K_0群与强不可约算子的近似相似不变量
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摘要
设H是可分的复Hilbert空间，L(H)表示H上的有界线性算子的全体。本文从逼近论的角度，证明了任何具有连通谱的有界线性算子T都在某个强不可约算子A的相似轨道闭包里，这里A的换位代数的半群同构于N，K_0群同构于Z，并且A′(A)/rad(A′(A))可交换，这里A′(A)表示A的换位代数，tad(A′(A))表示A′(A)的Jacobson根。
本文包含四章。第一章，介绍本文的选题背景，对已有的工作进行扼要的介绍；第二章，利用算子理论和Banach代数的综合技巧构造了几类特殊的强不可约算子；第三章，利用第二章的结果，对几类特殊的强不可约算子的直和在相似意义下分解的唯一性做了系统的研究，再利用算子逼近论的思想，K理论和相似轨道闭包定理证明了本文的主要结论；第四章，总结了本文的主要结果。
Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. In this paper, we show that: For any T∈L(H) with connected spectrum andε＞0, there exists a strongly irreducible operator A, such that‖A-T‖＜ε, V(A'(A))≌N, K_0(A'(A))≌Z, and A'(A)/rad A'(A) is commutative, where A'(A) denotes the commutant of A and rad(A'(A)) denotes the Jacobson radical of A'(A).
This paper includes four chapters. In chapter 1, we introduce the relative background in this paper and give some skeleton expressions of the original work. In chapter 2, we construct some kinds of strongly irreducible operators by operator theory and the technique of Banach algebra. In chapter 3, by some results of chapter 2, we characterize the unique strongly irreducible decomposition up to similarity of finitely direct sum of this operator. And by theory of approximation of operators, similarity orbit theorem and K theory, we get the main result of this paper. In chapter 4, we conclude the prime conclusion of this paper.
本文包含四章。第一章，介绍本文的选题背景，对已有的工作进行扼要的介绍；第二章，利用算子理论和Banach代数的综合技巧构造了几类特殊的强不可约算子；第三章，利用第二章的结果，对几类特殊的强不可约算子的直和在相似意义下分解的唯一性做了系统的研究，再利用算子逼近论的思想，K理论和相似轨道闭包定理证明了本文的主要结论；第四章，总结了本文的主要结果。
Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. In this paper, we show that: For any T∈L(H) with connected spectrum andε＞0, there exists a strongly irreducible operator A, such that‖A-T‖＜ε, V(A'(A))≌N, K_0(A'(A))≌Z, and A'(A)/rad A'(A) is commutative, where A'(A) denotes the commutant of A and rad(A'(A)) denotes the Jacobson radical of A'(A).
This paper includes four chapters. In chapter 1, we introduce the relative background in this paper and give some skeleton expressions of the original work. In chapter 2, we construct some kinds of strongly irreducible operators by operator theory and the technique of Banach algebra. In chapter 3, by some results of chapter 2, we characterize the unique strongly irreducible decomposition up to similarity of finitely direct sum of this operator. And by theory of approximation of operators, similarity orbit theorem and K theory, we get the main result of this paper. In chapter 4, we conclude the prime conclusion of this paper.
引文
[1]J.B.Conway, Subnormal Operators, Research Notes In Math, 51 Pitman Books Itd, 1981.
[2]A.L.Shields, Weight Shift Operator and Analytic Function Theory, Math.surveys 13, Amer.Math. 1974,51-128.
[3]江泽坚, Topics in Operator Theory, Seminar Report in Functional analysis, Jilin Unicversity, 1981.
[4]M.J.Cowen and R.G.Douglas, Complex Geometry and Operator theory, Acta Math, 141, 1978. 187-261.
[5]D.A.Herrero, Spectral Picture of Operator in the Cowen-Douglas Class and its closure, J.Operator Theory 10, 1987, 213-222.
[6]Q.Lin, Irreducible Cowen-Douglas Operator, Acta.Math.Sinica, New Series 10, 1991, 192-210.
[7]Zhu Kehe, Operators in Cowen-Douglas Classes, Illinois J.Math,44(2000), 767-783.
[8]C.L.Jiang and H.He, Quasisimilarity of Cowen-Douglas Operators, Science in China, Ser.A.Math.47, 2004. 297-310.
[9]G.A.Elliott, On the Classification of inductive limits of sequence of semisimple finit dimensional algebras, J.Algebra 38, 1976, 29-44.
[10]G.A.Elliott and G.H.Gong, On the Classification of C~*-algebra of real rank zeroⅡ, Annals of Mathematics., 144, 1996, 497-610.
[11]G.H.Gong, On the Classification of simple inductive limit C~*-algebras, Ⅰ.Documenta.Math. 7, 2002, 255-461.
[12]G.H.Gong, On the Classification of simple inductive limit C~*-algebrasⅡ, Preprint.
[13]Y.Cao, J.S.Fang and C.L.Jiang, K-Group of Banach Algebras and Strongly Irreducible Decompostion of Operators, J.Operator Theory, 2002(48), 235-253.
[14]C.L.Jiang, Similarity Classification of Cowen-Douglas Operators, Canad.J.Math, 2004, 56(4), 742-775.
[15]C.L.Jiang, X.Z.Guo and K.Ji, K-Group and Similarity Classifition of Cowen-Douglas operators, J.Funt.Anal.(accepted).
[16]D.A.Herrero, Approximation of Hilbert Space Operators Ⅰ, 2ed Research Notes in Math., 224 Pitman Books Itd, 1990.
[17]C.Apostol, L.A.Fialkow, D.A.Herrero and D.Voiculessu, Approximation of Hilbert Space Operators Ⅱ, Research Notes in Math., 102 Pitman Books Itd, 1984.
[18]C.L.Jiang and Z.Y.Wang, The Spectral picture and the closure of the similarity orbit of strongly irreducible operators, Intergral Equations and Operator Theory 24, 1996, 81-105.
[19]D.A.Herrero and C.L.Jiang, Limits of Strongly Irreducible Operators and the Riesz Decomposition Theorem, Michigan Math.J.37,1990.
[20]J.S.Fang, C.L.Jiang and P.Y.Wu, Direct Sums of Irreducible Operators, Studia Math., 155(1), 2003, 37-49.
[21]C.L.Jiang, S.Power and Z.Y.Wang, Biquasitriangular+small compact=strongly irreducible, Quartly, Math 18, 2001, 13-38.
[22]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
[23]K.R.Davidson, D.A.Herrero, The Jordan form of a Bitriangular Operator, J.Funct.Anal. 94, 1990, 27-73.
[24]C.L. Jiang and Z.Y. Wang,Strongly Irreducible Operators on Hilbert Space, Research Notes in Math., 389 Pitman Books, 1998.
[25]C.L. Jiang and Z.Y. Wang,Structure of Hilbert Space Operators,World Scientific,2006.
[26]Y.Cao,J.S.Fang and C.L.Jiang,K_0 Group of Banach algebras and Decomposition of strongly irreducible operators, J.Oper.Theory 48(2002),235-253.
[27]C.L. Jiang,Similarity Classitication of Cowen-Douglas Operators, Canad.J.Math.,Vol6(4),(2004),742-775.
[28]J．S．Vang and C．L．Jiang，有限强不可约分解算子，数学年刊，1999．
[29]J.B.Conway,A Course in Functional Analysis,Springer,世界图书出版公司，GTM(96)，1990．
[30]李炳仁，Banach代数，科学出版社，1997．
[31]Bernard Aupetit,A Primer on Spectral Theory, Springer-Verlag,1993.
[32]童裕孙，泛函分析教程，复旦大学出版社，2003．
[33]D.A. Herrero, Approximation of Hilbert space operators, Pitman, Research Notes in Mathematics, Series 72(1982).
[34]房军生博士学位论文，Hilbcrt空间的有界算子的Jordan标准型定理与Schur定理，2003．
[2]A.L.Shields, Weight Shift Operator and Analytic Function Theory, Math.surveys 13, Amer.Math. 1974,51-128.
[3]江泽坚, Topics in Operator Theory, Seminar Report in Functional analysis, Jilin Unicversity, 1981.
[4]M.J.Cowen and R.G.Douglas, Complex Geometry and Operator theory, Acta Math, 141, 1978. 187-261.
[5]D.A.Herrero, Spectral Picture of Operator in the Cowen-Douglas Class and its closure, J.Operator Theory 10, 1987, 213-222.
[6]Q.Lin, Irreducible Cowen-Douglas Operator, Acta.Math.Sinica, New Series 10, 1991, 192-210.
[7]Zhu Kehe, Operators in Cowen-Douglas Classes, Illinois J.Math,44(2000), 767-783.
[8]C.L.Jiang and H.He, Quasisimilarity of Cowen-Douglas Operators, Science in China, Ser.A.Math.47, 2004. 297-310.
[9]G.A.Elliott, On the Classification of inductive limits of sequence of semisimple finit dimensional algebras, J.Algebra 38, 1976, 29-44.
[10]G.A.Elliott and G.H.Gong, On the Classification of C~*-algebra of real rank zeroⅡ, Annals of Mathematics., 144, 1996, 497-610.
[11]G.H.Gong, On the Classification of simple inductive limit C~*-algebras, Ⅰ.Documenta.Math. 7, 2002, 255-461.
[12]G.H.Gong, On the Classification of simple inductive limit C~*-algebrasⅡ, Preprint.
[13]Y.Cao, J.S.Fang and C.L.Jiang, K-Group of Banach Algebras and Strongly Irreducible Decompostion of Operators, J.Operator Theory, 2002(48), 235-253.
[14]C.L.Jiang, Similarity Classification of Cowen-Douglas Operators, Canad.J.Math, 2004, 56(4), 742-775.
[15]C.L.Jiang, X.Z.Guo and K.Ji, K-Group and Similarity Classifition of Cowen-Douglas operators, J.Funt.Anal.(accepted).
[16]D.A.Herrero, Approximation of Hilbert Space Operators Ⅰ, 2ed Research Notes in Math., 224 Pitman Books Itd, 1990.
[17]C.Apostol, L.A.Fialkow, D.A.Herrero and D.Voiculessu, Approximation of Hilbert Space Operators Ⅱ, Research Notes in Math., 102 Pitman Books Itd, 1984.
[18]C.L.Jiang and Z.Y.Wang, The Spectral picture and the closure of the similarity orbit of strongly irreducible operators, Intergral Equations and Operator Theory 24, 1996, 81-105.
[19]D.A.Herrero and C.L.Jiang, Limits of Strongly Irreducible Operators and the Riesz Decomposition Theorem, Michigan Math.J.37,1990.
[20]J.S.Fang, C.L.Jiang and P.Y.Wu, Direct Sums of Irreducible Operators, Studia Math., 155(1), 2003, 37-49.
[21]C.L.Jiang, S.Power and Z.Y.Wang, Biquasitriangular+small compact=strongly irreducible, Quartly, Math 18, 2001, 13-38.
[22]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
[23]K.R.Davidson, D.A.Herrero, The Jordan form of a Bitriangular Operator, J.Funct.Anal. 94, 1990, 27-73.
[24]C.L. Jiang and Z.Y. Wang,Strongly Irreducible Operators on Hilbert Space, Research Notes in Math., 389 Pitman Books, 1998.
[25]C.L. Jiang and Z.Y. Wang,Structure of Hilbert Space Operators,World Scientific,2006.
[26]Y.Cao,J.S.Fang and C.L.Jiang,K_0 Group of Banach algebras and Decomposition of strongly irreducible operators, J.Oper.Theory 48(2002),235-253.
[27]C.L. Jiang,Similarity Classitication of Cowen-Douglas Operators, Canad.J.Math.,Vol6(4),(2004),742-775.
[28]J．S．Vang and C．L．Jiang，有限强不可约分解算子，数学年刊，1999．
[29]J.B.Conway,A Course in Functional Analysis,Springer,世界图书出版公司，GTM(96)，1990．
[30]李炳仁，Banach代数，科学出版社，1997．
[31]Bernard Aupetit,A Primer on Spectral Theory, Springer-Verlag,1993.
[32]童裕孙，泛函分析教程，复旦大学出版社，2003．
[33]D.A. Herrero, Approximation of Hilbert space operators, Pitman, Research Notes in Mathematics, Series 72(1982).
[34]房军生博士学位论文，Hilbcrt空间的有界算子的Jordan标准型定理与Schur定理，2003．