Cowen-Douglas算子的相似分类
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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
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 strong
irreducible 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
1978年,M.J.Cowen和R.G.Douglas把复几何工具引入了算子理论的
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 strong
irreducible 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
引文
[1] P. R. Halmos., Irreducible operators., Mich. Math. J., 15(1968), 215-223.
[2] A. Azoff, C. K. Fong and F. Gilfeather., A reduction theory for non-self-adjoint operator algebras., Trans. Amer. Math. Soc, 224 (1976), 351-366.
[3] D. Voiculescu., A non-commutative Weyl-von Neumann theorem., Rev. Roum. Math. Pures Appl., 21 (1976), 97-113.
[4] H. Grauert., Analytische Faserungen uber holomorph vollstandigen Raumen., Math. Ann., 135(1958), 263-273.
[5] K. R. Davidson and D. A. Herrero., The Jordan form of a bitriangular operator., J. Funct. Annal. 94(1990), 27-73.
[6] F. Gilfeather., Strong reducibility of operators., Ind. Univ. Math. J., 22(1972), 393-397.
[7] 江泽坚,关于BIR算子的若干问题Ⅰ,Ⅱ,在吉林大学泛函分析会议上的报告,长春,1979.
[8] A. L. Shields., Weighted Shift operators and analytic function theory., Math. Surveys, no. 13 Amer. Soc. Providence, R. I. 49-128.
[9] 郭献洲,一类算子的换位代数的K-群,待发表.
[10] 房军生,Hilbert空间上有界算子的Jordan标准型定理和Schur定理,中国科学院博士论文,2003.
[11] 江泽坚,关于线性算子的结构,在全国算子理论会议上的报告,九江.1981.
[12] 蒋春澜,正规算子相似于不可约算子的充要条件,在第五届全国泛函分析会议上的报告,南京.
[13] C. K. Fong and C. L. Jiang., Normal Operators Similar to Irreducible Operators., Acta Math. Sinica. 1994, Vol. 10, No. 2, 132-135.
[14] D. A. Herrero and C. J. Jiang., Limits of strongly irreducible operators and the Riesz decomposition theorem., Micah, Math. J, 37(1990), 283-291.
[15] Y. Q. Ji and C. L. Jiang., Operators with connected spectrum + compact operators = strongly irreducible operators., Sci. China Ser. A, 42 (1999), no. 9, 925-931.
[16] J. S. Fang, C. L. Jiang and P. Y. Wu., Direct Sum of irreducible operators., Studia Math., 155(1), 2003, 37-49.
[17] C. L. Jiang and Z. Y. Wang., A class of strongly irreducible operators with nice property., J. Operator Theory, 36 (1996), no. 1, 3-19.
[18] C. L. Jiang and J. X. Li., The irreducible decomposition of Cowen-Douglas operators and operator weighted shifts., Acta-Sci-Math. (Szeged), (2000) no 3-4, 679-695.
[19] K. R. Davidson and D. A. Herrero., The Jordan form of bitriangular operator, J. Funct. Anal. 94(1990), 27-73.
[20] D. A. Herrero., Approximation of Hilbert space operators., Pitman, Research Notes in Mathematics, Series 72(1982).
[21] K.R. Davidson., C*-algebra by example., Field institute Monographs 6, Amer. Math., Soc, Providence, RI, 1996.
[22] C. L. Jiang., Approximation of direct sum of strongly irreducible operators., Northeastern Math. J. 5(1989), 253-254.
[23] G. H. Gong., Banach reducibility of bilateral weighted shifts., Northeastern Math. J., 3(1987), 17-36.
[24] C.Z. Zou., Banach irreducibility of contraction operators of the class C0., Acta Sci. Nat. Univ. Jilinensis, (1981), No.2, 43-49.
[25] C.K. Fong and C.L. Jiang., Approximation by Jordan type operators., Houston. J. Math., 19(1993), 51-62.
[26] C.L. Jiang, S. Power and Z.Y. Wang., Biquasitriangular operators have strongly irreducible perturbations., Quarterly. J. Math(2000) 51, 353-369.
[27] C.L. Jiang and P.Y. Wu., Sums of strongly irreducible operators., Houston Journal of Mathematics, 24, (1998) 467-481.
[28] C.I. Hsin., Co-contractions quasisimilar to irreducible ones., Taiwanes J. Math. 2002, 6, 39-58.
[29] C.L. Jiang, S.H. Sun and Z.Y. Wang., Essentially normal operator + compact operator = strongly irreducible operator. Trans., Amer. Soci 349(1997), 217- 233.
[30] X.M. Chen and C.L. Jiang., The closure of unitary orbit for strongly irreducible operators in a class of nest algebras., Integral Equations Operator Theory 27(1997), 215-293.
[31] C.L. Jiang and K.Y. Guo., The strong irreducibility of hyponormal operators and BereZin perturbation. Acta. Sci. Math, (Szeged) 64(1998), 231-248.
[32] C.L. Jiang and Z.Y. Wang., Strongly irreducible operators on Hilbert space., Pitman Research Notes in Mathematics Series, 389(1998), Longman, Harlow.
[33] Y.Q. Ji, C.L. Jiang and and Z.Y. Wang., The closure of unitary orbit of the set of strongly irreducible operators in non-wellorded nest algebras, J. Operator Theory, 44(2000), Nol, 25-41.
[34] Y.Q. Ji, C.L. Jiang and and Z.Y. Wang., The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set., Mich, Math. J.44(1997), 85-98.
[35] Z.Y. Wang and Y.F. Xue., On the unit finite (SI) decomposition of the Jordan operator (?) S(?) for certain inner functions (?)., Integral Equation Operator Theory 36(2000), 370-377.
[36] Y.Q. Ji and C.L. Jiang., Small compact perturbation of strongly irreducible operators., Integral Equations Operator Theory 48(2001), 41-76.
[37] Y. Cao, J.S. Fang and C.L. Jiang., K0-Group of Banach algebras and Decomposition of strongly irreducible operators, J.Oper. Theory 48(2002), 235-253.
[38] J.S. Fang and C.L. Jiang., K0-Group of Banach algebras and Decomposition of strongly irreducible operators, (II), (preprint).
[39] Z.J. Jiang and S.L. Sun., On completely irreducible operators., Acta. Scientiarum Naturalium Univ. Jilinensis (4), 1992, 20-29.
[40] Z.Y. Wang and C.L. Jiang., The recent study of completely irreducible operators., Acta. Scientiarum, Naturalium University, Jilinenis, (3)(1995), 30-34.
[41] C.L. Jiang., Similarity, reducibility and approximation of Cowen-Douglas operators., J.Operator Theory, 22(1993), 361-397.
[42] C.L. Jiang and Z.Y. Wang., The spectral picture and the closure of similarity orbit of strongly irreducible operators., Integral Equation Operator Theory 24(1996), 81-105.
[43] C.L. Jiang., Similarity, Reducibility And Approximation of The Cowen-Douglas Operators., J.Operator Theory, 32(1994), 77-89.
[44] F.F. Bonsall and J. Duncan., Complete normed algebras., Springer-Verlag, Heidelberg, 1973.
[45] 李炳仁, Banach 代数,科学出版社, 1997.
[46] M.J. Cowen and R.G. Douglas., Complex geometry and operator theory., Acta. Math. 141(1978), 187-261.
[47] D.A. Herrero., Spectral pictures of operators in the Cowen-Douglas class B_n(?) and its closure., J. Operator Theory, 128(1987), 213-222.
[48] C.L. Jiang., Similarity Classification of Cowen-Douglas Operators., Canada J. Math.(2004), Vol 156, No.4, 742-775.
[49 ] C.L. Jiang., Commutant Algebras of strongly irreducible operators., J.Funct Anal, (to appear).
[50 ] B. Behncke., Structure of certain non-normal operators., J. Math, Mech, 18(1968), 103-107.
[51 ] T. Kato., Perturbation Theory for linear operator., Berlin, Heidellberg, New York, Tokyo; Springer-Verlag, 1964.
[52 ] C.Q. Yan., Irreducible decomposition of Cowen-Douglas opcrators., Northeast Math. J. 42(1993), 261-267.
[53 ] G.L. Yu., A completely unitary invariant and reducibility of a class of operators in B_2(?)., Northeast Math. J.3(1987), 410-417.
[54 ] B. Blackadar., K-theory for operator algebras., Springer-Verlag, Heidelberg, 1986.
[55 ] D.A. Herrero., Approximate of Hilbert space operators I, 2ed Research Notes in Math, 224 Pitman Books ltd, 1990.
[56 ] C. Apostol, L.A. Fialkow, D.A. Herrero and D. Voiculescu., Approximation of Hilbert space operators II., Research Notes in Math. 102 Pitman Booksltd, 1984.
[57 ] A.L. Shields., Weights shift operators and analytic function theory., Math. Surveys 13, Amer. Math. Soc. 1974, 51-128.
[58 ] J.B. Convey., Subnormal operators., Research Notes in Math., 51 Pitman Books ltd, 1981.
[59] G. A. Elliott., On the classification of inductive limits of sequence of semisimple finite-dimensional algebras., J. Algebra 38(1976), 29-44.
[60] G. A. Elliott., The classification problem for amenable C~*-algebras., Proceedings. of the International Coagress of Mathematics., Vol. 1, 2., Birkhauser, Bssel, 1995, 922-932.
[61] G. Elliott, and G. Gong, On the classification of C~*-algebras of real rank Zero Ⅱ., Annals of Mathematics., 144(1996), 497-610.
[62] G. Elliott, and G. Gong, Inductive limit of limit of matrix algebras over tovas, American J of Math., 1996, 187-205.
[63] G. Ell(?)ott, G. Gong and L. Li, On the classification of simple inductive limit C~*-algebras., Ⅱ: The Isomorphism Theorem., Preprint.
[64] C. R. Putnam., Commutation properties of Hilbert space operators and related topics., Ergebnisse. der, Math. Und. Threr Grenz 36(New, York), Springer-Valger, (1967).
[65] C. L. Jiang and H. He., Unitary Equivalence, Similarity and Calculation of K_0-Group., preprint.
[66] C. L. Jiang and H. He., Quasisimilarity of Cowen-Douglas operators., Science in China, Ser. A, Math. 47, 2004, 297-310.
[67] 蒋春澜,郭献洲,杨永法,幂零算子相似于不可约算子,中国科学,2001,6.
[2] A. Azoff, C. K. Fong and F. Gilfeather., A reduction theory for non-self-adjoint operator algebras., Trans. Amer. Math. Soc, 224 (1976), 351-366.
[3] D. Voiculescu., A non-commutative Weyl-von Neumann theorem., Rev. Roum. Math. Pures Appl., 21 (1976), 97-113.
[4] H. Grauert., Analytische Faserungen uber holomorph vollstandigen Raumen., Math. Ann., 135(1958), 263-273.
[5] K. R. Davidson and D. A. Herrero., The Jordan form of a bitriangular operator., J. Funct. Annal. 94(1990), 27-73.
[6] F. Gilfeather., Strong reducibility of operators., Ind. Univ. Math. J., 22(1972), 393-397.
[7] 江泽坚,关于BIR算子的若干问题Ⅰ,Ⅱ,在吉林大学泛函分析会议上的报告,长春,1979.
[8] A. L. Shields., Weighted Shift operators and analytic function theory., Math. Surveys, no. 13 Amer. Soc. Providence, R. I. 49-128.
[9] 郭献洲,一类算子的换位代数的K-群,待发表.
[10] 房军生,Hilbert空间上有界算子的Jordan标准型定理和Schur定理,中国科学院博士论文,2003.
[11] 江泽坚,关于线性算子的结构,在全国算子理论会议上的报告,九江.1981.
[12] 蒋春澜,正规算子相似于不可约算子的充要条件,在第五届全国泛函分析会议上的报告,南京.
[13] C. K. Fong and C. L. Jiang., Normal Operators Similar to Irreducible Operators., Acta Math. Sinica. 1994, Vol. 10, No. 2, 132-135.
[14] D. A. Herrero and C. J. Jiang., Limits of strongly irreducible operators and the Riesz decomposition theorem., Micah, Math. J, 37(1990), 283-291.
[15] Y. Q. Ji and C. L. Jiang., Operators with connected spectrum + compact operators = strongly irreducible operators., Sci. China Ser. A, 42 (1999), no. 9, 925-931.
[16] J. S. Fang, C. L. Jiang and P. Y. Wu., Direct Sum of irreducible operators., Studia Math., 155(1), 2003, 37-49.
[17] C. L. Jiang and Z. Y. Wang., A class of strongly irreducible operators with nice property., J. Operator Theory, 36 (1996), no. 1, 3-19.
[18] C. L. Jiang and J. X. Li., The irreducible decomposition of Cowen-Douglas operators and operator weighted shifts., Acta-Sci-Math. (Szeged), (2000) no 3-4, 679-695.
[19] K. R. Davidson and D. A. Herrero., The Jordan form of bitriangular operator, J. Funct. Anal. 94(1990), 27-73.
[20] D. A. Herrero., Approximation of Hilbert space operators., Pitman, Research Notes in Mathematics, Series 72(1982).
[21] K.R. Davidson., C*-algebra by example., Field institute Monographs 6, Amer. Math., Soc, Providence, RI, 1996.
[22] C. L. Jiang., Approximation of direct sum of strongly irreducible operators., Northeastern Math. J. 5(1989), 253-254.
[23] G. H. Gong., Banach reducibility of bilateral weighted shifts., Northeastern Math. J., 3(1987), 17-36.
[24] C.Z. Zou., Banach irreducibility of contraction operators of the class C0., Acta Sci. Nat. Univ. Jilinensis, (1981), No.2, 43-49.
[25] C.K. Fong and C.L. Jiang., Approximation by Jordan type operators., Houston. J. Math., 19(1993), 51-62.
[26] C.L. Jiang, S. Power and Z.Y. Wang., Biquasitriangular operators have strongly irreducible perturbations., Quarterly. J. Math(2000) 51, 353-369.
[27] C.L. Jiang and P.Y. Wu., Sums of strongly irreducible operators., Houston Journal of Mathematics, 24, (1998) 467-481.
[28] C.I. Hsin., Co-contractions quasisimilar to irreducible ones., Taiwanes J. Math. 2002, 6, 39-58.
[29] C.L. Jiang, S.H. Sun and Z.Y. Wang., Essentially normal operator + compact operator = strongly irreducible operator. Trans., Amer. Soci 349(1997), 217- 233.
[30] X.M. Chen and C.L. Jiang., The closure of unitary orbit for strongly irreducible operators in a class of nest algebras., Integral Equations Operator Theory 27(1997), 215-293.
[31] C.L. Jiang and K.Y. Guo., The strong irreducibility of hyponormal operators and BereZin perturbation. Acta. Sci. Math, (Szeged) 64(1998), 231-248.
[32] C.L. Jiang and Z.Y. Wang., Strongly irreducible operators on Hilbert space., Pitman Research Notes in Mathematics Series, 389(1998), Longman, Harlow.
[33] Y.Q. Ji, C.L. Jiang and and Z.Y. Wang., The closure of unitary orbit of the set of strongly irreducible operators in non-wellorded nest algebras, J. Operator Theory, 44(2000), Nol, 25-41.
[34] Y.Q. Ji, C.L. Jiang and and Z.Y. Wang., The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set., Mich, Math. J.44(1997), 85-98.
[35] Z.Y. Wang and Y.F. Xue., On the unit finite (SI) decomposition of the Jordan operator (?) S(?) for certain inner functions (?)., Integral Equation Operator Theory 36(2000), 370-377.
[36] Y.Q. Ji and C.L. Jiang., Small compact perturbation of strongly irreducible operators., Integral Equations Operator Theory 48(2001), 41-76.
[37] Y. Cao, J.S. Fang and C.L. Jiang., K0-Group of Banach algebras and Decomposition of strongly irreducible operators, J.Oper. Theory 48(2002), 235-253.
[38] J.S. Fang and C.L. Jiang., K0-Group of Banach algebras and Decomposition of strongly irreducible operators, (II), (preprint).
[39] Z.J. Jiang and S.L. Sun., On completely irreducible operators., Acta. Scientiarum Naturalium Univ. Jilinensis (4), 1992, 20-29.
[40] Z.Y. Wang and C.L. Jiang., The recent study of completely irreducible operators., Acta. Scientiarum, Naturalium University, Jilinenis, (3)(1995), 30-34.
[41] C.L. Jiang., Similarity, reducibility and approximation of Cowen-Douglas operators., J.Operator Theory, 22(1993), 361-397.
[42] C.L. Jiang and Z.Y. Wang., The spectral picture and the closure of similarity orbit of strongly irreducible operators., Integral Equation Operator Theory 24(1996), 81-105.
[43] C.L. Jiang., Similarity, Reducibility And Approximation of The Cowen-Douglas Operators., J.Operator Theory, 32(1994), 77-89.
[44] F.F. Bonsall and J. Duncan., Complete normed algebras., Springer-Verlag, Heidelberg, 1973.
[45] 李炳仁, Banach 代数,科学出版社, 1997.
[46] M.J. Cowen and R.G. Douglas., Complex geometry and operator theory., Acta. Math. 141(1978), 187-261.
[47] D.A. Herrero., Spectral pictures of operators in the Cowen-Douglas class B_n(?) and its closure., J. Operator Theory, 128(1987), 213-222.
[48] C.L. Jiang., Similarity Classification of Cowen-Douglas Operators., Canada J. Math.(2004), Vol 156, No.4, 742-775.
[49 ] C.L. Jiang., Commutant Algebras of strongly irreducible operators., J.Funct Anal, (to appear).
[50 ] B. Behncke., Structure of certain non-normal operators., J. Math, Mech, 18(1968), 103-107.
[51 ] T. Kato., Perturbation Theory for linear operator., Berlin, Heidellberg, New York, Tokyo; Springer-Verlag, 1964.
[52 ] C.Q. Yan., Irreducible decomposition of Cowen-Douglas opcrators., Northeast Math. J. 42(1993), 261-267.
[53 ] G.L. Yu., A completely unitary invariant and reducibility of a class of operators in B_2(?)., Northeast Math. J.3(1987), 410-417.
[54 ] B. Blackadar., K-theory for operator algebras., Springer-Verlag, Heidelberg, 1986.
[55 ] D.A. Herrero., Approximate of Hilbert space operators I, 2ed Research Notes in Math, 224 Pitman Books ltd, 1990.
[56 ] C. Apostol, L.A. Fialkow, D.A. Herrero and D. Voiculescu., Approximation of Hilbert space operators II., Research Notes in Math. 102 Pitman Booksltd, 1984.
[57 ] A.L. Shields., Weights shift operators and analytic function theory., Math. Surveys 13, Amer. Math. Soc. 1974, 51-128.
[58 ] J.B. Convey., Subnormal operators., Research Notes in Math., 51 Pitman Books ltd, 1981.
[59] G. A. Elliott., On the classification of inductive limits of sequence of semisimple finite-dimensional algebras., J. Algebra 38(1976), 29-44.
[60] G. A. Elliott., The classification problem for amenable C~*-algebras., Proceedings. of the International Coagress of Mathematics., Vol. 1, 2., Birkhauser, Bssel, 1995, 922-932.
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