1. Introduction

Let T be a bounded linear operator on a Hilbert space H; a subspace M of H is called an invariant subspace of T if T(M) [subset or equal to] M and a reducing subspace of T if M is an invariant subspace of T and [T.sup.*]. A reducing subspace M of T is called minimal if for every reducing subspace [??] of T such that [??] [subset or equal to] M then either [??] = M or [??] = 0. For a concrete bounded operator T on a separable Hilbert space H, it is important to determine invariant subspaces and reducing subspaces for T.

Let D be the unit open disc in the complex plane and dA(z) be the normalized area measure on D. The Bergman space [L.sup.2.sub.a](D) consists of all analytic functions in the Lebesgue space [L.sup.2](dA(z)). It is clear that [L.sup.2.sub.a](D) is a closed subspace of [L.sup.2](dA(z)), and let P denote the projection from [L.sup.2](dA(z)) onto [L.sup.2.sub.a](D). The Toeplitz operator [T.sub.[phi]] on [L.sup.2.sub.a](D) with symbol [phi] [member of] [L.sup.[infinity]](dA(z)) is defined by ([T.sub.[phi]]f)(z) = P([phi]f)(z); it is called an analytic Toeplitz operator if [phi] [member of] [H.sup.[infinity]](D).

An nth-order Blaschke product B is the analytic function on D given by

[mathematical expression not reproducible], (1)

where [theta] is a real number and [a.sub.i] [member of] D for 1 [less than or equal to] i [less than or equal to] n. A Blaschke product is very important in the theory of Hardy space.

Characterization of reducing subspaces of an analytic Toeplitz operator [T.sub.[phi]] on Bergman space has been of great interest for last two decades. Thomson [1,2] showed that it suffices to study reducing subspace of [T.sub.B] 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 [T.sub.B] for a Blaschke product of order 2 firstly and showed that [T.sub.B] has exactly two distinct minimal reducing subspaces (cf. [3]). Motivated by this fact, Zhu conjectured that the number of minimal reducing subspaces of [T.sub.B] equals the order of B (cf. [3]). Guo et al. showed that in general this is not true (cf. [4]), and they found that the number of minimal reducing subspaces of [T.sub.B] equals the number of connected components of the Riemann surface of B(z) = B(w) when the order of B is 3,4,6. Then they conjectured that the number of minimal reducing subspaces of [T.sub.B] equals the number of connected components of the Riemann surface of B(z) = B(w) for any finite Blaschke product (called the refined Zhu's conjecture, cf. [4]). Douglas et al. confirmed the conjecture in [5,6] by using local inverses of Blaschke products [7]. Tikaradze [8] generalized a part of results in [5] to bounded smooth pseudoconvex domains in [C.sup.N]. Douglas and Kim [9] studied reducing subspace [mathematical expression not reproducible] on Bergman space of the annulus; the case of Hardy space was summarized in [10].

In [11], Douglas et al. generalized the bundle shift [12] to the case of Bergman spaces, constructed a vector bundle model for analytic Toeplitz operator [T.sub.[phi]] on the Bergman space [L.sup.2.sub.a](D), and tried to build vector bundle models for restrictions of [T.sub.[phi]] to its minimal reducing subspaces, but it is not completed. Douglas [13] studied unitary equivalence of the restrictions by computing their curvatures of corresponding geometric models. Hu et al. [14] showed that, for [T.sub.[phi]], there is a distinguished reducing subspace [M.sub.0] such that the restriction of [T.sub.[phi]] on [M.sub.0] 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 S be an operator on a separable infinite-dimensional Hilbert space H. S is called a one-side weighted shift if there exist an orthonormal basis {[e.sub.i]}, i = 0, 1, 2, ... for H and a bounded sequence {[[alpha].sub.i]} of complex numbers such that S[e.sub.i] = [[alpha].sub.i][e.sub.i+1]. Similarly, S is called a two-side weighted shift if there exist an orthonormal basis {[e.sub.i]} , i = ..., -2, -1, 0, 1, 2, ... for H and a bounded sequence {[[alpha].sub.i]} of complex numbers such that S[e.sub.i] = [[alpha].sub.i][e.sub.i+1] for all i [member of] Z.

Lemma 1 (see [15]). Suppose that S and T are two injective one-side weighted shifts with weights {[[alpha].sub.i]} and {[[beta].sub.i]} respectively; then S is unitarily equivalent to T if and only if [absolute value of [[alpha].sub.i]] = [absolute value of [[beta].sub.i]] for all i.

Lemma 2 (see [15]). Suppose that S and T are two injective one-side weighted shifts with weights {[[alpha].sub.i]} and {[[beta].sub.i]}, respectively; then S is similar to T if and only if there exist two constants M and N such that

[mathematical expression not reproducible] (2)

for all i.

3. Models for Restriction of Toeplitz Operators on Their Minimal Reducing Subspaces

3.1. The Bergman Spaces on the Slit Disc. The domain G = D \ [0, 1) is called the slit disk. Let dA denote the normalized area measure on G. [L.sup.2.sub.a](G) is the set of analytic functions in the Lebesgue space [L.sup.2](G, dA). For a nonnegative measurable function g(w) on G, we can define the weighted Bergman space with respect to g(w)dA(w) to be the set of all analytic functions in the Lebesgue space [L.sup.2](G, g(w)dA). Ross studied invariant subspaces of Bergman spaces on slit domains in [16]. 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 [17] for details.

3.2. Slit Disc Models for [mathematical expression not reproducible]. It is easy to check that 1/4[absolute value of w] is a measurable function on G and that (1/4[absolute value of w])dA(w) is a probability measure on G.

Lemma 3. The Bergman space [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)) contains constants and power functions {[w.sup.n] | n [member of] N}. Furthermore, {[square root of (2(2n + 1)][w.sup.n]; n [member of] N} and {2 [square root of (n + 1][w.sup.n]; n [member of] N} are orthonormal basis of [L.sup.2.sub.a](G, (1/4)dA(w)) and [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)), respectively.

Proof. For the Bergman space [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)) and n [member of] N,

[mathematical expression not reproducible]. (3)

For the Bergman space [L.sup.2.sub.a](G, (1/4)dA(w)) and n [member of] N,

[mathematical expression not reproducible]. (4)

Lemma 4. The multiplication operator [T.sub.w] is a bounded operator on [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)) and [L.sup.2.sub.a](G, (1/4)dA(w)).

Proof. For the Bergman space [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)), it is easy to show that

[mathematical expression not reproducible]. (5)

So [T.sub.w] is a weighted shift with weight [{[square root of ((2n + 1)/(2n + 3)]}.sup.[infinity].sub.n=0] and [parallel][T.sub.w][parallel] [less than or equal to] 1. Similarly, on the Bergman space [L.sup.2.sub.a](G,(1/4)dA(w)), [T.sub.w] is a weighted shift with weight [{[square root of ((2n + 1)/(n + 2)]}.sup.[infinity].sub.n=0] and so bounded. ?

We know that

[M.sub.0] = V {[z.sup.k]; k [equivalent to] 0 mod 2},

[M.sub.1] = V {[z.sup.k]; k [equivalent to] 1 mod 2} (6)

are the only two minimal reducing subspaces of [mathematical expression not reproducible] on [L.sup.2.sub.a](D).

Theorem 5. [mathematical expression not reproducible] are unitarily equivalent to the multiplication operators [T.sub.w] on the following two spaces, respectively:

[mathematical expression not reproducible]. (7)

Proof. Define the following maps:

[mathematical expression not reproducible]. (8)

Then [V.sub.1], [V.sub.0] are two isometries on [M.sub.1], [M.sub.0], respectively. As a fact, by changing of variable [z.sup.2] = w, [absolute value of dz/dw] = 1/2 [square root of (w)], we have

[mathematical expression not reproducible], (9)

so [V.sub.1] is an isometry. And

[mathematical expression not reproducible]; (10)

it shows that [V.sub.0] is an isometry. The next is to show that [mathematical expression not reproducible] on [L.sup.2.sub.a](G,(1/4)dA(w)). For every function f(w) [member of] [L.sup.2.sub.a](G,(1/4)dA(w)), we have

[mathematical expression not reproducible]. (11)

One can show that [mathematical expression not reproducible] on [L.sup.2.sub.a](G,(1/4w)dA(w)) similarly.

Suppose that [phi] is a Blaschke product [phi] = ((a - z)/(1 - [bar.a]z)) ((b - z)/(1 - [bar.b]z)) = [[phi].sub.a](z)[[phi].sub.b](z) with two different zeroes a and b; let [k.sub.[lambda]](z) = (1 - [[absolute value of [lambda]].sup.2])/[(1 - [bar.[lambda]z).sup.2] be the normalized reproducing kernel at [lambda].

Lemma 6 (see [3]). Let [phi] = ((a - z)/(1 - [bar.a]z))((b - z)/(1 - [bar.b]z)) with two different zeroes a,b [member of] D. Let m be the geodesic midpoint between a and b. Then the operator

[M.sub.[phi]] : [L.sup.2.sub.a] (D) [right arrow] [L.sup.2.sub.a] (D) (12)

has

[mathematical expression not reproducible], (13)

as its only two proper reducing subspaces, where [[phi].sub.m] = (m - z)/ (1 - [bar.m]z).

Theorem 7. The restrictions of [T.sub.[phi]] on [X.sub.e] and [X.sub.0] are unitarily equivalent to [T.sub.[psi]([square root of (w)]w) on [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)) and [L.sup.2.sub.a](G,(1/4)dA(w)), respectively, where [psi] = [phi] x [[phi].sub.m].

Proof. Now we define the following operators:

[mathematical expression not reproducible]. (14)

Then [V.sub.0], [V.sub.e] are two isometries on [X.sub.0], [X.sub.e], respectively. As a fact, by changing of variables two times, (u = [[phi].sub.m](z), [[phi]'.sub.m](z) = (1 - [[absolute value of m].sup.2])/[(1 - [bar.m]z).sup.2] = [k.sub.m](z) for the first time; [z.sup.2] = w, [absolute value of dz/dw] = [absolute value of 1/2 [square root w] for the second time), we have

[mathematical expression not reproducible], (15)

so [V.sub.0] is a isometry. And

[mathematical expression not reproducible]; (16)

it shows that [V.sub.e] is a isometry. Moreover, for any f(w) [member of] [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)),we have

[mathematical expression not reproducible], (17)

where [psi](z) = [phi]([[phi].sub.m](z)) and m is the midpoint between a and b. On the other hand,

[mathematical expression not reproducible]. (18)

Note that [psi]([square root of (w)]) is a function in [L.sup.2.sub.a](G,(1/4[absolute value of w])dA(w)). For f(w) [member of] [L.sup.2.sub.a](G,(1/4)dA(w)), one can prove similarly that

[T.sub.[phi]][V.sub.0]f(w) = [V.sub.0][T.sub.[psi]([square root of (w)])f(w). (19)

3.3. Weighted Shift Models

Proposition 8. The restrictions of [mathematical expression not reproducible] on its minimal reducing subspaces are one-side weighted shifts, and they are not unitarily equivalent to each other.

Proof. We know that

[mathematical expression not reproducible] (20)

are the two minimal reducing subspaces of [mathematical expression not reproducible] on the Bergman space [L.sup.2.sub.a](D). It is clear that

[mathematical expression not reproducible], (21)

so [mathematical expression not reproducible] are weighted shifts with the weight sequences {[square root of ((2k + 1)/(2k + 3))]} and {[square root of ((k + 1)/(k + 2))]}, respectively. It is clear that [mathematical

expression not reproducible] is not unitarily equivalent to [mathematical expression not reproducible] by Lemma 1.

Proposition 9. The restrictions of [mathematical expression not reproducible] on its minimal reducing subspaces are one-side weighted shifts, and they are similar to each other.

Proof. Let [M.sub.0] and [M.sub.0] be the minimal reducing subspaces of [T.sub.[phi]]; we know that [mathematical expression not reproducible] is the weighted shift with weight [{[square root of ((2k + 1)/(2k + 3))]}.sup.[infinity].sub.k=0] and [mathematical expression not reproducible] is the weighted shift with weight [{[square root of ((k + 1)/(k + 2))]}.sup.[infinity].sub.k=0]. By a direct computation, we have

[mathematical expression not reproducible]. (22)

So for any n, we have

[mathematical expression not reproducible], (23)

since [square root of ((n + 2)/(2n + 3))] is decreasing as n increases to infinity. It implies that [mathematical expression not reproducible] is similar to [mathematical expression not reproducible] by Lemma 2.

Remark 10. [mathematical expression not reproducible] is just the Bergman shift, so we know that [T.sub.[phi]] is similar to the direct sum of two copies of the Bergman shift.

3.4. For Reducing Subspace of [mathematical expression not reproducible]

Theorem 11. The slit disc models for restrictions of [mathematical expression not reproducible] are the [T.sub.w] on the following n spaces:

[mathematical expression not reproducible]. (24)

Proof. Let

[M.sup.n.sub.k] = V {[z.sup.m]; m = k mod n} , k = 0, 1, ..., n-1. (25)

We define the following n operators, for k = 0, 1, ..., n-1:

[mathematical expression not reproducible], (26)

and then we can show that [V.sup.n.sub.k] are isometries; as a fact, by changing of variable [mathematical expression not reproducible], so

[mathematical expression not reproducible]. (27)

Moreover, for any f(w) [member of] [L.sup.2.sub.a](G,(1/[n.sup.2][[absolute value of w].sup.2(n-k-1)/n])dA(w)), we have

[mathematical expression not reproducible]. (28)

Remark 12. [r.sup.[alpha]]dA(z) is a finite measure on G for [alpha] > -2, so (1/[n.sup.2][[absolute value of w].sup.2(n-k-1)/n])dA(w) is a finite measure on G.

Proposition 13. The restrictions of [mathematical expression not reproducible] on its minimal reducing subspaces are one-side weighted shifts, and they are not unitarily equivalent to each other.

Proof. For the Bergman space [L.sup.2.sub.a](G,(1/ [n.sup.2][[absolute value of w].sup.2(n-k-1)/n])dA(w)) and any j [greater than or equal to] 0, we have

[mathematical expression not reproducible], (29)

that is, [parallel]w[parallel] = [square root of (1/n(nj + k + 1)]. It implies that

[mathematical expression not reproducible], (30)

that is, [mathematical expression not reproducible] is the weighted shift with weight [{[square root of ((nj + k + 1)/(n(j +1) + k+ 1))]}, and so [T.sub.w] is bounded and [mathematical expression not reproducible] is not unitarily equivalent to [mathematical expression not reproducible] when k [not equal to] j.

Proposition 14. The restrictions of [mathematical expression not reproducible] on its minimal reducing subspaces are one-side weighted shifts, and they are similar to each other.

Proof. [mathematical expression not reproducible] are the one-side weighted shift with weights {[square root of ((nj + k + 1)/(n(j + 1) + k + 1))]} and {[square root of ((nj + l + 1)/(n(j + 1) + l + 1))]}, where 0 [less than or equal to] k < l [less than or equal to] n - 1. So, for any m,

[mathematical expression not reproducible]. (31)

Then,

[mathematical expression not reproducible], (32)

since (n(m + 1) + l + 1)/(n(m+ 1) + k + 1) is decreasing as m increases to infinity. It implies that [mathematical expression not reproducible] is similar to [mathematical expression not reproducible] for any k [not equal to] l.

Remark 15. For any 0 [less than or equal to] l [less than or equal to] n - 1, [mathematical expression not reproducible] is similar to [mathematical expression not reproducible] which is the Bergman shift, and so [mathematical expression not reproducible] is similar to the direct sum of n copies of the Bergman shift.

https://doi.org/10.1155/2017/2190418

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

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).

References

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[3] K. Zhu, "Reducing subspaces for a class of multiplication operators," Journal of the London Mathematical Society. Second Series, vol. 62, no. 2, pp. 553-568, 2000.

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[8] A. Tikaradze, "Multiplication operators on the Bergman spaces of pseudoconvex domains," New York Journal of Mathematics, vol. 21, pp. 1327-1345, 2015.

[9] R. G. Douglas and Y.-S. Kim, "Reducing subspaces on the annulus," Integral Equations and Operator Theory, vol. 70, no. 1, pp. 1-15, 2011.

[10] S. D. Fisher, Function Theory on Planar Domains, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, NY, USA, 1983.

[11] R. G. Douglas, D. K. Keshari, and A. Xu, "Generalized bundle shift with application to multiplication operator on the Bergman space," Journal of Operator Theory, vol. 75, no. 1, pp. 3-19, 2016.

[12] M. B. Abrahamse and R. G. Douglas, "A class of subnormal operators related to multiply-connected domains," Advances in Mathematics, vol. 19, no. 1, pp. 106-148, 1976.

[13] R. G. Douglas, "Operator theory and complex geometry," Extracta Mathematicae, vol. 24, no. 2, pp. 135-165, 2009.

[14] J. Hu, S. Sun, X. Xu, and D. Yu, "Reducing subspace of analytic Toeplitz operators on the Bergman space," Integral Equations and Operator Theory, vol. 49, no. 3, pp. 387-395, 2004.

[15] R. Kelley, Weighted Shifts on Hilbert Space [Thesis], University of Michigan, Ann Arbor, Mich, USA, 1966.

[16] W. T. Ross, "Invariant subspaces of Bergman spaces on slit domains," The Bulletin of the London Mathematical Society, vol. 26, no. 5, pp. 472-482, 1994.

[17] A. Aleman, N. S. Feldman, and W. T. Ross, The Hardy Space of a Slit Domain, Frontiers in Mathematics, Birkhauser Basel, 2009.

Anjian Xu (1) and Yang Zou (2)

(1) College of Science, Chongqing University of Technology, Chongqing 400065, China

(2) School of Mathematics and Information, Chongqing University of Education, Chongqing 400054, China

Correspondence should be addressed to Anjian Xu; xuaj@cqut.edu.cn

Received 26 May 2017; Accepted 6 August 2017; Published 7 September 2017

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Title Annotation:Research Article
Author:Xu, Anjian; Zou, Yang
Publication:Journal of Function Spaces
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Date:Jan 1, 2017
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