【作者】 郭献洲；

【导师】 蒋春澜；

【作者基本信息】 河北师范大学 ， 基础数学， 2005， 博士

【摘要】 对一个复的、可分的Hilbert空间(?),设(?)((?))表示作用在(?)上的全体有界线性算子。算子理论中的一个最基本的问题是寻找两个算子的完全相似不变量,即对(?)((?))中的算子A和B,什么时候存在(?)((?))中的一个可逆算子X,使得XA=BX。当(?)是有限维空间时,由著名的Jordan标准型定理知道,算子的特征值及其广义特征子空间是算子的完全相似不变量。当(?)是无穷维空间时,寻找一般算子的完全相似不变量是几乎不可能的,人们只能对于不同的算子类寻找不同的完全相似不变量,或者寻找近似的完全相似不变量。20世纪70年代,J.B.Conway表明了对*-循环的正规算子和次正规算子,由其诱导的标量值谱测度等价性是其完全相似不变量。20世纪80年代,A.L.Shields表明,内射加权移位算子的权序列的比率是其完全相似不变量。20世纪70年代到80年代,以C.Apostol、L.A.Fialkow、D.A.Herrero和D.Voiculesu为代表的数学家通过引入指标理论及精细谱图形的工具,建立了著名的相似轨道闭包定理,证明了算子的精细谱图形是算子的相似轨道闭包意义下的完全不变量,但他们也表明精细谱图形不是算子的完全相似不变量。1979年,江泽坚提出强不可约算子可以被看成无穷维空间上的Jordan块的替代物。 20世纪90年代,蒋春澜、王宗尧、D.A.Herrero、纪友清、吴培元、C.K.Fong、S.Power、K.R.Davidson等五批海内外学者以强不可约算子为基本模型,获得了一系列算子近似相似不变量的结果。其代表性的工作是蒋春澜和王宗尧给出的强不可约算子的谱图象定理。寻找强不可约算子的完全相似不变量和算子的强不可约分解在相似意义下的唯一性是密切相关的,是算子理论的一个基本问题。 1978年,M.J.Cowen和R.G.Douglas把复几何工具引入了算子理论的更多还原

【Abstract】 Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. A basic problem in operator theory is to determine when two operators A and B in L(H) are similar, that is, when does there exist an invertible operator X on H satisfying XA = BX. One of the most important problems in operator theory is to find a similarity invariant which can be used to tell whether two operators are similar.When H is a finite-dimensional Hilbert space, we know from the theorem on Jordan forms that the eigenvalues and the generalized eigenspaces of an operator form a complete set of similarity invariants. When H is an infinite-dimensional Hilbert space, in a real sense the problem has no general solution, but one can restrict attention to special classes of operators. For two star-cyclic normal operators (or star-cyclic subnormal operators) A and B, J.B. Conwayl[58] showed that A and B are similar if and only if the scalar valued spectral measures induced by A and B are equivalent. A.L. Shields[57] characterized similarity for injective weighted shift operators. From 1970s to 1980s, using the index theory and sophisticated spectral graph, C. Apostol 、L.A. Fialkow 、 D.A. Herrero and D. Voiculesul[55],[56] established the famous closure theorem of similarity orbit. They proved that the sophisticated spectral graph of operator is the complete invariant up to closure of similarity orbit, but they also declared that it is not the complete similarity invariant. In 1979, Professor Zejian Jiang[7] brought up strong irreducible operator may be viewed as a countpart of Jordan block in the infinite-dimensional space. From then on, Chunlan Jiang, Zongyao Wang, D.A. Herrero, Youqing Ji, Peiyuan Wu, C.K. Fong, S.Power and K.R. Davidson[10][16][26][32][42] obtained a series of theorems of approximate similaxity invariant using the strongirreducible operator to be a model. To find the complete similarity invariants of operators is closely related to the uniqueness of strong irreducible decomposition up to similarity. It is a basic problem in operator theory.In 1978, M.J. Cowen and R.G. Douglas’46! drew complex geometry into the study of operator theory, and defined a class of operators-Cowen-Douglas operators by holomorphic bundle. Cowen-Douglas operators form an especially rich class. First of all, M.J. Cowen and R.G. Douglas established Clabi rigidity theorem in holomorphic complete bundle, and they defined a curvature function. And then they proved the curvature function is the complete unitary invariants for Cowen-Douglas operators. Further, M.J. Cowen and R.G. Douglas conjectured the curvature is the complete similarity invariants for Cowen-Douglas operators with index 1, and hoped to find the complete similarity invariants for holomorphic bundle by curvature function. But a counter-example proved that the conjecture is wrong. From 1980s to 1990s, one began to find the similarity invariants for Cowen-Douglas operators’47’1’48!’’66’. In 1970s, G. Elliott proved that ordered i^o-group is the complete similarity invariants for AF-algebra. In the effort of many mathematicians(especially H. Lin’s and D. Dadarlat’s outstanding work), G. Elliott, G. Gong and L. Li’60’’’61’-’62’’’63’ have successfully classified simple AH-algebras using the scaled order K-group. In the spirit of the above work, Chunlan Jiang and his cooperators drew the if-group into the classification study of commutants of operators, and proved the theorem CFJ’37’. They proved the relation of the uniqueness of strong irreducible decomposition up to similarity and the /Co-group of commutants of operators. In 2004, using techniques of complex geometry Chunlan Jiang’48’ computed the if-groups of commutants of strongly irreducible Cowen-Douglas operators, and proved that the ordered /^o-group of the commutant更多还原