【作者】 张潇；

【导师】 纪友清；

【作者基本信息】 吉林大学 ， 基础数学， 2004， 硕士

【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（l²（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IWl²（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IWl²（Cⁿ）。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∈IWl²（Cⁿ），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IWl²（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IWl²（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?更多还原

【Abstract】 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)) , andthen 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 Sis 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 lemmaLemma 2.1 For eachHi, there exists an OXB { } . such that every Wi can be represented asThus, 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 followingLemma 2.3 Let S IW2(H).S ~ {Wi}. then S is unitarily equivalent towhere 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 obtainLemma3.6 .A if 1(A) r(B] = o. then C RanTAB of each operator C of the followingformFrom 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 theoremtheorem 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 remainedquestion (1) for 5" G I\V(2(H).th更多还原