| 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...
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