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Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra
J. Math. Anal. Appl. 455 (2017) 1249–1256
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra ✩ Yong Chen ∗ , Chuntao Qin, Qi Wu Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, PR China
a r t i c l e
i n f o
Article history: Received 5 April 2017 Available online 15 June 2017 Submitted by J.A. Ball Keywords: Reducibility Unitary equivalence Multiplier Finite Blaschke product Sobolev disk algebra
a b s t r a c t It is proved that Mφ with 2-Blaschke product φ is reducible on Sobolev disk algebra α−z 2 R(D) if and only if φ = β 1−αz 2 for some α ∈ D and unimodular constant β. Also, an analytic multiplier Mφ is unitary equivalent to Mzn on R(D) if and only if φ = λz n for some unimodular constant λ. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Let D be the unit disk and T the unit circle. Sobolev disk algebra R(D) consists of all analytic functions f on D satisfying f = 2
(|f |2 + |f |2 + |f |2 )dA < ∞,
D
where dA is the normalized area measure on D. Obviously f ∈ R(D) if and only if f ∈ D , the classical Dirichlet space on D. By Sobolev imbedding theorem ([1]), R(D) ⊂ A(D) ∩ C(D), here A(D) is the disk algebra on D. R(D) is a reproducing analytic Hilbert space with respect to the inner product f, g = f, g2 + f, g∗ + f , g ∗ for functions f, g ∈ R(D), where ✩
The work was supported by NSFC (No. 11471113) and ZJNSFC (No. LY14A010013).
* Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Chen), [email protected] (C. Qin), [email protected] (Q. Wu). http://dx.doi.org/10.1016/j.jmaa.2017.06.030 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
f, g2 =
f g dA,
f, g∗ =
D
and the reproducing kernel Kw (z) = ej (z) = βj z j with βj =
∞
f g dA,
D
j=0 ej (w)ej (z),
here {en }+∞ j=0 is an orthonormal basis of R(D),
12 j+1 , 3j 4 − j 2 + 2j + 1
j = 0, 1, 2 . . . .
Moreover, R(D) is a subalgebra generated by the rational functions with poles outside D. See [9] for more details of Sobolev disk algebra. If a function ψ defined on D satisfies ψR(D) ⊂ R(D), then we call ψ is a multiplier on the Sobolev disk algebra. Since 1 ∈ R(D), so ψ ∈ R(D). By the closed graph theorem we see that for a multiplier ψ, the multiplication operator Mψ : f → ψf , is bounded on R(D). Since each multiplier is a bounded analytic function on D, so Mψ is also bounded on the Hardy space H 2 or on the Bergman space L2a . λ−z Given λ ∈ D, let ϕλ (z) = 1−λz be the Mobius transformation of D. For finitely many points λ1 , · · · , λn ∈ D, φ = ϕλ1 ϕλ2 · · · ϕλn is called the n-Blaschke product with order n. It’s obvious that each n-Blaschke product is a multiplier of R(D). For a given operator T on a Hilbert space H, a closed subspace M of H is called invariant for T if T M ⊂ M. If in addition T ∗ M ⊂ M, then M is called a reducing subspace for T . Equivalently, M is a reducing subspace of T if and only if T M ⊂ M and T M⊥ ⊂ M⊥ , where M⊥ = H M. We say T is reducible if T has a nontrivial reducing subspace. A reducing subspace M for T is called minimal if the reducing subspaces for T contained in M are only M and {0}. The characterization for reducing subspaces problem for certain multiplication operators has been studied extensively. Let {Mφ } be the commutant algebra of Mφ . The problem of classifying the reducing subspace of Mφ is equivalent to finding the projection in {Mφ } . This classification problem in the case of the Hardy space H 2 was the motivation of the highly original works by Thomson and Cowen (see [5,15,16]). In the case of the Bergman space L2a , given a positive integer n, Stessin and Zhu ([12]) proved that Mzn has exactly 2n reducing subspaces. A notable result is that there always exists a nontrivial minimal reducing subspace M0 (φ) for Mφ with n-Blaschke product φ (n ≥ 2) on L2a ([8,13]). As an example, if φ = zϕλ1 · · · ϕλn−1 with λ1 , · · · , λn−1 nonzero and distinct, then M0 (φ) is generated by φ φj :
j ∈ N = {0, 1, 2, 3, . . .}
and its orthogonal complement in L2a is generated by ([13]) φm : 1 − λk z
m ∈ N, 1 ≤ k ≤ n − 1.
The recently striking paper [6] proved that for a finite Blaschke product φ, the number of the minimal reducing subspaces (are orthogonal with each other) of Mφ on the Bergman space is equal to the number of connected components of the Riemann surface of φ−1 ◦ φ. The case of reducibility on the Dirichlet space D is quite different. The results in [2–4,11,18] showed that for finite Blaschke product φ with the order not greater than 4 is not always reducible on D and the reducible case seems very few. For the case on Sobolev disk algebra R(D), recently Li ([10]) characterized that Mzn (n ≥ 2) has 2n reducing subspace on R(D), in contrast to the assertion that there are uncountably many Banach reducing subspace of Mφ with n-Blaschke product φ when n ≥ 2 (see [17]). Motivated by the above works, in this paper we first investigate the reducibility of Mφ with finite Blaschke product φ on R(D). We will show that the number of the reducing subspaces of Mφ with finite Blaschke
Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
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product φ on R(D) is not greater than the number of connected components of the Riemann surface of φ−1 ◦φ, see Proposition 1. We also show Li’s result with different method for the case of Mzn , see Proposition 2. For 2-Blaschke product symbol, we have the following completely characterization. 2
α−z Theorem 1. Let φ be a 2-Blaschke product, then Mφ is reducible on R(D) if and only if φ = β 1−αz 2 for some α ∈ D and unimodular constant β. In this case, there are only two nontrivial reducing subspaces of Mφ .
This result also shows that the reducible case is very few for analytic multiplier Mφ with 2-Blaschke product φ on R(D). We next consider the unitary equivalence of analytic multipliers on the Sobolev disk algebra R(D). On the Bergman space L2a , it was showed in [7,14] that Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n if and only φ = λϕnc for some c ∈ D and constant λ. But it is not the case on the Dirichlet space D (see [3,11]). On the Sobolev disk algebra R(D), we give the following assertion. Theorem 2. Let φ be an analytic multiplier, then Mφ is unitary equivalent to Mzn on R(D) if and only if φ = λz n for some unimodular constant λ. 2. The proof of main results We start with the following lemma which is straightforward and will be used frequently in this paper. In what follows we denote ·, ·H 2 as the inner product of the Hardy space H 2 . Lemma 1. Let p and q be two polynomials. Then p, q∗ = zp , qH 2 . By the above lemma, we have the following formula which is also very useful to our problem. Lemma 2. Let f, g ∈ D and φ be finite Blaschke product, then φm f, φm g∗ = mzφ f, φgH 2 + f, g∗ ,
m ∈ N.
(1)
Proof. It is straight to verify the above to be true when f and g are polynomials using Lemma 1. Since all polynomials are dense in D , the result holds by a routine argument. The proof is complete. 2 The following lemma is the key which establishes the relation of the reducibility problem of analytic multiplier between on the Sobolev disk algebra R(D) and the Bergman spaces L2a . Lemma 3. Let φ be n-Blaschke product (n ≥ 2) and M be a nontrivial reducing subspace of Mφ on R(D), then the following statements hold: ⊥ (a) for each integer k, T ξφ |φ |2 φk f g |dξ| 2π = 0 for f ∈ M and g ∈ M . ⊥
(b) M is a nontrivial reducing subspace of Mφ on L2a , where M is the closure of M in L2a , and M = M⊥ in L2a . Proof. Since M is a nontrivial reducing subspace, then for each f ∈ M and g ∈ M⊥ , we have φm+t f, φm+ g = 0 for every m, t, ∈ N. Hence
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Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
0 = φm+t f, φm+ g2 + φm+t f, φm+ g∗ + (φm+t f ) , (φm+ g) ∗ = φm+t f, φm+ g2 + φm+t f, φm+ g∗ + (m + t)(m + )φm+t−1 φ f, φm+−1 φ g∗ + (m + t)φm+t−1 φ f, φm+ g ∗
(2)
+ (m + )φm+t f , φm+−1 φ g∗ + φm+t f , φm+ g ∗ for every m, t, ∈ N. When m ≥ 1, making use of (1) we have φm+t f, φm+ g∗ = mzφ f, φ+1−t gH 2 + φt f, φ g∗ , φm+t−1 φ f, φm+−1 φ g∗ = (m − 1)z(φ )2 f, φ+1−t φ gH 2 + φt φ f, φ φ g∗ , φm+t−1 φ f, φm+ g ∗ = (m − 1)zφ f, φ+2−t g H 2 + φt φ f, φ+1 g ∗ , φm+t f , φm+−1 φ g∗ = (m − 1)zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ and φm+t f , φm+ g ∗ = mzφ f , φ+1−t g H 2 + φt f , φ g ∗ . Put the above five equalities into (2) and then both sides of the equality are divided by m3 , when letting m → ∞ we will get z(φ )2 f, φ+1−t φ gH 2 = 0, which gives (a). Further, with the same arguments, after dividing both sides of (2) by m2 and m respectively and letting m → ∞ twice, we then obtain 0 = φm+t f, φm+ g2 + φt f, φ g∗ + t φt φ f, φ φ g∗ − tzφ f, φ+2−t g H 2 + tφt φ f, φ+1 g ∗ − zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ + φt f , φ g ∗ . It’s easy to see that φm+t f, φm+ g2 → 0 as m → ∞, so the above yields φm+t f, φm+ g2 = 0,
t, ∈ N, m ≥ 1
(3)
and 0 = φt f, φ g∗ + t φt φ f, φ φ g∗ − tzφ f, φ+2−t g H 2 + tφt φ f, φ+1 g ∗ − zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ + φt f , φ g ∗ ,
t, ∈ N.
(4)
Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
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Let t = = 0 in (4), then we get 0 = f, g∗ + f , g ∗ . Because 0 = f, g = f, g2 + f, g∗ + f , g ∗ , so we have f, g2 = 0,
(5)
which also means f ⊥ g in the Bergman space L2a . Let t = 0, ≥ 1 in (4) and using (1) for φf , φ φ g∗ , then we can get 0 = f, φ g∗ − zφ f , φ φ gH 2 + φf , φ φ g∗ + f , φ g ∗ = f, φ g∗ + f , φ−1 φ g∗ + f , φ g ∗ = f, φ g − f, φ g2 . Since f, φ g = 0, so the above induces f, φ g2 = 0,
≥ 1.
Therefore, it follows from the above and the equalities (3) and (5) we have φt f, φ g2 = 0,
t, ∈ N
(6)
for any f ∈ M, g ∈ M⊥ . Now we use (6) to show the statement of (b). Let M and M⊥ be the closure of M and M⊥ in L2a respectively, then it is easy to see that M ⊥ M⊥ and M ⊕ M⊥ = L2a . In addition, we have that φM ⊂ M and φM⊥ ⊂ M⊥ , so M is a nontrivial reducing subspace for Mφ on the Bergman space L2a , as desired. The proof is complete. 2 The conclusion of Lemma 3 (a) implies that in the Hardy space H 2 , two invariant subspaces of Mφ which are generated respectively by z(φ )2 f and φ g are orthogonal with each other. It seems that the reducibility problem of Mφ on R(D) is closely connected with the invariant subspaces problem of Mφ on the Hardy space. Since for a n-Blaschke product φ with n ≥ 2, the number of minimal reducing subspaces of Mφ on the Bergman space is equal to the number of connected components of the Riemann surface of φ−1 ◦ φ (see [6]), then from the above lemma (b), we get the following result immediately. Proposition 1. Let φ be n-Blaschke product with n ≥ 2. Then the number of the minimal reducing subspaces of Mφ on R(D) is not greater than the number of connected components of the Riemann surface φ−1 ◦ φ. Recall that on the Bergman space L2a ([12]), Mzn (n ≥ 2) has n minimal reducing subspaces Mj (j = 1, 2, · · · , n) generated by {z mn+j−1 : m ∈ N}.
Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
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Let Mj be the closure of the above set in R(D), then clearly each Mj is a reducing subspace of Mzn on R(D). Also by Lemma 3 (b) we can easily obtain the below result which was proved in [10] using matrix manipulation combing with operator theory method. Proposition 2. On R(D), Mzn (n ≥ 2) has 2n reducing subspaces with the n minimal reducing subspaces Mj (j = 1, 2, · · · , n). Now we prove the first main result. Proof of Theorem 1. Suppose φ = ϕa ϕb for a, b ∈ D, then ϕab ◦ φ = dzϕc for some c ∈ D and unimodular constant d. Let ψ = zϕc . Then Mφ and Mψ are mutually analytic function calculus of each other and hence have same reducing subspaces. Now we only need to show that Mψ is reducible if and only if c = 0, which will claim the theorem with α = −abd¯ and β = −d. If c = 0, then we have done. If c = 0, in this case we know there are only two nontrivial reducing subspaces M0 (ψ) and M0 (ψ)⊥ of Mψ on L2a and they are generated respectively by {ψ ψ j : j ∈ N} and {
ψm : m ∈ N} 1 − c¯z
in L2a . Let M0 (ψ) and M0 (ψ)⊥ respectively be the closures of the above sets in R(D), then Lemma 3 (b) tells that M0 (ψ) and M0 (ψ)⊥ are the only possible reducing subspaces of Mψ on R(D). We would show they are not orthogonal with each other. 1 ⊥ In fact, choose ψ ∈ M0 (ψ) and 1−¯ cz ∈ M0 (ψ) . First note that
ψ ,
1 1 1 c¯ = ψ, + ψ , + ψ , 1 − c¯z 1 − c¯z 2 1 − c¯z ∗ (1 − c¯z)2 ∗ c¯ c¯ = ψ , + ψ , 2 (1 − c¯z) 2 (1 − c¯z)2 ∗
1 since ψ , 1−¯ cz 2 = 0. By the reproducing property of Bergman kernel
ψ ,
1 (1−¯ c)2
we have
c¯ = cψ (c). (1 − c¯z)2 2
On the other hand, using Lemma 1 we have
ψ ,
c¯ c¯ z 2 = zψ , = c z ψ , (1 − c¯z)2 ∗ (1 − c¯z)2 H 2 (1 − c¯z)2 H 2 = c(z 2 ψ ) (c) = 2c2 ψ (c) + c3 ψ (4) (c),
where the third equality comes from the reproducing property of kernel f (k) (c) = f, ∂c¯k (
1 1−¯ cz
1 ) 1 − c¯z H 2
for integer k ≥ 0. Therefore, it follows from the above we get
ψ ,
1 = cψ (c) + 2c2 ψ (c) + c3 ψ (4) (c) 1 − c¯z −2c = (7|c|4 + 4|c|2 + 1). (1 − |c|2 )4
in the Hardy space H 2 (7)
Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
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1 Since 7|c|4 + 4|c|2 + 1 = 0 and c = 0, so we get ψ , 1−¯ cz = 0, which means M0 (ψ) is not orthogonal to
M0 (ψ)⊥ , hence these subspaces are not the reducing subspaces of Mψ on R(D), which implies Mψ is not reducible on R(D) when c = 0. The proof is complete. 2 In order to prove the second main result, we recall some facts. For each λ ∈ D, define a unitary operator Uλ : L2a → L2a by Uλ f = −f ◦ ϕλ ϕλ . It is easy to verify that on the Bergman space, Uλ∗ Mzn Uλ = Mϕnλ . Since for all j = 1, 2 · · · , n, the subspaces generated by {z mn+j−1 : m ∈ N} are n minimal reducing subspaces of Mzn on L2a , then subspaces generated by {Uλ z mn+j−1 : m ∈ N} = {ϕmn+j−1 ϕλ : m ∈ N} λ for j = 1, 2 · · · , n are n minimal reducing subspaces of Mϕnλ and orthogonal with each other in L2a . Proof of Theorem 2. Suppose Mφ is unitary equivalent to Mzn on R(D). It is known that φ must be a n-Blaschke product because in this case Mφ is also similar to Mzn (see [17]). Noting that Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n, there is an orthogonal basis {f1 , · · · , fn } of R(D) φR(D) such that for j, k = 1, · · · , n and , m ∈ N, φ fj , φm fk = 0, j = k and φ fj , φm fj = 0, = m. It follows from the above equations and the arguments as in the proof of Lemma 3, we get that for j, k = 1, · · · , n and , m ∈ N, φ fj , φm fk 2 = 0, j = k and φ fj , φm fj 2 = 0, = m. The above two equations imply that in the Bergman space L2a , Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n, thus by Theorem 6.1 in [7], we have φ = λϕnc for some c ∈ D and some unimodular constant λ. Noting Mϕnc is unitary equivalent to Mzn on L2a under the unitary operator Uc , then by Proposition 2, Lemma 3 (b) and the fact recalled before the proof we have fj = cj ϕj−1 ϕc , c
j = 1, 2, · · · , n
for some nonzero constants cj . Now it’s left to show c = 0 from the fact f2 ⊥ f1 in R(D). Indeed, 0 = ϕc ϕc , ϕc = ϕc ϕc , ϕc 2 + ϕc ϕc , ϕc ∗ + (ϕc ϕc ) , ϕc ∗ = ϕc ϕc , ϕc ∗ + ϕc ϕc + ϕc ϕc , ϕc ∗ since ϕc ϕc , ϕc 2 = 0. Using Lemma 1 we have ϕc ϕc , ϕc ∗ = z(ϕc )2 , ϕc H 2 + zϕc ϕc , ϕc H 2 and ϕc ϕc + ϕc ϕc , ϕc ∗ = 3zϕc ϕc , ϕc H 2 + zϕc ϕ c , ϕc H 2 .
By the above two equalities and using the property (7), the equality (8) will then induce
(8)
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Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
0 = (|c|2 − 1)(z 2 (ϕc )2 ) (c) + c2 ϕc (c) + 3c(|c|2 − 1)(z 3 ϕc ϕc ) (c) + c(|c|2 − 1)(z 3 ϕc ϕ c ) (c)
=
−2|c|2 c − 2c 2|c|4 c − 2|c|2 c + (1 − |c|2 )2 (1 − |c|2 )3 +
=
−36|c|2 c(1 + 3|c|2 + |c|4 ) −24|c|6 c − 36|c|4 c + (1 − |c|2 )4 (1 − |c|2 )4
−2c(1 + 18|c|2 + 69|c|4 + 32|c|6 ) , (1 − |c|2 )4
which gives c = 0 and we finish the proof. 2 References [1] R. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975. [2] Y. Chen, H. Xu, Reducibility and unitarily equivalence for a class of analytic multipliers on the Dirichlet space, Complex Anal. Oper. Theory 7 (6) (2013) 1897–1908. [3] Y. Chen, Y. Lee, T. Yu, Reducibility and unitary equivalence for a class of analytic multiplication operator on the Dirichlet space, Studia Math. 220 (2) (2014) 141–156. [4] Y. Chen, X. Xu, Y. Zhao, Reducibility for a class of analytic multipliers on the Dirichlet space, Complex Anal. Oper. Theory (2017), http://dx.doi.org/10.1007/s11785-017-0661-9. [5] C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978) 1–31. [6] R. Douglas, M. Putinar, K. Wang, Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal. 263 (6) (2012) 1744–1765. [7] K. Guo, D. Zheng, Rudin orthogonality problem on the Bergman space, J. Funct. Anal. 261 (1) (2011) 51–68. [8] J. Hu, S. Sun, X. Xu, D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004) 387–395. [9] C. Jiang, Z. Wang, Structure of Hilbert Space Operators, World Scientific, Singapore, 2006. [10] Y. Li, Q. Liu, W. Lan, On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra, J. Math. Anal. Appl. 419 (2) (2014) 1161–1167. [11] S. Luo, Reducing subspaces of multiplication operators on the Dirichlet space, Integral Equations Operator Theory 85 (4) (2016) 539–554. [12] M. Stessin, K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002) 2631–2639. [13] M. Stessin, K. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad. J. Math. 55 (2) (2003) 379–400. [14] S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space and weighted shifts, J. Operator Theory 59 (2) (2008) 435–454. [15] J. Thomson, The commutant of a class of analytic Toeplitz operators II, Indiana Univ. Math. J. 25 (1976) 793–800. [16] J. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977) 522–529. [17] Z. Wang, R. Zhao, Y. Jin, Finite Blaschke product and the multiplication operators on Sobolev disk algebra, Sci. China Ser. A 52 (1) (2009) 142–146. [18] L. Zhao, Reducing subspaces for a class of multiplication operators on the Dirichlet space, Proc. Amer. Math. Soc. 137 (2009) 3091–3097.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra ✩ Yong Chen ∗ , Chuntao Qin, Qi Wu Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, PR China
a r t i c l e
i n f o
Article history: Received 5 April 2017 Available online 15 June 2017 Submitted by J.A. Ball Keywords: Reducibility Unitary equivalence Multiplier Finite Blaschke product Sobolev disk algebra
a b s t r a c t It is proved that Mφ with 2-Blaschke product φ is reducible on Sobolev disk algebra α−z 2 R(D) if and only if φ = β 1−αz 2 for some α ∈ D and unimodular constant β. Also, an analytic multiplier Mφ is unitary equivalent to Mzn on R(D) if and only if φ = λz n for some unimodular constant λ. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Let D be the unit disk and T the unit circle. Sobolev disk algebra R(D) consists of all analytic functions f on D satisfying f = 2
(|f |2 + |f |2 + |f |2 )dA < ∞,
D
where dA is the normalized area measure on D. Obviously f ∈ R(D) if and only if f ∈ D , the classical Dirichlet space on D. By Sobolev imbedding theorem ([1]), R(D) ⊂ A(D) ∩ C(D), here A(D) is the disk algebra on D. R(D) is a reproducing analytic Hilbert space with respect to the inner product f, g = f, g2 + f, g∗ + f , g ∗ for functions f, g ∈ R(D), where ✩
The work was supported by NSFC (No. 11471113) and ZJNSFC (No. LY14A010013).
* Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Chen), [email protected] (C. Qin), [email protected] (Q. Wu). http://dx.doi.org/10.1016/j.jmaa.2017.06.030 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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Y. Chen et al. / J. Math. Anal. Appl. 455 (2017) 1249–1256
f, g2 =
f g dA,
f, g∗ =
D
and the reproducing kernel Kw (z) = ej (z) = βj z j with βj =
∞
f g dA,
D
j=0 ej (w)ej (z),
here {en }+∞ j=0 is an orthonormal basis of R(D),
12 j+1 , 3j 4 − j 2 + 2j + 1
j = 0, 1, 2 . . . .
Moreover, R(D) is a subalgebra generated by the rational functions with poles outside D. See [9] for more details of Sobolev disk algebra. If a function ψ defined on D satisfies ψR(D) ⊂ R(D), then we call ψ is a multiplier on the Sobolev disk algebra. Since 1 ∈ R(D), so ψ ∈ R(D). By the closed graph theorem we see that for a multiplier ψ, the multiplication operator Mψ : f → ψf , is bounded on R(D). Since each multiplier is a bounded analytic function on D, so Mψ is also bounded on the Hardy space H 2 or on the Bergman space L2a . λ−z Given λ ∈ D, let ϕλ (z) = 1−λz be the Mobius transformation of D. For finitely many points λ1 , · · · , λn ∈ D, φ = ϕλ1 ϕλ2 · · · ϕλn is called the n-Blaschke product with order n. It’s obvious that each n-Blaschke product is a multiplier of R(D). For a given operator T on a Hilbert space H, a closed subspace M of H is called invariant for T if T M ⊂ M. If in addition T ∗ M ⊂ M, then M is called a reducing subspace for T . Equivalently, M is a reducing subspace of T if and only if T M ⊂ M and T M⊥ ⊂ M⊥ , where M⊥ = H M. We say T is reducible if T has a nontrivial reducing subspace. A reducing subspace M for T is called minimal if the reducing subspaces for T contained in M are only M and {0}. The characterization for reducing subspaces problem for certain multiplication operators has been studied extensively. Let {Mφ } be the commutant algebra of Mφ . The problem of classifying the reducing subspace of Mφ is equivalent to finding the projection in {Mφ } . This classification problem in the case of the Hardy space H 2 was the motivation of the highly original works by Thomson and Cowen (see [5,15,16]). In the case of the Bergman space L2a , given a positive integer n, Stessin and Zhu ([12]) proved that Mzn has exactly 2n reducing subspaces. A notable result is that there always exists a nontrivial minimal reducing subspace M0 (φ) for Mφ with n-Blaschke product φ (n ≥ 2) on L2a ([8,13]). As an example, if φ = zϕλ1 · · · ϕλn−1 with λ1 , · · · , λn−1 nonzero and distinct, then M0 (φ) is generated by φ φj :
j ∈ N = {0, 1, 2, 3, . . .}
and its orthogonal complement in L2a is generated by ([13]) φm : 1 − λk z
m ∈ N, 1 ≤ k ≤ n − 1.
The recently striking paper [6] proved that for a finite Blaschke product φ, the number of the minimal reducing subspaces (are orthogonal with each other) of Mφ on the Bergman space is equal to the number of connected components of the Riemann surface of φ−1 ◦ φ. The case of reducibility on the Dirichlet space D is quite different. The results in [2–4,11,18] showed that for finite Blaschke product φ with the order not greater than 4 is not always reducible on D and the reducible case seems very few. For the case on Sobolev disk algebra R(D), recently Li ([10]) characterized that Mzn (n ≥ 2) has 2n reducing subspace on R(D), in contrast to the assertion that there are uncountably many Banach reducing subspace of Mφ with n-Blaschke product φ when n ≥ 2 (see [17]). Motivated by the above works, in this paper we first investigate the reducibility of Mφ with finite Blaschke product φ on R(D). We will show that the number of the reducing subspaces of Mφ with finite Blaschke
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product φ on R(D) is not greater than the number of connected components of the Riemann surface of φ−1 ◦φ, see Proposition 1. We also show Li’s result with different method for the case of Mzn , see Proposition 2. For 2-Blaschke product symbol, we have the following completely characterization. 2
α−z Theorem 1. Let φ be a 2-Blaschke product, then Mφ is reducible on R(D) if and only if φ = β 1−αz 2 for some α ∈ D and unimodular constant β. In this case, there are only two nontrivial reducing subspaces of Mφ .
This result also shows that the reducible case is very few for analytic multiplier Mφ with 2-Blaschke product φ on R(D). We next consider the unitary equivalence of analytic multipliers on the Sobolev disk algebra R(D). On the Bergman space L2a , it was showed in [7,14] that Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n if and only φ = λϕnc for some c ∈ D and constant λ. But it is not the case on the Dirichlet space D (see [3,11]). On the Sobolev disk algebra R(D), we give the following assertion. Theorem 2. Let φ be an analytic multiplier, then Mφ is unitary equivalent to Mzn on R(D) if and only if φ = λz n for some unimodular constant λ. 2. The proof of main results We start with the following lemma which is straightforward and will be used frequently in this paper. In what follows we denote ·, ·H 2 as the inner product of the Hardy space H 2 . Lemma 1. Let p and q be two polynomials. Then p, q∗ = zp , qH 2 . By the above lemma, we have the following formula which is also very useful to our problem. Lemma 2. Let f, g ∈ D and φ be finite Blaschke product, then φm f, φm g∗ = mzφ f, φgH 2 + f, g∗ ,
m ∈ N.
(1)
Proof. It is straight to verify the above to be true when f and g are polynomials using Lemma 1. Since all polynomials are dense in D , the result holds by a routine argument. The proof is complete. 2 The following lemma is the key which establishes the relation of the reducibility problem of analytic multiplier between on the Sobolev disk algebra R(D) and the Bergman spaces L2a . Lemma 3. Let φ be n-Blaschke product (n ≥ 2) and M be a nontrivial reducing subspace of Mφ on R(D), then the following statements hold: ⊥ (a) for each integer k, T ξφ |φ |2 φk f g |dξ| 2π = 0 for f ∈ M and g ∈ M . ⊥
(b) M is a nontrivial reducing subspace of Mφ on L2a , where M is the closure of M in L2a , and M = M⊥ in L2a . Proof. Since M is a nontrivial reducing subspace, then for each f ∈ M and g ∈ M⊥ , we have φm+t f, φm+ g = 0 for every m, t, ∈ N. Hence
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0 = φm+t f, φm+ g2 + φm+t f, φm+ g∗ + (φm+t f ) , (φm+ g) ∗ = φm+t f, φm+ g2 + φm+t f, φm+ g∗ + (m + t)(m + )φm+t−1 φ f, φm+−1 φ g∗ + (m + t)φm+t−1 φ f, φm+ g ∗
(2)
+ (m + )φm+t f , φm+−1 φ g∗ + φm+t f , φm+ g ∗ for every m, t, ∈ N. When m ≥ 1, making use of (1) we have φm+t f, φm+ g∗ = mzφ f, φ+1−t gH 2 + φt f, φ g∗ , φm+t−1 φ f, φm+−1 φ g∗ = (m − 1)z(φ )2 f, φ+1−t φ gH 2 + φt φ f, φ φ g∗ , φm+t−1 φ f, φm+ g ∗ = (m − 1)zφ f, φ+2−t g H 2 + φt φ f, φ+1 g ∗ , φm+t f , φm+−1 φ g∗ = (m − 1)zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ and φm+t f , φm+ g ∗ = mzφ f , φ+1−t g H 2 + φt f , φ g ∗ . Put the above five equalities into (2) and then both sides of the equality are divided by m3 , when letting m → ∞ we will get z(φ )2 f, φ+1−t φ gH 2 = 0, which gives (a). Further, with the same arguments, after dividing both sides of (2) by m2 and m respectively and letting m → ∞ twice, we then obtain 0 = φm+t f, φm+ g2 + φt f, φ g∗ + t φt φ f, φ φ g∗ − tzφ f, φ+2−t g H 2 + tφt φ f, φ+1 g ∗ − zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ + φt f , φ g ∗ . It’s easy to see that φm+t f, φm+ g2 → 0 as m → ∞, so the above yields φm+t f, φm+ g2 = 0,
t, ∈ N, m ≥ 1
(3)
and 0 = φt f, φ g∗ + t φt φ f, φ φ g∗ − tzφ f, φ+2−t g H 2 + tφt φ f, φ+1 g ∗ − zφ f , φ−t φ gH 2 + φt+1 f , φ φ g∗ + φt f , φ g ∗ ,
t, ∈ N.
(4)
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Let t = = 0 in (4), then we get 0 = f, g∗ + f , g ∗ . Because 0 = f, g = f, g2 + f, g∗ + f , g ∗ , so we have f, g2 = 0,
(5)
which also means f ⊥ g in the Bergman space L2a . Let t = 0, ≥ 1 in (4) and using (1) for φf , φ φ g∗ , then we can get 0 = f, φ g∗ − zφ f , φ φ gH 2 + φf , φ φ g∗ + f , φ g ∗ = f, φ g∗ + f , φ−1 φ g∗ + f , φ g ∗ = f, φ g − f, φ g2 . Since f, φ g = 0, so the above induces f, φ g2 = 0,
≥ 1.
Therefore, it follows from the above and the equalities (3) and (5) we have φt f, φ g2 = 0,
t, ∈ N
(6)
for any f ∈ M, g ∈ M⊥ . Now we use (6) to show the statement of (b). Let M and M⊥ be the closure of M and M⊥ in L2a respectively, then it is easy to see that M ⊥ M⊥ and M ⊕ M⊥ = L2a . In addition, we have that φM ⊂ M and φM⊥ ⊂ M⊥ , so M is a nontrivial reducing subspace for Mφ on the Bergman space L2a , as desired. The proof is complete. 2 The conclusion of Lemma 3 (a) implies that in the Hardy space H 2 , two invariant subspaces of Mφ which are generated respectively by z(φ )2 f and φ g are orthogonal with each other. It seems that the reducibility problem of Mφ on R(D) is closely connected with the invariant subspaces problem of Mφ on the Hardy space. Since for a n-Blaschke product φ with n ≥ 2, the number of minimal reducing subspaces of Mφ on the Bergman space is equal to the number of connected components of the Riemann surface of φ−1 ◦ φ (see [6]), then from the above lemma (b), we get the following result immediately. Proposition 1. Let φ be n-Blaschke product with n ≥ 2. Then the number of the minimal reducing subspaces of Mφ on R(D) is not greater than the number of connected components of the Riemann surface φ−1 ◦ φ. Recall that on the Bergman space L2a ([12]), Mzn (n ≥ 2) has n minimal reducing subspaces Mj (j = 1, 2, · · · , n) generated by {z mn+j−1 : m ∈ N}.
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Let Mj be the closure of the above set in R(D), then clearly each Mj is a reducing subspace of Mzn on R(D). Also by Lemma 3 (b) we can easily obtain the below result which was proved in [10] using matrix manipulation combing with operator theory method. Proposition 2. On R(D), Mzn (n ≥ 2) has 2n reducing subspaces with the n minimal reducing subspaces Mj (j = 1, 2, · · · , n). Now we prove the first main result. Proof of Theorem 1. Suppose φ = ϕa ϕb for a, b ∈ D, then ϕab ◦ φ = dzϕc for some c ∈ D and unimodular constant d. Let ψ = zϕc . Then Mφ and Mψ are mutually analytic function calculus of each other and hence have same reducing subspaces. Now we only need to show that Mψ is reducible if and only if c = 0, which will claim the theorem with α = −abd¯ and β = −d. If c = 0, then we have done. If c = 0, in this case we know there are only two nontrivial reducing subspaces M0 (ψ) and M0 (ψ)⊥ of Mψ on L2a and they are generated respectively by {ψ ψ j : j ∈ N} and {
ψm : m ∈ N} 1 − c¯z
in L2a . Let M0 (ψ) and M0 (ψ)⊥ respectively be the closures of the above sets in R(D), then Lemma 3 (b) tells that M0 (ψ) and M0 (ψ)⊥ are the only possible reducing subspaces of Mψ on R(D). We would show they are not orthogonal with each other. 1 ⊥ In fact, choose ψ ∈ M0 (ψ) and 1−¯ cz ∈ M0 (ψ) . First note that
ψ ,
1 1 1 c¯ = ψ, + ψ , + ψ , 1 − c¯z 1 − c¯z 2 1 − c¯z ∗ (1 − c¯z)2 ∗ c¯ c¯ = ψ , + ψ , 2 (1 − c¯z) 2 (1 − c¯z)2 ∗
1 since ψ , 1−¯ cz 2 = 0. By the reproducing property of Bergman kernel
ψ ,
1 (1−¯ c)2
we have
c¯ = cψ (c). (1 − c¯z)2 2
On the other hand, using Lemma 1 we have
ψ ,
c¯ c¯ z 2 = zψ , = c z ψ , (1 − c¯z)2 ∗ (1 − c¯z)2 H 2 (1 − c¯z)2 H 2 = c(z 2 ψ ) (c) = 2c2 ψ (c) + c3 ψ (4) (c),
where the third equality comes from the reproducing property of kernel f (k) (c) = f, ∂c¯k (
1 1−¯ cz
1 ) 1 − c¯z H 2
for integer k ≥ 0. Therefore, it follows from the above we get
ψ ,
1 = cψ (c) + 2c2 ψ (c) + c3 ψ (4) (c) 1 − c¯z −2c = (7|c|4 + 4|c|2 + 1). (1 − |c|2 )4
in the Hardy space H 2 (7)
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1 Since 7|c|4 + 4|c|2 + 1 = 0 and c = 0, so we get ψ , 1−¯ cz = 0, which means M0 (ψ) is not orthogonal to
M0 (ψ)⊥ , hence these subspaces are not the reducing subspaces of Mψ on R(D), which implies Mψ is not reducible on R(D) when c = 0. The proof is complete. 2 In order to prove the second main result, we recall some facts. For each λ ∈ D, define a unitary operator Uλ : L2a → L2a by Uλ f = −f ◦ ϕλ ϕλ . It is easy to verify that on the Bergman space, Uλ∗ Mzn Uλ = Mϕnλ . Since for all j = 1, 2 · · · , n, the subspaces generated by {z mn+j−1 : m ∈ N} are n minimal reducing subspaces of Mzn on L2a , then subspaces generated by {Uλ z mn+j−1 : m ∈ N} = {ϕmn+j−1 ϕλ : m ∈ N} λ for j = 1, 2 · · · , n are n minimal reducing subspaces of Mϕnλ and orthogonal with each other in L2a . Proof of Theorem 2. Suppose Mφ is unitary equivalent to Mzn on R(D). It is known that φ must be a n-Blaschke product because in this case Mφ is also similar to Mzn (see [17]). Noting that Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n, there is an orthogonal basis {f1 , · · · , fn } of R(D) φR(D) such that for j, k = 1, · · · , n and , m ∈ N, φ fj , φm fk = 0, j = k and φ fj , φm fj = 0, = m. It follows from the above equations and the arguments as in the proof of Lemma 3, we get that for j, k = 1, · · · , n and , m ∈ N, φ fj , φm fk 2 = 0, j = k and φ fj , φm fj 2 = 0, = m. The above two equations imply that in the Bergman space L2a , Mφ is unitary equivalent to a weighted unilateral shift of finite multiplicity n, thus by Theorem 6.1 in [7], we have φ = λϕnc for some c ∈ D and some unimodular constant λ. Noting Mϕnc is unitary equivalent to Mzn on L2a under the unitary operator Uc , then by Proposition 2, Lemma 3 (b) and the fact recalled before the proof we have fj = cj ϕj−1 ϕc , c
j = 1, 2, · · · , n
for some nonzero constants cj . Now it’s left to show c = 0 from the fact f2 ⊥ f1 in R(D). Indeed, 0 = ϕc ϕc , ϕc = ϕc ϕc , ϕc 2 + ϕc ϕc , ϕc ∗ + (ϕc ϕc ) , ϕc ∗ = ϕc ϕc , ϕc ∗ + ϕc ϕc + ϕc ϕc , ϕc ∗ since ϕc ϕc , ϕc 2 = 0. Using Lemma 1 we have ϕc ϕc , ϕc ∗ = z(ϕc )2 , ϕc H 2 + zϕc ϕc , ϕc H 2 and ϕc ϕc + ϕc ϕc , ϕc ∗ = 3zϕc ϕc , ϕc H 2 + zϕc ϕ c , ϕc H 2 .
By the above two equalities and using the property (7), the equality (8) will then induce
(8)
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0 = (|c|2 − 1)(z 2 (ϕc )2 ) (c) + c2 ϕc (c) + 3c(|c|2 − 1)(z 3 ϕc ϕc ) (c) + c(|c|2 − 1)(z 3 ϕc ϕ c ) (c)
=
−2|c|2 c − 2c 2|c|4 c − 2|c|2 c + (1 − |c|2 )2 (1 − |c|2 )3 +
=
−36|c|2 c(1 + 3|c|2 + |c|4 ) −24|c|6 c − 36|c|4 c + (1 − |c|2 )4 (1 − |c|2 )4
−2c(1 + 18|c|2 + 69|c|4 + 32|c|6 ) , (1 − |c|2 )4
which gives c = 0 and we finish the proof. 2 References [1] R. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975. [2] Y. Chen, H. Xu, Reducibility and unitarily equivalence for a class of analytic multipliers on the Dirichlet space, Complex Anal. Oper. Theory 7 (6) (2013) 1897–1908. [3] Y. Chen, Y. Lee, T. Yu, Reducibility and unitary equivalence for a class of analytic multiplication operator on the Dirichlet space, Studia Math. 220 (2) (2014) 141–156. [4] Y. Chen, X. Xu, Y. Zhao, Reducibility for a class of analytic multipliers on the Dirichlet space, Complex Anal. Oper. Theory (2017), http://dx.doi.org/10.1007/s11785-017-0661-9. [5] C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978) 1–31. [6] R. Douglas, M. Putinar, K. Wang, Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal. 263 (6) (2012) 1744–1765. [7] K. Guo, D. Zheng, Rudin orthogonality problem on the Bergman space, J. Funct. Anal. 261 (1) (2011) 51–68. [8] J. Hu, S. Sun, X. Xu, D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004) 387–395. [9] C. Jiang, Z. Wang, Structure of Hilbert Space Operators, World Scientific, Singapore, 2006. [10] Y. Li, Q. Liu, W. Lan, On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra, J. Math. Anal. Appl. 419 (2) (2014) 1161–1167. [11] S. Luo, Reducing subspaces of multiplication operators on the Dirichlet space, Integral Equations Operator Theory 85 (4) (2016) 539–554. [12] M. Stessin, K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002) 2631–2639. [13] M. Stessin, K. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad. J. Math. 55 (2) (2003) 379–400. [14] S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space and weighted shifts, J. Operator Theory 59 (2) (2008) 435–454. [15] J. Thomson, The commutant of a class of analytic Toeplitz operators II, Indiana Univ. Math. J. 25 (1976) 793–800. [16] J. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977) 522–529. [17] Z. Wang, R. Zhao, Y. Jin, Finite Blaschke product and the multiplication operators on Sobolev disk algebra, Sci. China Ser. A 52 (1) (2009) 142–146. [18] L. Zhao, Reducing subspaces for a class of multiplication operators on the Dirichlet space, Proc. Amer. Math. Soc. 137 (2009) 3091–3097.