算子权移位的Banach可约性
摘要
设H是复可分的Hilbert空间，若{W_i}_(i=1)~(+x)是一列H上的一致有界的线性算子，S∈L(l~2(H))。且有那么称S为一个单边算子加权移位，简记为所有这样定义的算子组成的集合，记为IWl~2(H)。特别的，若设C代表复平面，那么S就称为一个n重单边算子加权移位，所有这样的算子的集合记为IWl~2(C~n)。算子权移位一直是人们关心的重要的具体算子类，人们对这类算子感兴趣主要因为它经常用于构造正反两方面的例子，而且算子理论中的某些一般性的问题都与其密切相关，因而一直受到重视。
设T∈L(H)．M∈LatT，如果存在N∈LatT使得，M∩N={0}且M+N=H。则称M是T的一个Banach约化子空间。若T有非平凡的Banach约化子空间就称T是Banach可约的，否则称T是Banach不可约的。T是Banach可约的当且仅当存在非平凡的幂等算子与之交换当且仅当T相似于可约算子。
算子的约化问题在整个算子理论中有很重要的意义，当H是有限维空间的时候，T是强不可约算子，即Banach不可约算子当且仅当T在某个基底下的矩阵表示是Jordan块，因此强不可约算子是Jordan块在无穷维空间的自然推广。这已被江泽坚，蒋春澜及其合作者所证实。
前向的纯量单边移位是强不可约的。在[1]中李觉先等人证明了若S∈IWl~2(C~n)，且σ_e(T)不连通，那么S是Banach可约的，那么这种性质对于更大范围IWl~2(H)中的算子是否成立?本文主要探讨了这个问题并得到了肯定的答复。
对于IWl~2(H)中的算子本文首先证明了
引理2．1对于每个H_i，都存在，使得每个W_i都能表成
吉林大学硕士学位论文
算子权移位的Banad，可约性
。{‘一，)
。{犷一‘，
一
(1)
州
‘/了‘..皿...lweweee、、、
南了、上
r
(i2
亡
这样对于每个多任川’(z(H)都可以酉等价一个上三角的单边算子权移
位.设V={人:rl<入卜r:}c脚，(匀是。。(S)中的一个洞，取、。任V.我
们可以得到(R。川亏一入。))上限制在H。上的一个，l维基底，利用这个基底我
们把5~{H一忿化成上三角算子权移位，其中每个H弓都表成(1)的形式.
那么有
引理2.3设S任川’(z( H).S、{忧}彗.那么g酉等价于
、一厂一“‘丫渺
\0刀/州HO川)
(2)
其中‘2(，侧)=0几〔，，划;·州，=v{。}‘，:‘三.，三，，}·，全O·‘2(H二、“)-
O霆。(H，。川，).一攻任粼自州))·B任自/z(H。州))是权可逆的上三角算子权
移位.
接着对入。与注.B的谱，本性谱的位置关系进行分析有
命题3.15任月I’尸(H)，万、{I卜}彗，入。任v.s酉等价于(2)的形式，
那么有入《，锗。(B)==汀，·(B).入。)>，·(B).
从引理2.3我们不妨设且任111一尸(Cn).对于月f尸(C叻中的算子，由!11
中的某些结果，我们可以得到
命题3.25任111一了“(H).万~{H诗彗.入。〔V.S酉等价于(2).的形式，
那么有.\(，贾二，(一们.入，<，·1(勺.
吉林大学硕士学位论文
算子权移位的Bol，。c.1l可约性
对1任乙(H，).B任乙(从).工、。是定义在乙(从，万，)上的R()s一111山川l算
子，私浏工)二一去Y一工B.对于任意的一Y任粼负，Hl).引入ROSoll)11，111算子
得到
引理3.6‘4任111一/Zf〔”，):一4~{42}彗.B任I下下丫2(H).B~{B，}彗.若
州勺自氏(B)二0.那么具有
CC……C
、、、龟lwe百weeeesseeeeeeeeeee
()‘’*
oC
了/了，les诬esesles.ee..l.怪es..、、、
HH…H
形式的算子C任R(II，:、，了.
根据引理3.6我们可以找出可逆算子，使得川一(2(H)中的算子相似干一
个可约算子，我们得到主要结果
定理3.1，任月I一了2(H).三~{H，，}彗.若氏(S)是不连通的，那么s是
Ba‘la(·11可约的.
本文还得到关于Co二·(’l1一Dol:glas算子的一结论，设贝是C的一个连通开
子集，尽，(卿代表侧H)中的算子且满足
(。，)茵2仁二(B):
(b)Rol}(B一入)=H、丫入任贝:
(..)V{肠以B一入):入任互砰二H:
(d)。11?，，大·。，(B一.\)二，，.丫入任f2.
那么称B。(卿中的算子为C二甲。n一Douglas算子.
推论3.1，任111一尸(H).5~{H，t}彗.人。任V.S酉等价于(2)的形式，
那么有一扩任尽，(贝).其中n为，，，.l(S一入。)的负值.
类似于尽，(贝)本文新定义了一个B二(Q)的概念，提出
吉林大学硕士学位论文
算子权移位的Bol招曲可约性
问题当S任
。，三+戈是否和
川一(z(H)时，存在一个连通开集几使得夕任氏;(卿·0<
:了(S)>是充要条件呢?
Lot H denote the complex separable Hilbert space. 2(H)=i=0xH. if {Wi}i=1+x is a sequence of uniformly bounded linear operators on H .S 6 (2(H)) , and
then S is called a unilateral operator weighted shift,denoted by S ~ {Wi} . the set of all this kind of operators is denoted by IW2(H).Particularly,let C be the complex plane.Cn - k=1n, C. 2(Cn) = Cn.then S is called a n-multiple unilateral operator weighted shift . the set of all this kind of operators is denoted by IW2(Cn).
Operator weighted shifts form an important class of operators that people are interested in . One pay more attention to them because they are often used to make examples and counter examples, moreover they are closely related with some general problems in operator theory .
Let T (H).M LatT. if there exists a N Latr.such that M N = {0} and M + N = H.then we call .M a Banach reducible subspace of T. T is said to be Banach reducible if there exist a nontrivial Banach reducible subspace,if not T is said to be Banach irreducible. T is Banach reducible if and only if there exists a nontrivial idempotent operator that commutes with it, if and only if T is similar to a reducible operator.
The problem of operator redudbility pay a significant role in operator theory.When H is a finitely dimensional space, T is strongly irreducible.that is Banach irreducible, if and only if it can be represented as a Jordan block under some OXB,so strongly irreducible operator is a natural generalization in infinitely dimensional space.This idea has been demenstrated being intelligent by Zejian Jiang .Chunlan Jiang and their partners.
The forward unilateral scalar weighted shift is strongly irreducible. In[1], Juexian Li proved that if S IW2(Cn). and e(T) is not connected .then S
is Banach redudble.Does this property hold for more extensive operators in IW('2(H) ? In this paper, we consider this question and give a positive answer. First.for IW 2(H) we prove the following lemma
Lemma 2.1 For eachHi, there exists an OXB { } . such that every Wi can be represented as
Thus, every S IW2(H) is unitarily equivalent to an upper triangular unilateral operator weighted shift.Let V = { : r1 < | | < r2 } F(S) be a hole of (S).hx 0 V.we can obtain an n dimensional OXB of (Ran/(S -0)) restricting on H.Every S ~ {Wi} can be represented as an upper triangular unilateral operator weighted shift under this OXB where every IF, has the form of (1) .Then we can show the following
Lemma 2.3 Let S IW2(H).S ~ {Wi}. then S is unitarily equivalent to
where 2(M)).B 2(H M)) are upper triangular unilateral operator weighted shifts with invertible multiplicity.
Then by analyzing the relations of position among A0 and .4. B. we obtain.
Proposition 3.1 S IW2(H). S ~ {Wi). 0 V. S is unitarily equivalent to the form of (2) .then 0 (B) = (B). 0 > r(B).
From lemma 2.3. we may let A IW72(Cn).for the operators in IW72(Cn).we get to the following propositions from some conclusions in [l].
Proposition 3.2 5 IW2(H). S ~ {W'i} . A0 V. S is unitarily equivalent to (2). then A0 1(A). 0 < r1(A).
For A (H1). B (H2).Rosenblum operator AB is defined on (H2, H1) as TAB(X) = AX - XB. for any A (H2,H1). By using this operator we obtain
Lemma3.6 .A if 1(A) r(B] = o. then C RanTAB of each operator C of the following
form
From lemma3.6. we can find an invortible operator such that an operator in IW2(H) is similar to a reducible operator, so we have the main theorem
theorem 3.1 S IW2(H). S ~ {Wi} . if e(5) is not comected.then 5 is Banach reducible.
In this paper, we also get some conclusions about the Cowen-Douglas operators, let be a connected open subset of C. Bn( ) denoted the set of operators B in (H) satisfying
(a) (B):
(b)Ran(B -) = H. :
(d) dimker(B - ) = . . Then call an operator in Bn( ) a Cowen-Douglas operator.
Corollary3.1 S IW 2(H)..S - {Wi}+ - 0 V. 5 is Militarily equivalent to (2).then A Bn( ).where n is the negative vahie of iucI(S - 0).
Analogously. we define a new concept Bx( ).tlie following remained
question (1) for 5" G I\V(2(H).th
设T∈L(H)．M∈LatT，如果存在N∈LatT使得，M∩N={0}且M+N=H。则称M是T的一个Banach约化子空间。若T有非平凡的Banach约化子空间就称T是Banach可约的，否则称T是Banach不可约的。T是Banach可约的当且仅当存在非平凡的幂等算子与之交换当且仅当T相似于可约算子。
算子的约化问题在整个算子理论中有很重要的意义，当H是有限维空间的时候，T是强不可约算子，即Banach不可约算子当且仅当T在某个基底下的矩阵表示是Jordan块，因此强不可约算子是Jordan块在无穷维空间的自然推广。这已被江泽坚，蒋春澜及其合作者所证实。
前向的纯量单边移位是强不可约的。在[1]中李觉先等人证明了若S∈IWl~2(C~n)，且σ_e(T)不连通，那么S是Banach可约的，那么这种性质对于更大范围IWl~2(H)中的算子是否成立?本文主要探讨了这个问题并得到了肯定的答复。
对于IWl~2(H)中的算子本文首先证明了
引理2．1对于每个H_i，都存在，使得每个W_i都能表成
吉林大学硕士学位论文
算子权移位的Banad，可约性
。{‘一，)
。{犷一‘，
一
(1)
州
‘/了‘..皿...lweweee、、、
南了、上
r
(i2
亡
这样对于每个多任川’(z(H)都可以酉等价一个上三角的单边算子权移
位.设V={人:rl<入卜r:}c脚，(匀是。。(S)中的一个洞，取、。任V.我
们可以得到(R。川亏一入。))上限制在H。上的一个，l维基底，利用这个基底我
们把5~{H一忿化成上三角算子权移位，其中每个H弓都表成(1)的形式.
那么有
引理2.3设S任川’(z( H).S、{忧}彗.那么g酉等价于
、一厂一“‘丫渺
\0刀/州HO川)
(2)
其中‘2(，侧)=0几〔，，划;·州，=v{。}‘，:‘三.，三，，}·，全O·‘2(H二、“)-
O霆。(H，。川，).一攻任粼自州))·B任自/z(H。州))是权可逆的上三角算子权
移位.
接着对入。与注.B的谱，本性谱的位置关系进行分析有
命题3.15任月I’尸(H)，万、{I卜}彗，入。任v.s酉等价于(2)的形式，
那么有入《，锗。(B)==汀，·(B).入。)>，·(B).
从引理2.3我们不妨设且任111一尸(Cn).对于月f尸(C叻中的算子，由!11
中的某些结果，我们可以得到
命题3.25任111一了“(H).万~{H诗彗.入。〔V.S酉等价于(2).的形式，
那么有.\(，贾二，(一们.入，<，·1(勺.
吉林大学硕士学位论文
算子权移位的Bol，。c.1l可约性
对1任乙(H，).B任乙(从).工、。是定义在乙(从，万，)上的R()s一111山川l算
子，私浏工)二一去Y一工B.对于任意的一Y任粼负，Hl).引入ROSoll)11，111算子
得到
引理3.6‘4任111一/Zf〔”，):一4~{42}彗.B任I下下丫2(H).B~{B，}彗.若
州勺自氏(B)二0.那么具有
CC……C
、、、龟lwe百weeeesseeeeeeeeeee
()‘’*
oC
了/了，les诬esesles.ee..l.怪es..、、、
HH…H
形式的算子C任R(II，:、，了.
根据引理3.6我们可以找出可逆算子，使得川一(2(H)中的算子相似干一
个可约算子，我们得到主要结果
定理3.1，任月I一了2(H).三~{H，，}彗.若氏(S)是不连通的，那么s是
Ba‘la(·11可约的.
本文还得到关于Co二·(’l1一Dol:glas算子的一结论，设贝是C的一个连通开
子集，尽，(卿代表侧H)中的算子且满足
(。，)茵2仁二(B):
(b)Rol}(B一入)=H、丫入任贝:
(..)V{肠以B一入):入任互砰二H:
(d)。11?，，大·。，(B一.\)二，，.丫入任f2.
那么称B。(卿中的算子为C二甲。n一Douglas算子.
推论3.1，任111一尸(H).5~{H，t}彗.人。任V.S酉等价于(2)的形式，
那么有一扩任尽，(贝).其中n为，，，.l(S一入。)的负值.
类似于尽，(贝)本文新定义了一个B二(Q)的概念，提出
吉林大学硕士学位论文
算子权移位的Bol招曲可约性
问题当S任
。，三+戈是否和
川一(z(H)时，存在一个连通开集几使得夕任氏;(卿·0<
:了(S)>是充要条件呢?
Lot H denote the complex separable Hilbert space. 2(H)=i=0xH. if {Wi}i=1+x is a sequence of uniformly bounded linear operators on H .S 6 (2(H)) , and
then S is called a unilateral operator weighted shift,denoted by S ~ {Wi} . the set of all this kind of operators is denoted by IW2(H).Particularly,let C be the complex plane.Cn - k=1n, C. 2(Cn) = Cn.then S is called a n-multiple unilateral operator weighted shift . the set of all this kind of operators is denoted by IW2(Cn).
Operator weighted shifts form an important class of operators that people are interested in . One pay more attention to them because they are often used to make examples and counter examples, moreover they are closely related with some general problems in operator theory .
Let T (H).M LatT. if there exists a N Latr.such that M N = {0} and M + N = H.then we call .M a Banach reducible subspace of T. T is said to be Banach reducible if there exist a nontrivial Banach reducible subspace,if not T is said to be Banach irreducible. T is Banach reducible if and only if there exists a nontrivial idempotent operator that commutes with it, if and only if T is similar to a reducible operator.
The problem of operator redudbility pay a significant role in operator theory.When H is a finitely dimensional space, T is strongly irreducible.that is Banach irreducible, if and only if it can be represented as a Jordan block under some OXB,so strongly irreducible operator is a natural generalization in infinitely dimensional space.This idea has been demenstrated being intelligent by Zejian Jiang .Chunlan Jiang and their partners.
The forward unilateral scalar weighted shift is strongly irreducible. In[1], Juexian Li proved that if S IW2(Cn). and e(T) is not connected .then S
is Banach redudble.Does this property hold for more extensive operators in IW('2(H) ? In this paper, we consider this question and give a positive answer. First.for IW 2(H) we prove the following lemma
Lemma 2.1 For eachHi, there exists an OXB { } . such that every Wi can be represented as
Thus, every S IW2(H) is unitarily equivalent to an upper triangular unilateral operator weighted shift.Let V = { : r1 < | | < r2 } F(S) be a hole of (S).hx 0 V.we can obtain an n dimensional OXB of (Ran/(S -0)) restricting on H.Every S ~ {Wi} can be represented as an upper triangular unilateral operator weighted shift under this OXB where every IF, has the form of (1) .Then we can show the following
Lemma 2.3 Let S IW2(H).S ~ {Wi}. then S is unitarily equivalent to
where 2(M)).B 2(H M)) are upper triangular unilateral operator weighted shifts with invertible multiplicity.
Then by analyzing the relations of position among A0 and .4. B. we obtain.
Proposition 3.1 S IW2(H). S ~ {Wi). 0 V. S is unitarily equivalent to the form of (2) .then 0 (B) = (B). 0 > r(B).
From lemma 2.3. we may let A IW72(Cn).for the operators in IW72(Cn).we get to the following propositions from some conclusions in [l].
Proposition 3.2 5 IW2(H). S ~ {W'i} . A0 V. S is unitarily equivalent to (2). then A0 1(A). 0 < r1(A).
For A (H1). B (H2).Rosenblum operator AB is defined on (H2, H1) as TAB(X) = AX - XB. for any A (H2,H1). By using this operator we obtain
Lemma3.6 .A if 1(A) r(B] = o. then C RanTAB of each operator C of the following
form
From lemma3.6. we can find an invortible operator such that an operator in IW2(H) is similar to a reducible operator, so we have the main theorem
theorem 3.1 S IW2(H). S ~ {Wi} . if e(5) is not comected.then 5 is Banach reducible.
In this paper, we also get some conclusions about the Cowen-Douglas operators, let be a connected open subset of C. Bn( ) denoted the set of operators B in (H) satisfying
(a) (B):
(b)Ran(B -) = H. :
(d) dimker(B - ) = . . Then call an operator in Bn( ) a Cowen-Douglas operator.
Corollary3.1 S IW 2(H)..S - {Wi}+ - 0 V. 5 is Militarily equivalent to (2).then A Bn( ).where n is the negative vahie of iucI(S - 0).
Analogously. we define a new concept Bx( ).tlie following remained
question (1) for 5" G I\V(2(H).th
引文
[1] J. X. Li. Y. Q. Ji and S. L. Sun, The essential spectrum and Banach reducibility of operator weighted shifts. Acta Mathematics Sinica. English Series. 17(2001)no.3.413-424
[2] 李觉先，关于算子权移位的若干结果，吉林大学博士学位论文辑，长春，1999
[3] Y. Q. Ji. J.X. Li and Y. Yang, A characterization of bilateral operator weighted shifts being Cowen-Douglas operator. Proceeding of the American Mathematical Society.129(2001).no.11.3205-3210.
[4] A.Lambert. Unitary equivalence and reducibility of inueribly weighted shifts. Bull.Austial Math.Soc..1971.5.157-174
[5] B.Morrel.P.Muhly. Centered operators. Studia Math. 51 (1974).251-263.
[6] C.M.Pearcy. S.Petrovic.On polynomially bounded weighted shifts. Houstou J. Math..20(1994). 27-45.
[7] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory. Acta Math..1978,141. 187-261
[8] A. L. Shields. Weighted, shifts operators and analytic function theory. Math.Surveys.vol.13.Amer.Math.Soc.Providence.RI.1974.49-148
[9] P. R. Halmos.Irreducible operators, Mich.Math.J.15(1968).215-223.
[10] D. V. Voiculescu.A non-commutative Weyl-von Neumann theorem, Rev. Roum. math.Pures Appl. 21(1976), 99-113.
[11] 江泽坚，关于完全不可约算子，第五届全国泛函分析会议报告，南京，1990．
[12] 江泽坚，孙善利，关于完全不可约算子，吉林大学自然科学学报，1992，No．4．20-28．
[13] D. A.Herrero, Jiang C. L., Limits of strongly irreducible operators and the Riesz decomposition theorem Michigan Math. J.. 37(1990), 283-291.
[14] 纪友清，BIR算子的紧扰动与套代数中的BIR算子，吉林大学博士学位论文辑，长春，1998．
[15] A. Ben-Artzi. I.Gohberg,Dichotomy, discrete exponent and spectrum of block weighted shifts, Integral Equations and Operator Theory 14(1991). 613-676.
[16] D. A. Herrero. Approximation of Hilbert space operators.J. Res. Notes Math.. 72. Pitman. Boston. 1982.
[2] 李觉先，关于算子权移位的若干结果，吉林大学博士学位论文辑，长春，1999
[3] Y. Q. Ji. J.X. Li and Y. Yang, A characterization of bilateral operator weighted shifts being Cowen-Douglas operator. Proceeding of the American Mathematical Society.129(2001).no.11.3205-3210.
[4] A.Lambert. Unitary equivalence and reducibility of inueribly weighted shifts. Bull.Austial Math.Soc..1971.5.157-174
[5] B.Morrel.P.Muhly. Centered operators. Studia Math. 51 (1974).251-263.
[6] C.M.Pearcy. S.Petrovic.On polynomially bounded weighted shifts. Houstou J. Math..20(1994). 27-45.
[7] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory. Acta Math..1978,141. 187-261
[8] A. L. Shields. Weighted, shifts operators and analytic function theory. Math.Surveys.vol.13.Amer.Math.Soc.Providence.RI.1974.49-148
[9] P. R. Halmos.Irreducible operators, Mich.Math.J.15(1968).215-223.
[10] D. V. Voiculescu.A non-commutative Weyl-von Neumann theorem, Rev. Roum. math.Pures Appl. 21(1976), 99-113.
[11] 江泽坚，关于完全不可约算子，第五届全国泛函分析会议报告，南京，1990．
[12] 江泽坚，孙善利，关于完全不可约算子，吉林大学自然科学学报，1992，No．4．20-28．
[13] D. A.Herrero, Jiang C. L., Limits of strongly irreducible operators and the Riesz decomposition theorem Michigan Math. J.. 37(1990), 283-291.
[14] 纪友清，BIR算子的紧扰动与套代数中的BIR算子，吉林大学博士学位论文辑，长春，1998．
[15] A. Ben-Artzi. I.Gohberg,Dichotomy, discrete exponent and spectrum of block weighted shifts, Integral Equations and Operator Theory 14(1991). 613-676.
[16] D. A. Herrero. Approximation of Hilbert space operators.J. Res. Notes Math.. 72. Pitman. Boston. 1982.