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On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra
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Doctopic: Functional Analysis
[m3L; v 1.134; Prn:28/05/2014; 11:30] P.1 (1-7)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra ✩ Yucheng Li ∗ , Qiuju Liu, Wenhua Lan Department of Mathematics, Hebei Normal University, Hebei Province Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, PR China
a r t i c l e
i n f o
Article history: Received 3 March 2014 Available online xxxx Submitted by J.A. Ball Keywords: Sobolev disk algebra Multiplication operator Similarity Reducing subspaces
a b s t r a c t Let D be the unit disk and SA(D) denote the Sobolev disk algebra which consists of all analytic functions in the Sobolev space W 2,2 (D). In this note, we prove that Mzn is similar to n 1 Mz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D). © 2014 Elsevier Inc. All rights reserved.
1. Introduction Let D be the unit disk and W 2,2 (D) denote the Sobolev space (see [2]). W 2,2 (D) = {f ∈ L2 (D, dA)| the distributional partial derivatives of the first and second orders of f belong to L2 (D, dA)}, where dA is the planar Lebesgue measure. We denote by SA(D) the Sobolev disk algebra which consists of all analytic functions in W 2,2 (D). W 2,2 (D) is a Hilbert space. If f, g ∈ W 2,2 (D), the inner product of f and g is defined α ∂ |α| f 2 α α by f, g = α2 , for 1 |α|≤2 D ∂ f ∂ gdA, where α = (α1 , α2 ) ∈ N , |α| = α1 + α2 , and ∂ f = ∂xα 1 ∂x2 1 2 α 2 (x1 , x2 ) ∈ R . The corresponding norm of f is defined by f 2,D = ( |α|≤2 D |∂ f | dA) 2 . For a bounded linear operator T on a Hilbert space H, let A (T ) denote the commutant of T , i.e., A (T ) = {Q ∈ L(H)|QT = T Q} (see [10]). In studying an operator on a Hilbert space, it is of interest to characterize the commutant of a given operator, as such a characterization should help in understanding the structure of the operator. The commutant of an analytic Toeplitz operator and the corresponding reducing subspaces on function spaces such as the Dirichlet space, the Hardy space and the Bergman space has been studied extensively in the literature. We mention here that the papers [3,5,6,8,9,11–15,17–20] and the ✩
This research was supported by NNSF of China (11371119, 140101).
* Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (Q. Liu), [email protected] (W. Lan). http://dx.doi.org/10.1016/j.jmaa.2014.05.052 0022-247X/© 2014 Elsevier Inc. All rights reserved.
JID:YJMAA
AID:18554 /FLA
Doctopic: Functional Analysis
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Y. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••
2
books [7,16] include a lot of analysis of the operator theory associated with the Hardy and the Bergman spaces; while the work on the corresponding operator theory associated with the Dirichlet or the Sobolev spaces is not as extensive, we note that the paper [4] delves into the operator theory associated with the Dirichlet/Sobolev space context. J.A. Ball (see [3]) and E. Nordgren (see [14]) studied the problem of determining the reducing subspaces for an analytic Toeplitz operator on the Hardy space. In [18], M. Stessin and K.H. Zhu described the properties of the commutant of analytic Toeplitz operators with inner function symbols on the Hardy space and the Bergman space. In 2007, Jiang and Li (see [9]) proved that each analytic Toeplitz operator MB(z) is similar to n copies of the Bergman shift if and only if B(z) is an n-Blaschke product. Jiang and Zheng in [11] showed that the main result in [9] holds on the weighted Bergman space. On n the weighted Bergman space, Li (see [12]) proved that multiplication operator Mzn is similar to 1 Mz . In 2012, Ahmadi and Hedayatian (see [1]) generalized this result to bilateral shift operators. In 2011, Douglas and Kim in [8] studied the reducing subspaces for an analytic multiplication operator Mzn on the Bergman space A2α (Ar ) of the annulus Ar . Recently, Li, Lan and Liu (see [13]) proved that multiplication operator n Mzn is quasi-similar to 1 Mz on the Fock space. n Based on the above works, in this note, we prove that multiplication operator Mzn is similar to 1 Mz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D). 2. The similarity of multiplication operator Lemma 2.1. Let ek (z) =
k+1 k (3k4 −k2 +2k+1)π z
(k = 0, 1, · · ·). Then {ek }∞ k=0 forms an orthonormal basis of
SA(D). Proof. Note that
k
z ,z
m
=
∂ α z k ∂ α z m dA
|α|≤2 D
= I1 + I2 + I3 + I4 + I5 + I6 .
(2.1)
Let z = x + yi = reiθ , (x, y) ∈ R2 , 0 < r < 1, 0 ≤ θ ≤ 2π. If m = k, we have I1 =
π k+1
I2 = kπ I3 = k2 (k − 1)π I4 = kπ I5 = k2 (k − 1)π I6 = k2 (k − 1)π Thus, z k , z k =
3k4 −k2 +2k+1 π. k+1
for α = (0, 0) ,
for α = (0, 1) ,
for α = (0, 2) ,
for α = (1, 0) ,
for α = (1, 1) ,
for α = (2, 0) .
Therefore, ek , em =
1, 0,
k = m, k = m.
2
(2.2)
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Lemma 2.2. Let Sj = span{enk+j } (j = 0, 1, · · · , n − 1). Then (1) {enk+j }∞ k=0 forms an orthonormal basis of Sj . (2) SA(D) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 . (3) Sj is a reducing subspace of Mzn . Proof. (1) Computation shows that 1, 0,
enk+j , enm+j =
k = m, k = m.
(2.3)
(2) It is easy to prove that Sj ⊥St , 0 ≤ j = t ≤ n − 1. Next, for f ∈ SA(D), we have that f has the form f=
∞
a0k enk + · · · +
k=0
∞
an−1,k enk+n−1 .
k=0
Suppose that f = 0. Then from
∞ n−1
ajk enk+j , el
=0
(l = 0, 1, · · ·),
(2.4)
k=0 j=0 n
we get that ajk = 0 (j = 0, · · · , n − 1, k = 0, 1, · · ·). Thus, 0 = 0 ⊕ 0 ⊕ · · · ⊕ 0. This means that SA(D) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 . (3) It is obvious that both Sj and Sj⊥ are the invariant subspaces of Mzn . 2 Theorem 2.3. The multiplication operator Mzn is similar to Proof. In the following, for simplicity, we set βk =
n 1
Mz on SA(D).
k+1 (3k4 −k2 +2k+1)π .
Mz ek = zβk z k =
βk ek+1 . βk+1
Note that
(2.5)
Set Mj = Mzn |Sj (j = 0, 1, · · · , n − 1). Then Mj enk+j = z n βnk+j z nk+j =
βnk+j enk+n+j . βnk+n+j
Define Xj : SA(D) −→ Sj such that Xj ek = ckj enk+j . Then we have Xj Mz ek = Mj Xj ek . In fact, Xj and
βk ek+1 = Mj ckj enk+j , βk+1
(2.6)
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Doctopic: Functional Analysis
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βk βnk+j ck+1,j enk+n+j = ckj enk+n+j . βk+1 βnk+n+j From βk+1 βnk+j ck+1,j = , ck,j βk βnk+n+j we obtain ckj =
βk βj βnk+j
(k ≥ 0).
(2.7)
Next, we will compute the limit of sequence {ckj } as k −→ +∞. lim ckj = lim
k→∞
k→∞
=
(k + 1)(j + 1)[3(nk + j)4 − (nk + j)2 + 2(nk + j) + 1]π (3k 4 − k2 + 2k + 1)π 2 (3j 4 − j 2 + 2j + 1)(nk + j + 1)
3 j+1 n2 . (3j 4 − j 2 + 2j + 1)π
(2.8)
From the above discussion, we know that the operator Xj is bounded and invertible. So Mj ∼ Mz
(j = 0, 1, · · · , n − 1).
Moreover, Mzn |SA(D) = M0 ⊕ M1 ⊕ · · · ⊕ Mn−1 ∼
n
Mz .
2
(2.9)
1
3. The reducing subspaces of multiplication operator Mzn Lemma 3.1. Let SA(D) be the Sobolev disk algebra. If the operator P has the matrix representation ⎛p
11
⎜p ⎜ 21 ⎜ ⎜p ⎜ 31 ⎜ P = ⎜ .. ⎜ . ⎜ ⎜ ⎜ pk1 ⎝ .. .
p12
p13
...
p1k
p22
p23
...
p2k
p32 .. .
p33 .. .
... .. .
p3k .. .
pk2 .. .
pk3 .. .
... .. .
pkk .. .
...⎞ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟ ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
with respect to the orthonormal basis of SA(D), where pjk ∈ C (j, k ≥ 1), then P ∈ A (Mz ) if and only if the entries of P satisfy the following equalities: ⎧ pij = 0, ⎪ ⎪ ⎨ pii = p11 , ⎪ ⎪ ⎩ βj+k−1 p βj−1
j+k,j
i < j, i = 2, 3, · · · , =
βj+k−2 βj−2 pj+k−1,j−1 ,
(j ≥ 2, k ≥ 1).
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Proof. Denote the orthonormal basis of SA(D) by {ek (z) = βk z k }∞ k=0 . From (2.5), we know that the operator Mz admits the following matrix representation with respect to the above basis ⎛
0
⎜ ββ0 ⎜ 1 ⎜ 0 ⎜ ⎜ Mz = ⎜ .. ⎜ . ⎜ ⎜ 0 ⎝ .. .
0 0
... ...
0 0
0 0
β1 β2
... .. . ... .. .
0 .. .
0 .. . 0
.. . 0 .. .
βk−1 βk
.. .
..
⎞ ... ...⎟ ⎟ ...⎟ ⎟ ⎟ ⎟. ···⎟ ⎟ ...⎟ ⎠ .. .
.
(3.1)
From Mz P = P Mz , we have ⎛
0
⎜ β0 p ⎜ β1 11 ⎜ ⎜ β1 p ⎜ β2 21 ⎜ ⎜ .. ⎜ . ⎜ ⎜ βk−2 ⎜ ⎝ βk−1 pk−1,1 .. . ⎛ β0 β1 p12 ⎜ β0 ⎜ β p22 ⎜ 1 ⎜β ⎜ 0 p32 ⎜ β1 =⎜ ⎜ .. ⎜ . ⎜ ⎜ β0 ⎜ β pk2 ⎝ 1 .. .
0
...
0
β0 β1 p12 β1 β2 p22
... ...
β0 β1 p1k β1 β2 p2k
.. .
.. .
.. .
βk−2 βk−1 pk−1,2
... .. .
βk−2 βk−1 pk−1,k
.. .
.. .
β1 β2 p13
...
βk−2 βk−1 p1k
β1 β2 p23
...
βk−2 βk−1 p2k
β1 β2 p33
...
βk−2 βk−1 p3k
.. .
.. .
.. .
β1 β2 pk3
... .. .
βk−2 βk−1 pkk
.. .
.. .
...
...⎞ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟ ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
(3.2)
So we obtain ⎧ ⎪ ⎨ pij = 0, pii = p11 , ⎪ ⎩ βj+k−1 βj−1 pj+k,j =
i < j, i = 2, 3, · · · , βj+k−2 βj−2 pj+k−1,j−1 ,
(3.3)
(j ≥ 2, k ≥ 1).
Conversely, if the entries of P satisfy (3.3), that is, P admits the following matrix representation with respect to the above basis ⎛
p11
⎜ β0 ⎜ β1 p21 ⎜ ⎜ β0 ⎜ β2 p31 ⎜ P =⎜ .. ⎜ . ⎜ ⎜ β ⎜ 0 p ⎜ βk−1 k1 ⎝ .. .
0
0
...
0
p11
0
...
0
β1 β2 p21
p11
...
0
.. .
.. .
..
.. .
β1 βk−1 pk−1,1
β2 βk−1 pk−2,1
.. .
.. .
Simple computation shows that Mz P = P Mz . So P ∈ A (Mz ).
.
. . . p11 .. . 2
.. .
...
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ...⎟ ⎟ ⎠ .. .
(3.4)
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Lemma 3.2. Let SA(D) be the Sobolev disk algebra. If G is a projection, then G ∈ A ( ⎛
Mz ) if and only if
⎞
G0
⎜ ⎜ G=⎜ ⎜ ⎝
1
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
. Gn−1
where Gi is I or 0. Proof. From Lemma 3.1, we know that P ∈ A (Mz ) if and only if P has the form (3.4). Moreover, if P is a projection, we claim that p11 = 1 or 0, pj1 = 0 (j = 2, 3, · · ·). In fact, from P = P ∗ = P 2 , we have ⎛ ⎜ ⎜ ⎜ ⎜ P =⎜ ⎜ ⎜ ⎜ ⎝
⎞
p11
⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
p11 ..
. p11 ..
(3.5)
. n
where p11 = 1 or 0. By the above analysis, we get that if G is a projection, and G ∈ A ( if G has the following form ⎛ ⎜ ⎜ G=⎜ ⎜ ⎝
1
Mz ) if and only
⎞
G0
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
.
(3.6)
Gn−1 where Gi (i = 0, 1, · · · , n − 1) is I or 0. 2 From [7], we know that determining the reducing subspaces of Mzn is equivalent to finding the projection in the commutant of Mzn . Thus we have the following conclusion. Theorem 3.3. Let SA(D) be the Sobolev disk algebra. For n ≥ 2, then the multiplication operator Mzn has 2n reducing subspaces with minimal reducing subspaces S0 , S1 , · · · , Sn−1 . Proof. By Theorem 2.3, the multiplication operator Mzn is similar to bounded and invertible operator ⎛ ⎜ ⎜ X=⎜ ⎜ ⎝
n 1
Mz on SA(D), so there exists a
⎞
X0
⎟ ⎟ ⎟ ⎟ ⎠
X1 ..
. Xn−1
such that n 1
Mz = X −1 Mzn X.
(3.7)
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Doctopic: Functional Analysis
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Y. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••
n
Note that G is a projection and G ∈ A ( G
n 1
Mz =
n
1
7
Mz ), G has the form described as (3.6), applying (3.7) we have
Mz G =⇒ GX −1 Mzn X = X −1 Mzn XG
1
=⇒ XGX −1 Mzn = Mzn XGX −1 =⇒ GXX −1 Mzn = Mzn GXX −1 =⇒ GMzn = Mzn G,
(3.8)
where ⎛ ⎜ ⎜ G=⎜ ⎜ ⎝
⎞
G0
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
. Gn−1
and Gi (i = 0, 1, · · · , n − 1) is ISi or 0. By Lemma 2.2, we have that the reducing subspaces of Mzn are c0 S0 ⊕ c1 S1 ⊕ · · · ⊕ cn−1 Sn−1 ,
where ci = 0 or 1 (i = 0, 1, · · · , n − 1),
and the minimal reducing subspaces are S0 , S1 , · · · , Sn−1 .
2
Acknowledgments The authors would like to thank the referees for their useful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
M.F. Ahmadi, K. Hedayatian, On similarity of powers of shift operators, Turkish J. Math. 36 (2012) 596–600. R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975. J.A. Ball, Hardy space expectation operators and reducing subspaces, Proc. Amer. Math. Soc. 47 (1975) 351–357. J.A. Ball, T.R. Fanney, Pure sub-Jordan operators and simultaneous approximation by a polynomial and its derivative, J. Operator Theory 32 (1995) 43–78. C.C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978) 1–31. J.A. Deddens, T.K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973) 261–273. R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972. R.G. Douglas, Y.S. Kim, Reducing subspaces on the annulus, Integral Equations Operator Theory 70 (2011) 1–15. C.L. Jiang, Y.C. Li, The commutant and similarity invariant of analytic Toeplitz operators on Bergman space, Sci. China Ser. A 50 (5) (2007) 651–664. C.L. Jiang, Z.Y. Wang, Strongly Irreducible Operators on Hilbert Space, Pitman Research Notes in Mathematics Series, vol. 389, Longman, London, 1998. C.L. Jiang, D.C. Zheng, Similarity of analytic Toeplitz operators on the Bergman spaces, J. Funct. Anal. 258 (2010) 2961–2982. Y.C. Li, On similarity of multiplication operator on weighted Bergman space, Integral Equations Operator Theory 63 (2009) 95–102. Y.C. Li, W.H. Lan, J.L. Liu, On quasi-similarity and reducing subspaces of multiplication operator on the Fock space, J. Math. Anal. Appl. 409 (2014) 899–905. E. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967) 175–181. R. Rochberg, Z. Wu, Toeplitz operators on Dirichlet spaces, Integral Equations Operator Theory 15 (2) (1992) 325–342. A.L. Shields, Weighted Shift Operators and Analytic Function Theory, Mathematical Survey Series, vol. 13, Springer, Berlin, 1974. M. Stessin, K.H. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002) 2631–2639. M. Stessin, K.H. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad. J. Math. 55 (2) (2003) 379–400. J. Thomson, The commutant of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976) 165–169. K.H. Zhu, The Reducing subspaces of a class of multiplication operator, J. Lond. Math. Soc. (2) 62 (2000) 553–568.
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[m3L; v 1.134; Prn:28/05/2014; 11:30] P.1 (1-7)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra ✩ Yucheng Li ∗ , Qiuju Liu, Wenhua Lan Department of Mathematics, Hebei Normal University, Hebei Province Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, PR China
a r t i c l e
i n f o
Article history: Received 3 March 2014 Available online xxxx Submitted by J.A. Ball Keywords: Sobolev disk algebra Multiplication operator Similarity Reducing subspaces
a b s t r a c t Let D be the unit disk and SA(D) denote the Sobolev disk algebra which consists of all analytic functions in the Sobolev space W 2,2 (D). In this note, we prove that Mzn is similar to n 1 Mz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D). © 2014 Elsevier Inc. All rights reserved.
1. Introduction Let D be the unit disk and W 2,2 (D) denote the Sobolev space (see [2]). W 2,2 (D) = {f ∈ L2 (D, dA)| the distributional partial derivatives of the first and second orders of f belong to L2 (D, dA)}, where dA is the planar Lebesgue measure. We denote by SA(D) the Sobolev disk algebra which consists of all analytic functions in W 2,2 (D). W 2,2 (D) is a Hilbert space. If f, g ∈ W 2,2 (D), the inner product of f and g is defined α ∂ |α| f 2 α α by f, g = α2 , for 1 |α|≤2 D ∂ f ∂ gdA, where α = (α1 , α2 ) ∈ N , |α| = α1 + α2 , and ∂ f = ∂xα 1 ∂x2 1 2 α 2 (x1 , x2 ) ∈ R . The corresponding norm of f is defined by f 2,D = ( |α|≤2 D |∂ f | dA) 2 . For a bounded linear operator T on a Hilbert space H, let A (T ) denote the commutant of T , i.e., A (T ) = {Q ∈ L(H)|QT = T Q} (see [10]). In studying an operator on a Hilbert space, it is of interest to characterize the commutant of a given operator, as such a characterization should help in understanding the structure of the operator. The commutant of an analytic Toeplitz operator and the corresponding reducing subspaces on function spaces such as the Dirichlet space, the Hardy space and the Bergman space has been studied extensively in the literature. We mention here that the papers [3,5,6,8,9,11–15,17–20] and the ✩
This research was supported by NNSF of China (11371119, 140101).
* Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (Q. Liu), [email protected] (W. Lan). http://dx.doi.org/10.1016/j.jmaa.2014.05.052 0022-247X/© 2014 Elsevier Inc. All rights reserved.
JID:YJMAA
AID:18554 /FLA
Doctopic: Functional Analysis
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Y. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••
2
books [7,16] include a lot of analysis of the operator theory associated with the Hardy and the Bergman spaces; while the work on the corresponding operator theory associated with the Dirichlet or the Sobolev spaces is not as extensive, we note that the paper [4] delves into the operator theory associated with the Dirichlet/Sobolev space context. J.A. Ball (see [3]) and E. Nordgren (see [14]) studied the problem of determining the reducing subspaces for an analytic Toeplitz operator on the Hardy space. In [18], M. Stessin and K.H. Zhu described the properties of the commutant of analytic Toeplitz operators with inner function symbols on the Hardy space and the Bergman space. In 2007, Jiang and Li (see [9]) proved that each analytic Toeplitz operator MB(z) is similar to n copies of the Bergman shift if and only if B(z) is an n-Blaschke product. Jiang and Zheng in [11] showed that the main result in [9] holds on the weighted Bergman space. On n the weighted Bergman space, Li (see [12]) proved that multiplication operator Mzn is similar to 1 Mz . In 2012, Ahmadi and Hedayatian (see [1]) generalized this result to bilateral shift operators. In 2011, Douglas and Kim in [8] studied the reducing subspaces for an analytic multiplication operator Mzn on the Bergman space A2α (Ar ) of the annulus Ar . Recently, Li, Lan and Liu (see [13]) proved that multiplication operator n Mzn is quasi-similar to 1 Mz on the Fock space. n Based on the above works, in this note, we prove that multiplication operator Mzn is similar to 1 Mz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D). 2. The similarity of multiplication operator Lemma 2.1. Let ek (z) =
k+1 k (3k4 −k2 +2k+1)π z
(k = 0, 1, · · ·). Then {ek }∞ k=0 forms an orthonormal basis of
SA(D). Proof. Note that
k
z ,z
m
=
∂ α z k ∂ α z m dA
|α|≤2 D
= I1 + I2 + I3 + I4 + I5 + I6 .
(2.1)
Let z = x + yi = reiθ , (x, y) ∈ R2 , 0 < r < 1, 0 ≤ θ ≤ 2π. If m = k, we have I1 =
π k+1
I2 = kπ I3 = k2 (k − 1)π I4 = kπ I5 = k2 (k − 1)π I6 = k2 (k − 1)π Thus, z k , z k =
3k4 −k2 +2k+1 π. k+1
for α = (0, 0) ,
for α = (0, 1) ,
for α = (0, 2) ,
for α = (1, 0) ,
for α = (1, 1) ,
for α = (2, 0) .
Therefore, ek , em =
1, 0,
k = m, k = m.
2
(2.2)
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Lemma 2.2. Let Sj = span{enk+j } (j = 0, 1, · · · , n − 1). Then (1) {enk+j }∞ k=0 forms an orthonormal basis of Sj . (2) SA(D) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 . (3) Sj is a reducing subspace of Mzn . Proof. (1) Computation shows that 1, 0,
enk+j , enm+j =
k = m, k = m.
(2.3)
(2) It is easy to prove that Sj ⊥St , 0 ≤ j = t ≤ n − 1. Next, for f ∈ SA(D), we have that f has the form f=
∞
a0k enk + · · · +
k=0
∞
an−1,k enk+n−1 .
k=0
Suppose that f = 0. Then from
∞ n−1
ajk enk+j , el
=0
(l = 0, 1, · · ·),
(2.4)
k=0 j=0 n
we get that ajk = 0 (j = 0, · · · , n − 1, k = 0, 1, · · ·). Thus, 0 = 0 ⊕ 0 ⊕ · · · ⊕ 0. This means that SA(D) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 . (3) It is obvious that both Sj and Sj⊥ are the invariant subspaces of Mzn . 2 Theorem 2.3. The multiplication operator Mzn is similar to Proof. In the following, for simplicity, we set βk =
n 1
Mz on SA(D).
k+1 (3k4 −k2 +2k+1)π .
Mz ek = zβk z k =
βk ek+1 . βk+1
Note that
(2.5)
Set Mj = Mzn |Sj (j = 0, 1, · · · , n − 1). Then Mj enk+j = z n βnk+j z nk+j =
βnk+j enk+n+j . βnk+n+j
Define Xj : SA(D) −→ Sj such that Xj ek = ckj enk+j . Then we have Xj Mz ek = Mj Xj ek . In fact, Xj and
βk ek+1 = Mj ckj enk+j , βk+1
(2.6)
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βk βnk+j ck+1,j enk+n+j = ckj enk+n+j . βk+1 βnk+n+j From βk+1 βnk+j ck+1,j = , ck,j βk βnk+n+j we obtain ckj =
βk βj βnk+j
(k ≥ 0).
(2.7)
Next, we will compute the limit of sequence {ckj } as k −→ +∞. lim ckj = lim
k→∞
k→∞
=
(k + 1)(j + 1)[3(nk + j)4 − (nk + j)2 + 2(nk + j) + 1]π (3k 4 − k2 + 2k + 1)π 2 (3j 4 − j 2 + 2j + 1)(nk + j + 1)
3 j+1 n2 . (3j 4 − j 2 + 2j + 1)π
(2.8)
From the above discussion, we know that the operator Xj is bounded and invertible. So Mj ∼ Mz
(j = 0, 1, · · · , n − 1).
Moreover, Mzn |SA(D) = M0 ⊕ M1 ⊕ · · · ⊕ Mn−1 ∼
n
Mz .
2
(2.9)
1
3. The reducing subspaces of multiplication operator Mzn Lemma 3.1. Let SA(D) be the Sobolev disk algebra. If the operator P has the matrix representation ⎛p
11
⎜p ⎜ 21 ⎜ ⎜p ⎜ 31 ⎜ P = ⎜ .. ⎜ . ⎜ ⎜ ⎜ pk1 ⎝ .. .
p12
p13
...
p1k
p22
p23
...
p2k
p32 .. .
p33 .. .
... .. .
p3k .. .
pk2 .. .
pk3 .. .
... .. .
pkk .. .
...⎞ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟ ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
with respect to the orthonormal basis of SA(D), where pjk ∈ C (j, k ≥ 1), then P ∈ A (Mz ) if and only if the entries of P satisfy the following equalities: ⎧ pij = 0, ⎪ ⎪ ⎨ pii = p11 , ⎪ ⎪ ⎩ βj+k−1 p βj−1
j+k,j
i < j, i = 2, 3, · · · , =
βj+k−2 βj−2 pj+k−1,j−1 ,
(j ≥ 2, k ≥ 1).
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Proof. Denote the orthonormal basis of SA(D) by {ek (z) = βk z k }∞ k=0 . From (2.5), we know that the operator Mz admits the following matrix representation with respect to the above basis ⎛
0
⎜ ββ0 ⎜ 1 ⎜ 0 ⎜ ⎜ Mz = ⎜ .. ⎜ . ⎜ ⎜ 0 ⎝ .. .
0 0
... ...
0 0
0 0
β1 β2
... .. . ... .. .
0 .. .
0 .. . 0
.. . 0 .. .
βk−1 βk
.. .
..
⎞ ... ...⎟ ⎟ ...⎟ ⎟ ⎟ ⎟. ···⎟ ⎟ ...⎟ ⎠ .. .
.
(3.1)
From Mz P = P Mz , we have ⎛
0
⎜ β0 p ⎜ β1 11 ⎜ ⎜ β1 p ⎜ β2 21 ⎜ ⎜ .. ⎜ . ⎜ ⎜ βk−2 ⎜ ⎝ βk−1 pk−1,1 .. . ⎛ β0 β1 p12 ⎜ β0 ⎜ β p22 ⎜ 1 ⎜β ⎜ 0 p32 ⎜ β1 =⎜ ⎜ .. ⎜ . ⎜ ⎜ β0 ⎜ β pk2 ⎝ 1 .. .
0
...
0
β0 β1 p12 β1 β2 p22
... ...
β0 β1 p1k β1 β2 p2k
.. .
.. .
.. .
βk−2 βk−1 pk−1,2
... .. .
βk−2 βk−1 pk−1,k
.. .
.. .
β1 β2 p13
...
βk−2 βk−1 p1k
β1 β2 p23
...
βk−2 βk−1 p2k
β1 β2 p33
...
βk−2 βk−1 p3k
.. .
.. .
.. .
β1 β2 pk3
... .. .
βk−2 βk−1 pkk
.. .
.. .
...
...⎞ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟ ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ⎟ ...⎟ ⎠ .. .
(3.2)
So we obtain ⎧ ⎪ ⎨ pij = 0, pii = p11 , ⎪ ⎩ βj+k−1 βj−1 pj+k,j =
i < j, i = 2, 3, · · · , βj+k−2 βj−2 pj+k−1,j−1 ,
(3.3)
(j ≥ 2, k ≥ 1).
Conversely, if the entries of P satisfy (3.3), that is, P admits the following matrix representation with respect to the above basis ⎛
p11
⎜ β0 ⎜ β1 p21 ⎜ ⎜ β0 ⎜ β2 p31 ⎜ P =⎜ .. ⎜ . ⎜ ⎜ β ⎜ 0 p ⎜ βk−1 k1 ⎝ .. .
0
0
...
0
p11
0
...
0
β1 β2 p21
p11
...
0
.. .
.. .
..
.. .
β1 βk−1 pk−1,1
β2 βk−1 pk−2,1
.. .
.. .
Simple computation shows that Mz P = P Mz . So P ∈ A (Mz ).
.
. . . p11 .. . 2
.. .
...
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ...⎟ ⎟ ⎠ .. .
(3.4)
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Lemma 3.2. Let SA(D) be the Sobolev disk algebra. If G is a projection, then G ∈ A ( ⎛
Mz ) if and only if
⎞
G0
⎜ ⎜ G=⎜ ⎜ ⎝
1
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
. Gn−1
where Gi is I or 0. Proof. From Lemma 3.1, we know that P ∈ A (Mz ) if and only if P has the form (3.4). Moreover, if P is a projection, we claim that p11 = 1 or 0, pj1 = 0 (j = 2, 3, · · ·). In fact, from P = P ∗ = P 2 , we have ⎛ ⎜ ⎜ ⎜ ⎜ P =⎜ ⎜ ⎜ ⎜ ⎝
⎞
p11
⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
p11 ..
. p11 ..
(3.5)
. n
where p11 = 1 or 0. By the above analysis, we get that if G is a projection, and G ∈ A ( if G has the following form ⎛ ⎜ ⎜ G=⎜ ⎜ ⎝
1
Mz ) if and only
⎞
G0
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
.
(3.6)
Gn−1 where Gi (i = 0, 1, · · · , n − 1) is I or 0. 2 From [7], we know that determining the reducing subspaces of Mzn is equivalent to finding the projection in the commutant of Mzn . Thus we have the following conclusion. Theorem 3.3. Let SA(D) be the Sobolev disk algebra. For n ≥ 2, then the multiplication operator Mzn has 2n reducing subspaces with minimal reducing subspaces S0 , S1 , · · · , Sn−1 . Proof. By Theorem 2.3, the multiplication operator Mzn is similar to bounded and invertible operator ⎛ ⎜ ⎜ X=⎜ ⎜ ⎝
n 1
Mz on SA(D), so there exists a
⎞
X0
⎟ ⎟ ⎟ ⎟ ⎠
X1 ..
. Xn−1
such that n 1
Mz = X −1 Mzn X.
(3.7)
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Note that G is a projection and G ∈ A ( G
n 1
Mz =
n
1
7
Mz ), G has the form described as (3.6), applying (3.7) we have
Mz G =⇒ GX −1 Mzn X = X −1 Mzn XG
1
=⇒ XGX −1 Mzn = Mzn XGX −1 =⇒ GXX −1 Mzn = Mzn GXX −1 =⇒ GMzn = Mzn G,
(3.8)
where ⎛ ⎜ ⎜ G=⎜ ⎜ ⎝
⎞
G0
⎟ ⎟ ⎟, ⎟ ⎠
G1 ..
. Gn−1
and Gi (i = 0, 1, · · · , n − 1) is ISi or 0. By Lemma 2.2, we have that the reducing subspaces of Mzn are c0 S0 ⊕ c1 S1 ⊕ · · · ⊕ cn−1 Sn−1 ,
where ci = 0 or 1 (i = 0, 1, · · · , n − 1),
and the minimal reducing subspaces are S0 , S1 , · · · , Sn−1 .
2
Acknowledgments The authors would like to thank the referees for their useful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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