算子权移位的不可约性和分类
|
摘要
设 H 是复可分 Hilbert 空间，{W1,W2,···} 是 H 上一致有界的算子列．定
义 Hilbert 空间 (H) 上的线性变换 S：{x0,x1,...} → {0,W1x0,W2x1,...}．那
2
么 S = sup Wk ，称 S 为 (H) 上具有权 {Wk}∞ 的 单侧算子权移位，简记
2
k=1
k
为 S ～ {Wk}．类似的，我们可以定义 双侧算子权移位．如果 dimH < ∞，称
S 是 有限重的；如果 dimH = ?0，则称 S 是 无穷重的．
算子权移位的研究始于 Lambert (1971)．它是重要的具体算子类――数值加
权移位算子的一个自然推广，拥有许多与数值加权移位相近的性质，有许多重
要的应用．近些年来，算子权移位一直是受人们关注的算子类．
Lambert 证明了下面的结果：每个权为可逆算子的移位都与一个权为正算子
的移位酉等价．在本文第二部分中，我们把 Lambert 的结果扩展到权为单射稠
值域算子的移位算子类上，并且给出了酉等价的两个算子关于权的更进一步的
信息．在后文中也将利用到这些结果：
定理 1: 每个单侧算子权移位 S ～ {Wk} 酉等价于一个以单射的正算子为权
的移位 T ～ {Vk}，且相同标号的权 Vk 与 |Wk| 是酉等价的．具体的，Vk =
Uk |Wk|Uk ，其中 U0 = I，Uk 是 WkUk
? 做极分解得到的酉算子．
?1 ?1 ?1
对双侧情形也有类似结果，不同处是当 k 0 时，Vk 与 |Wk| 酉等价．
?
第三部分中，我们研究了权为单射稠值域算子的权移位与不可约算子的相
似关系．从 Halmos 的不可约算子稠密定理和 Voiculescu 的非交换的 Weyl-von
Neumann 定理可见，不可约算子是重要的算子类，在算子的结构和算子逼近理
论的研究中，从某种意义上讲是很基本的“元素”．算子的不可约性是酉等
价下的不变量．但很不幸，它并不是相似变换下的不变量．Gilfeather 和江泽
坚 [2] 曾给出不可约算子，相似于一个可约算子例子．在 1988 年，Herrero [18]
提出一个猜想：算子 T ∈ B(H) 相似于不可约算子的充要条件是下面三个条件
同时成立：(1) 对 ?λ ∈ C, T ? λ 不是有限秩算子；(2) T 不满足任意的二次方程
ax2 + bx + c = 0，这里 |a| + |b| + |c| = 0；(3) 对 ?λ ∈ C, T ? λ 不能写成一个有
限秩算子和一个满足二次方程的算子之直和形式．
许多学者研究了具体的算子类（如正规算子，幂零算子，拟正规算子，矩
阵等等）相似于一个不可约算子的充要条件．本文中证明了结果：
定理 2: 每个单侧算子权移位 S ～ {Wk} 必相似于一个不可约算子．进而 Sn 也
相似于一个不可约算子．
–26 –
中 文 摘 要
在前言中我们提到 Gilfeather 和江泽坚都曾举出了不可约算子和不可约算子的例
子．上面提到的二者的例子都是在正规算子情形下给出的，本节中将给出一些
非正规算子情形下的例子．这些例子是用算子权移位构造的．首先，介绍一个
需要用到的引理：
引理 1 (Faour 和 Khalil,1987): 算子权移位 S ～ {Wk} 是紧算子当且仅当每个权
Wk 是紧算子且 limk →∞ Wk = 0．进一步，对任意的 1 ≤ p < ∞，算子权移位
S 是 Cp 类的当且仅当每个权 Wk 是 Cp 类的且数列 { Wk } 是 p-可和的．
例 1: 存在可约算子权移位相似于一个不可约算子．
例 2: 存在紧的和 Cp (1 ≤ p < ∞) 类的可约算子权移位相似于一个不可约算子．
定理 3: 每个单侧算子权移位 S ～ {Wk} 都满足 Herrero 的猜想中的条件 (1)，(2)
和 (3) ．
由前一个定理，每个算子权移位（以单射，稠值域算子作为权）都可以相
似于一个不可约算子，因此这些结果是对 Herrero 的猜想的一个支持．
定义 1: 设 T 是 Hilbert 空间 H 上的压缩算子，
记 T ∈ C0 ，如果 ? f ∈ H, Tmf ?→ 0 (m → ∞)．
?
记 T ∈ C1 ，如果 ? f ∈ H,f = 0, Tmf ?→ 0 (m → ∞)．
?
记 T ∈ C? ，如果 ? f ∈ H, T? f ?→ 0 (m → ∞)．
m
0
记 T ∈ C? ，如果 ? f ∈ H, f = 0, T? f ?→ 0 (m → ∞)．
m
1
由于 T 是压缩的，所以上面的极限存在．记 Cαβ = Cα ∩ C? ，对 α,β = 0,1．
? β
压缩算子 Cαβ 分类的概念是由 Sz.-Nagy 和 Foias 提出的并建立了基本理论．Cαβ
分类的思想有助于研究线性算子的结构．例如，它推动了算子理论中最受关
注的不变子空间问题的研究．对任意非零的 T ∈ B(H)，T/ T 是一个压缩算
子，所以压缩算子的性质反映了一般算子的性质．特别的，压缩算子的不变子
空间问题等价于全体算子的不变子空间问题．利用压缩算子的 Cαβ 的思想，
Brown，Chevreau 和 Pearcy 证明了重要定理：每个谱包含单位圆周的压缩算子
都存在非平凡的不变子空间． 迄今为止，这个?
Let H be a complex separable Hilbert space and let {W1,W2,...} be a
uniformly bounded sequence of operators on H. The operator S on 2(H) =
H ⊕ H ⊕ ··· de?ned by
S : {x0,x1,...} → {0,W1x0,W2x1,...}
is called an operator weighted shift on 2(H) with the weight sequence {Wk},
denoted by S ～ {Wk}. Then S is bounded and S = sup Wk . The operator
k
weighted shift was ?rst studied by Lambert. It is a natural generalization of
the scalar weighted shift operator and owns many properties similar to those
of the scalar one. In this thesis, operator weighted shifts with one-to-one and
dense-range operators as weights are main objects which are studied.
This thesis consists of four chapters.
In the ?rst chapter, the basic concepts, the background of problems, and
the work ?nished in past are introduced.
In the second chapter, we proved that every operator weighted shift S ～
{Wk} with one-to-one and dense-range operators {Wk} as weights is similar to
an operator weighted shift T ～ {Vk} whose weights {Vk} are one-to-one and
positive operators. Moreover, the weight Vk is unitarily equivalent to |Wk| for
each ?xed k.
In the third chapter, we proved that every operator weighted shift S ～ {Wk},
–29 –
英 文 摘 要
whose weight sequence {Wk} consists of one-to-one and dense-range opera-
tors, is similar to an irreducible operator. In addition, we show a class of exam-
ples that reducible operators are similar to irreducible ones. In these examples,
compact operators and Cp (1 ≤ p < ∞) ones are also contained. All these
examples show that irreducibility is not an invariant under similar transforms.
In the ?nal chapter, the suf?cient and necessary conditions of Cαβ classi?-
cation of any multiple contracted operator weighted shifts are given and proved.
Especially, in the case of ?nite multiple, some of these conditions can be sim-
pli?ed. Finally, two examples are provided to show why these conditions can
not be simpli?ed in general cases.
义 Hilbert 空间 (H) 上的线性变换 S：{x0,x1,...} → {0,W1x0,W2x1,...}．那
2
么 S = sup Wk ，称 S 为 (H) 上具有权 {Wk}∞ 的 单侧算子权移位，简记
2
k=1
k
为 S ～ {Wk}．类似的，我们可以定义 双侧算子权移位．如果 dimH < ∞，称
S 是 有限重的；如果 dimH = ?0，则称 S 是 无穷重的．
算子权移位的研究始于 Lambert (1971)．它是重要的具体算子类――数值加
权移位算子的一个自然推广，拥有许多与数值加权移位相近的性质，有许多重
要的应用．近些年来，算子权移位一直是受人们关注的算子类．
Lambert 证明了下面的结果：每个权为可逆算子的移位都与一个权为正算子
的移位酉等价．在本文第二部分中，我们把 Lambert 的结果扩展到权为单射稠
值域算子的移位算子类上，并且给出了酉等价的两个算子关于权的更进一步的
信息．在后文中也将利用到这些结果：
定理 1: 每个单侧算子权移位 S ～ {Wk} 酉等价于一个以单射的正算子为权
的移位 T ～ {Vk}，且相同标号的权 Vk 与 |Wk| 是酉等价的．具体的，Vk =
Uk |Wk|Uk ，其中 U0 = I，Uk 是 WkUk
? 做极分解得到的酉算子．
?1 ?1 ?1
对双侧情形也有类似结果，不同处是当 k 0 时，Vk 与 |Wk| 酉等价．
?
第三部分中，我们研究了权为单射稠值域算子的权移位与不可约算子的相
似关系．从 Halmos 的不可约算子稠密定理和 Voiculescu 的非交换的 Weyl-von
Neumann 定理可见，不可约算子是重要的算子类，在算子的结构和算子逼近理
论的研究中，从某种意义上讲是很基本的“元素”．算子的不可约性是酉等
价下的不变量．但很不幸，它并不是相似变换下的不变量．Gilfeather 和江泽
坚 [2] 曾给出不可约算子，相似于一个可约算子例子．在 1988 年，Herrero [18]
提出一个猜想：算子 T ∈ B(H) 相似于不可约算子的充要条件是下面三个条件
同时成立：(1) 对 ?λ ∈ C, T ? λ 不是有限秩算子；(2) T 不满足任意的二次方程
ax2 + bx + c = 0，这里 |a| + |b| + |c| = 0；(3) 对 ?λ ∈ C, T ? λ 不能写成一个有
限秩算子和一个满足二次方程的算子之直和形式．
许多学者研究了具体的算子类（如正规算子，幂零算子，拟正规算子，矩
阵等等）相似于一个不可约算子的充要条件．本文中证明了结果：
定理 2: 每个单侧算子权移位 S ～ {Wk} 必相似于一个不可约算子．进而 Sn 也
相似于一个不可约算子．
–26 –
中 文 摘 要
在前言中我们提到 Gilfeather 和江泽坚都曾举出了不可约算子和不可约算子的例
子．上面提到的二者的例子都是在正规算子情形下给出的，本节中将给出一些
非正规算子情形下的例子．这些例子是用算子权移位构造的．首先，介绍一个
需要用到的引理：
引理 1 (Faour 和 Khalil,1987): 算子权移位 S ～ {Wk} 是紧算子当且仅当每个权
Wk 是紧算子且 limk →∞ Wk = 0．进一步，对任意的 1 ≤ p < ∞，算子权移位
S 是 Cp 类的当且仅当每个权 Wk 是 Cp 类的且数列 { Wk } 是 p-可和的．
例 1: 存在可约算子权移位相似于一个不可约算子．
例 2: 存在紧的和 Cp (1 ≤ p < ∞) 类的可约算子权移位相似于一个不可约算子．
定理 3: 每个单侧算子权移位 S ～ {Wk} 都满足 Herrero 的猜想中的条件 (1)，(2)
和 (3) ．
由前一个定理，每个算子权移位（以单射，稠值域算子作为权）都可以相
似于一个不可约算子，因此这些结果是对 Herrero 的猜想的一个支持．
定义 1: 设 T 是 Hilbert 空间 H 上的压缩算子，
记 T ∈ C0 ，如果 ? f ∈ H, Tmf ?→ 0 (m → ∞)．
?
记 T ∈ C1 ，如果 ? f ∈ H,f = 0, Tmf ?→ 0 (m → ∞)．
?
记 T ∈ C? ，如果 ? f ∈ H, T? f ?→ 0 (m → ∞)．
m
0
记 T ∈ C? ，如果 ? f ∈ H, f = 0, T? f ?→ 0 (m → ∞)．
m
1
由于 T 是压缩的，所以上面的极限存在．记 Cαβ = Cα ∩ C? ，对 α,β = 0,1．
? β
压缩算子 Cαβ 分类的概念是由 Sz.-Nagy 和 Foias 提出的并建立了基本理论．Cαβ
分类的思想有助于研究线性算子的结构．例如，它推动了算子理论中最受关
注的不变子空间问题的研究．对任意非零的 T ∈ B(H)，T/ T 是一个压缩算
子，所以压缩算子的性质反映了一般算子的性质．特别的，压缩算子的不变子
空间问题等价于全体算子的不变子空间问题．利用压缩算子的 Cαβ 的思想，
Brown，Chevreau 和 Pearcy 证明了重要定理：每个谱包含单位圆周的压缩算子
都存在非平凡的不变子空间． 迄今为止，这个?
Let H be a complex separable Hilbert space and let {W1,W2,...} be a
uniformly bounded sequence of operators on H. The operator S on 2(H) =
H ⊕ H ⊕ ··· de?ned by
S : {x0,x1,...} → {0,W1x0,W2x1,...}
is called an operator weighted shift on 2(H) with the weight sequence {Wk},
denoted by S ～ {Wk}. Then S is bounded and S = sup Wk . The operator
k
weighted shift was ?rst studied by Lambert. It is a natural generalization of
the scalar weighted shift operator and owns many properties similar to those
of the scalar one. In this thesis, operator weighted shifts with one-to-one and
dense-range operators as weights are main objects which are studied.
This thesis consists of four chapters.
In the ?rst chapter, the basic concepts, the background of problems, and
the work ?nished in past are introduced.
In the second chapter, we proved that every operator weighted shift S ～
{Wk} with one-to-one and dense-range operators {Wk} as weights is similar to
an operator weighted shift T ～ {Vk} whose weights {Vk} are one-to-one and
positive operators. Moreover, the weight Vk is unitarily equivalent to |Wk| for
each ?xed k.
In the third chapter, we proved that every operator weighted shift S ～ {Wk},
–29 –
英 文 摘 要
whose weight sequence {Wk} consists of one-to-one and dense-range opera-
tors, is similar to an irreducible operator. In addition, we show a class of exam-
ples that reducible operators are similar to irreducible ones. In these examples,
compact operators and Cp (1 ≤ p < ∞) ones are also contained. All these
examples show that irreducibility is not an invariant under similar transforms.
In the ?nal chapter, the suf?cient and necessary conditions of Cαβ classi?-
cation of any multiple contracted operator weighted shifts are given and proved.
Especially, in the case of ?nite multiple, some of these conditions can be sim-
pli?ed. Finally, two examples are provided to show why these conditions can
not be simpli?ed in general cases.
引文
[1] 孙善利 , 于涛, 多重加权移位算子的谱图形, Cαβ分类和弱压缩性, 吉林大学自然科学学
报3 (1992), 25–30.
[2] 江泽坚, 关于线性算子的结构, 全国算子理论会议报告, 九江, 1981.
[3] 李觉先, 算子权移位的超自反性, 辽宁大学学报(自然科学版) 20 (1993), 29–34.
[4] , 算子权移位的 Banach 约化性, 数学研究与评论14 (1994), 97–100.
[5] , 算子权移位与 Cowen-Douglas 算子的若干结果, 吉林大学博士学位论文, 长春,
1999.
[6] Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland
Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR
90d:47001
[7] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block
weighted shifts, Integral Equations Operator Theory 14 (1991), no. 5, 613–677. MR 93e:47033
[8] S. Brown, B. Chevreau, and C. Pearcy, Contractions with rich spectrum have invariant sub-
spaces, J. Operator Theory 1 (1979), no. 1, 123–136. MR 80m:47002
[9] , On the structure of contraction operators. II, J. Funct. Anal. 76 (1988), no. 1, 30–55.
MR 90b:47030b
[10] J. B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Ad-
vanced Publishing Program), Boston, Mass., 1981. MR 83i:47030
[11] R. G. Douglas, Canonical models, Topics in operator theory, Amer. Math. Soc., Providence,
R.I., 1974, pp. 161–218. Math. Surveys, No. 13. MR 50 #8121
[12] N. Faour and R. Khalil, On the spectrum of weighted operator shifts, Houston J. Math. 13
(1987), no. 4, 549–556. MR 89d:47049
[13] C.K. Fong and C.L. Jiang, Normal operators similar to irreducible operators, Acta Math.
Sinica (N.S.) 10 (1994), no. 2, 132–135. MR 97g:47016
[14] F. Gilfeather, Strong reducibility of operators, Indiana Univ. Math. J. 22 (1972/73), 393–397.
MR 46 #2460
[15] P. R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215–223. MR 37 #6788
– 23 –
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#5066
[17] , A Hilbert space problem book, Graduate Texts in Mathematics, vol. 19, Springer-
Verlag, New York, 1982. MR 84e:47001
[18] D. A. Herrero, The personal communication with C.L. Jiang, 1988.
[19] , Spectral pictures of hyponormal bilateral operator weighted shifts, Proc. Amer.
Math. Soc. 109 (1990), no. 3, 753–763. MR 90k:47050
[20] C.I. Hsin, Finite matrices similar to irreducible ones, Taiwanese J. Math. 4 (2000), no. 3,
457–477. MR 2001f:15016
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(2000), no. 3, 923–931. MR 2001k:47027
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ator weighted shifts, Acta Sci. Math. (Szeged) 66 (2000), no. 3-4, 679–695. MR 2001m:47059
[23] Chunlan Jiang, Xianzhou Guo, and Yongfa Yang, Nilpotent operators similar to irreducible
operators, Sci. China Ser. A 44 (2001), no. 12, 1544–1557. MR 2003c:47038
[24] A. Lambert, Unitary equivalence and reducibility of invertibly weighted shifts, Bull. Austral.
Math. Soc. 5 (1971), 157–173. MR 45 #4196
[25] , The algebra generated by an invertibly weighted shift, J. London Math. Soc. (2) 5
(1972), 741–747. MR 47 #5628
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(1994), no. 1, 27–45. MR 95f:47051
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[32] A. L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory,
Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR 50 #14341
[33] Shan-li Sun and Ru-ning Ma, Cαβ classi?cation of analytic Toeplitz operators, Northeast.
Math. J. 16 (2000), no. 1, 67–74. MR 2001m:47053
[34] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, Translated from
the French and revised, North-Holland Publishing Co., Amsterdam, 1970. MR 43 #947
[35] D. Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures
Appl. 21 (1976), no. 1, 97–113. MR 54 #3427
– 25 –
报3 (1992), 25–30.
[2] 江泽坚, 关于线性算子的结构, 全国算子理论会议报告, 九江, 1981.
[3] 李觉先, 算子权移位的超自反性, 辽宁大学学报(自然科学版) 20 (1993), 29–34.
[4] , 算子权移位的 Banach 约化性, 数学研究与评论14 (1994), 97–100.
[5] , 算子权移位与 Cowen-Douglas 算子的若干结果, 吉林大学博士学位论文, 长春,
1999.
[6] Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland
Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR
90d:47001
[7] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block
weighted shifts, Integral Equations Operator Theory 14 (1991), no. 5, 613–677. MR 93e:47033
[8] S. Brown, B. Chevreau, and C. Pearcy, Contractions with rich spectrum have invariant sub-
spaces, J. Operator Theory 1 (1979), no. 1, 123–136. MR 80m:47002
[9] , On the structure of contraction operators. II, J. Funct. Anal. 76 (1988), no. 1, 30–55.
MR 90b:47030b
[10] J. B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Ad-
vanced Publishing Program), Boston, Mass., 1981. MR 83i:47030
[11] R. G. Douglas, Canonical models, Topics in operator theory, Amer. Math. Soc., Providence,
R.I., 1974, pp. 161–218. Math. Surveys, No. 13. MR 50 #8121
[12] N. Faour and R. Khalil, On the spectrum of weighted operator shifts, Houston J. Math. 13
(1987), no. 4, 549–556. MR 89d:47049
[13] C.K. Fong and C.L. Jiang, Normal operators similar to irreducible operators, Acta Math.
Sinica (N.S.) 10 (1994), no. 2, 132–135. MR 97g:47016
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