根算子及相关问题探讨
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摘要
设X是一个无限维Banach空间,B(X)是X上的有界线性算子全体构成的Banach代数,K(X)是B(X)中的紧算子理想,π：B(X)→C(X)表示由B(X)到Calkin代数C(X)：=B(X)/K(X)的典则商同态映射,于是产生了B(X)中的一个子集一根算子集I(X),其中每个T∈I(X),就定义为π(T)是Calkin代数C(X)中的Jacobson根者.已知I(X)形成了B(X)中的一个理想.本文主要就是围绕着根算子I(X)的等价特征和性质这一主题而展开的.
全文共分为四章,第一章主要是介绍该课题研究的现状,本文所采用的研究思想方法及所得的主要结果.第二章主要是从一般代数A的角度出发来研究它的Jacobson根Rad(A)的性质与特征,完善了文献[25]中的定理1.4.14的证明,并从Z(A)(Z(A)={a∈A：ab-ba∈Rad(A)})的角度对Rad(A)进行研究,得到了Z(A)∩Q(A)=Rad(A)这一主要结果.在这章的最后我们推广了Rad(B(X))={O}这一命题,得到了相关的结论.在第三章中,一方面我们在第一章研究的基础上得到根算子相应的等价命题,另一方面我们从广义算子理想、算子半群及空间理想的角度出发来探讨根算子的性质特征,并探讨了I(X),R(X),Q(X)三者的包含关系.在这章的最后,我们应用Z(A)∩Q(A)=Rad(A)这一结果来研究遗传不可分解空间上的强不可约算子的性质.在第四章中,我们为了解决Gonzalez等在文献[35]中的问题2.10(设A为一算子理想,若Space(A)=F,是否A(?)I?其中I为根算子理想),我们定义了空间理想Z类可补奇异算子的概念,得到了一个与文献[35]中的问题2.10等价的命题,接着我们定义了与空间理想Z类可补奇异算子类似的弱空间理想Z类奇异算子及弱空间理想Z类余奇异算子的概念,并应用前面所得的结论分别探讨了它们根的性质.
Let X be a an infinite dimensional Banach space, and B(X) is the Banach al-gebra formed by the whole of bounded linear operators on X. K(X) is the compact operator ideal in B(X).π:B(X)→C(X) represents the canonical quotient ho-momorphism from B(X) to C(X),so it generates a subset in B(X)-radical operators set(I(X)).For each T∈I(X),π(T) belongs to the Jacobson radical of C(X). we know I(X) forming a operator ideal in B(X). This thesis is organized around the equivalent characteristics and the nature of radical operators.
The full-text is divided into four chapters. The first chapter mainly introduces the topic of the status quo, the approach to the study methods and main results of our research. The second chapter studies the nature of the general algebraic Jacobson radical. We complete the proof of the theorem 1.4.14 in the literature [25],and research the nature of Jacobson radical from the perspective of Z(A),so we obtain the result that Z(A)∩Q(A)=Rad(A).In the end of this chapter, we promote the proposition of Rad(B(X))={0}.In the third chapter,on the one hand, we discuss the inclusive relation among I(X)、R(X) and Q(X).On the other hand,we study the nature of radical operators from operator ideals、operator semigroups and space ideal.In the fourth chapter,we define the concept of space ideal Z complement singluar operators in order to slove the question 2.10 in literature[35](Let A is a operator ideal,if Space(A)=F,A(?)I?)and we obtain a equivalent proposition about question 2.10 in literature[35].Then,we define the concept of weak space ideal Z singluar operators and weak space ideal Z cosingluar operators. We research the nature of the two operators and the radical of the two operators respectively.
全文共分为四章,第一章主要是介绍该课题研究的现状,本文所采用的研究思想方法及所得的主要结果.第二章主要是从一般代数A的角度出发来研究它的Jacobson根Rad(A)的性质与特征,完善了文献[25]中的定理1.4.14的证明,并从Z(A)(Z(A)={a∈A：ab-ba∈Rad(A)})的角度对Rad(A)进行研究,得到了Z(A)∩Q(A)=Rad(A)这一主要结果.在这章的最后我们推广了Rad(B(X))={O}这一命题,得到了相关的结论.在第三章中,一方面我们在第一章研究的基础上得到根算子相应的等价命题,另一方面我们从广义算子理想、算子半群及空间理想的角度出发来探讨根算子的性质特征,并探讨了I(X),R(X),Q(X)三者的包含关系.在这章的最后,我们应用Z(A)∩Q(A)=Rad(A)这一结果来研究遗传不可分解空间上的强不可约算子的性质.在第四章中,我们为了解决Gonzalez等在文献[35]中的问题2.10(设A为一算子理想,若Space(A)=F,是否A(?)I?其中I为根算子理想),我们定义了空间理想Z类可补奇异算子的概念,得到了一个与文献[35]中的问题2.10等价的命题,接着我们定义了与空间理想Z类可补奇异算子类似的弱空间理想Z类奇异算子及弱空间理想Z类余奇异算子的概念,并应用前面所得的结论分别探讨了它们根的性质.
Let X be a an infinite dimensional Banach space, and B(X) is the Banach al-gebra formed by the whole of bounded linear operators on X. K(X) is the compact operator ideal in B(X).π:B(X)→C(X) represents the canonical quotient ho-momorphism from B(X) to C(X),so it generates a subset in B(X)-radical operators set(I(X)).For each T∈I(X),π(T) belongs to the Jacobson radical of C(X). we know I(X) forming a operator ideal in B(X). This thesis is organized around the equivalent characteristics and the nature of radical operators.
The full-text is divided into four chapters. The first chapter mainly introduces the topic of the status quo, the approach to the study methods and main results of our research. The second chapter studies the nature of the general algebraic Jacobson radical. We complete the proof of the theorem 1.4.14 in the literature [25],and research the nature of Jacobson radical from the perspective of Z(A),so we obtain the result that Z(A)∩Q(A)=Rad(A).In the end of this chapter, we promote the proposition of Rad(B(X))={0}.In the third chapter,on the one hand, we discuss the inclusive relation among I(X)、R(X) and Q(X).On the other hand,we study the nature of radical operators from operator ideals、operator semigroups and space ideal.In the fourth chapter,we define the concept of space ideal Z complement singluar operators in order to slove the question 2.10 in literature[35](Let A is a operator ideal,if Space(A)=F,A(?)I?)and we obtain a equivalent proposition about question 2.10 in literature[35].Then,we define the concept of weak space ideal Z singluar operators and weak space ideal Z cosingluar operators. We research the nature of the two operators and the radical of the two operators respectively.
引文
[1]钟怀杰.巴拿赫空间结构和算子理想.北京：科学出版社,2005.
[2]H.P.Rosenthal.On totally incomparable Banach spaces,J.Punct.Anal.4.167-175(1969).
[3]T.Alvarez,M.Gonzalez and V.M.Onieva.Totally incomparable Banach spaces and three-space Banach space ideals,Math.Nachr.131 (1987).
[4]M.Gonzalez.On essentially incomparable Banach spaces,Math.Z.215,621-629(1994).
[5]P.Aiena,and Manuel Gonzalez.Examples of improjective operators,MathZ.233,471-479(2000).
[6]W.B.Johnson and Lindenstrauss J.eds.handbook of Geometry of Banach spaces, Volume 1. Amsterdam:North-Holland,2001.
[7]W. J. Bicker. Well-bounded operators of type(B) in H. I. spaces. Acta Sci. Math. (Szeged),1994,59:475-488.
[8]F.Rabiger and W.J.Ricker.Aspects of operator theory in hereditarily indecomposable Banach spaces.Rend.Sem.Math.Fis.Milano,1997,67:139-149.
[9]G.Androulakis and T.Schlumprecht.Strictly singular,non-compact operators exist on the space of Gowers and Maurey. J.London Math.Soc.2001,64(2):655-674.
[10]N.J.Laustsen.K-theory for algebras of operators on Banach spaces.J.London Math.Soc,1999,59(2):715-728.
[11]A.Zsak.A Banach space whose operator algebra has-group Q.Proc.London Math.Soc.2002,84(3):747-768.
[12]Aupetit. A Primer on Spectral Theory. Springer-verlag. Springer-Verlag New York,Inc.1991.
[13]张云南,钟怀杰,苏维钢.Banach空间上算子代数的群.中国科学,2005,A35(9)：1008-1018(英文版2006,49(2)：233-244).
[14]R.J.Whitley.Strictly singular operators and their conjugates,T.A.M.S.1964(113):252-261.
[15]J.Lindenstrauss.and L.Tzafrifi.1977 Classical Banach spaces I.Sequence spaces.New York.Springer.
[16]P.Aiena and M.Gonzalez.Essentially incomparable Banach spaces and Fredholm theory.Pro.R.Ir.Acad.Vol.93A,No.l,49-59(1993).
[17]俞鑫泰.Banach空间选论,华东师范大学出版社.
[18]Zhong huaijie.Significant development in the structural theory of Banach spaces based on the series of results by Gowers and Maurey, Adv. Math(China)20(2000),1-18. MR.98c:46013.
[19]A.Pietro,M.Gonzalez and M.Antonio.Operator semigroups in Banach space the-ory.Bullettino U.M.I.(8)4-B(2001),157-205.
[20]A.Pietsh.1980 Operator Ideals, Amsterdam-New York-Oxford. North-Holland.
[21]Chen Dongyang and Zhong Huaijie.The Classification of G-M type Banach spaces And structure of operators on them.
[22]刘绍学.近世代数基础.高等教育出版社.1999.
[23]H. R. Dowson,Spectral theory of linear operators. London Math. Soc-.Mono-graph,1978. Bull.London. Math. Soc.29(1997)338-344.
[24]定光桂.巴拿赫空间引论,科学出版社,1983.
[25]李炳仁.Banach代数,科学出版社,1997.
[26]邓生华.Banach代数中理想概念的推广,西南师范大学学报,1999.
[27]A.Pietsch.Inessential operators in Banach spaces.Integral Equations and Operator Theory,l(1978):589-591.
[28]Yu.V.Turovskii and V.S.Shulman,Radicals in Banach Algebras and Some Problems. in the Theory of Radical Banach Algebras, Functional Analysis and Its Applica-tions,Vol.35,No.4, pp.312-314,2001.
[29]A. Pietro and M.Gonzalez. Essentially incomparable B.S and Fredholm theory.Proc. R. Ir. Acad.39A:1(1993):49-59.
[30]M.Schechter.Riesz operators and Fredholm perturbations.Bull.AMS.74(1968):1139-1144.
[31]P.Aiena.On Riesz and Inessential Operators,Math.Z.201,521-528(1989).
[32]W. T. Gowers,B.Maurey,Banach spaces with small spaces of operators,Math. Ann. 307,543-568(1997).
[33]陈东阳,钟怀杰.本性不可比的Banach空间,应用泛函分析学报,第3卷,.第3期,2001年9月.
[34]M.Gonzalez.The perturbation classes problem in Fredholm theory,Journal of Func-tional Analysis 200 (2003)65-70.
[35]P. Aiena and M.Gonzalez.On the operators which are invertible modulo an operator ideal,Bull.Austral.Math.Soc.Vol.64(2001) [233-243].
[36]陈剑岚,钟怀杰.超投影空问与严格余奇异算子,福建师范大学学报,1997,13(4)：19-21.
[37]J.Zemanek.A note on the radical of a Banach algebra,manuscripta math.20,191-196 (1977).
[38]钟怀杰.关于非平凡的黎斯算子,福建师范大学学报,1993,9(3)：10-14.
[39]林丽琼,钟怀杰.关于弱自反严格奇异算子,福建师范大学学报,第3卷,第34期,2006年6月.
[40]林丽琼,钟怀杰.关于弱自反严格余奇异算子,福州大学学报,第35卷,第1期,2007年2月.
[41]Manuel Gonzalez.The perturbation classes problem in Fredholm theory,Journal of Functional Analysis 200 (2003)65-70.
[42]P.Aiena and M.Gonzalez.On inessential and improjective operators. Studia Math.l31(1998),271-287.
[43]钟怀杰,陈东晓,陈剑岚.某些G-M型Banach空间及其上算子代数的K群.中国科学,2003,A33(4)：338-353(英文版2004,47(3)：372-392).
[44]P. Aiena,M. Gonzalez,Incomparable Banach spaces and operator semigroups. Arch. Math.79(2002)372-378.
[45]P. Aiena,An internal characterization of inessential operators,Proc. Amer. Math. Soc. 102(1988),625-626.
[46]T. Alvarez and M. Gonzalez. Some examples of tauberian operators,Proc. Amer. Math. Soc.111(1991),1023-1027.
[47]M.Basallote.Representabilidad finita por cocientes y operators,Doctoral Thesis,Univ. Sevilla,1998.
[48]W. T. Gowers and B. Maurey. The unconditional basic sequence problem[J]. JAMS,1993,6(3):851-874.
[49]Pietro Aiena.Fredholm and local spectral theory with applications to multipli-ers [M].Kluwer Academic Publishers,2004.
[50]Rudin W.Functional analysis[M].2nd ed.New York:McGraw-Hill,1991.
[51]陈焕艮.关于Jacobson根的特征.广西师范大学学报,第11卷,第2期,1993.
[52][苏]O.A.德洛兹德,B.B.基里钦柯著,刘绍学.张英伯译.有限维代数[M].北京师范大学出版社1984.
[53]S.R.Caradus and W.E.Pfaffenberger and Bertram Yood.Calkin Algebras and Alge-bras of Operators on Banach Spaces.Marcel Dekker,INC. New York 1974.
[54]Nathan Jacobson. Basic Algebra.W.H.FREEMAN AND COMPANY.San Fran-cisco.1980.
[55]张长耀.Banach空间上强不可约算子的初步探讨.福建师范大学硕士学位论文2008.
[2]H.P.Rosenthal.On totally incomparable Banach spaces,J.Punct.Anal.4.167-175(1969).
[3]T.Alvarez,M.Gonzalez and V.M.Onieva.Totally incomparable Banach spaces and three-space Banach space ideals,Math.Nachr.131 (1987).
[4]M.Gonzalez.On essentially incomparable Banach spaces,Math.Z.215,621-629(1994).
[5]P.Aiena,and Manuel Gonzalez.Examples of improjective operators,MathZ.233,471-479(2000).
[6]W.B.Johnson and Lindenstrauss J.eds.handbook of Geometry of Banach spaces, Volume 1. Amsterdam:North-Holland,2001.
[7]W. J. Bicker. Well-bounded operators of type(B) in H. I. spaces. Acta Sci. Math. (Szeged),1994,59:475-488.
[8]F.Rabiger and W.J.Ricker.Aspects of operator theory in hereditarily indecomposable Banach spaces.Rend.Sem.Math.Fis.Milano,1997,67:139-149.
[9]G.Androulakis and T.Schlumprecht.Strictly singular,non-compact operators exist on the space of Gowers and Maurey. J.London Math.Soc.2001,64(2):655-674.
[10]N.J.Laustsen.K-theory for algebras of operators on Banach spaces.J.London Math.Soc,1999,59(2):715-728.
[11]A.Zsak.A Banach space whose operator algebra has-group Q.Proc.London Math.Soc.2002,84(3):747-768.
[12]Aupetit. A Primer on Spectral Theory. Springer-verlag. Springer-Verlag New York,Inc.1991.
[13]张云南,钟怀杰,苏维钢.Banach空间上算子代数的群.中国科学,2005,A35(9)：1008-1018(英文版2006,49(2)：233-244).
[14]R.J.Whitley.Strictly singular operators and their conjugates,T.A.M.S.1964(113):252-261.
[15]J.Lindenstrauss.and L.Tzafrifi.1977 Classical Banach spaces I.Sequence spaces.New York.Springer.
[16]P.Aiena and M.Gonzalez.Essentially incomparable Banach spaces and Fredholm theory.Pro.R.Ir.Acad.Vol.93A,No.l,49-59(1993).
[17]俞鑫泰.Banach空间选论,华东师范大学出版社.
[18]Zhong huaijie.Significant development in the structural theory of Banach spaces based on the series of results by Gowers and Maurey, Adv. Math(China)20(2000),1-18. MR.98c:46013.
[19]A.Pietro,M.Gonzalez and M.Antonio.Operator semigroups in Banach space the-ory.Bullettino U.M.I.(8)4-B(2001),157-205.
[20]A.Pietsh.1980 Operator Ideals, Amsterdam-New York-Oxford. North-Holland.
[21]Chen Dongyang and Zhong Huaijie.The Classification of G-M type Banach spaces And structure of operators on them.
[22]刘绍学.近世代数基础.高等教育出版社.1999.
[23]H. R. Dowson,Spectral theory of linear operators. London Math. Soc-.Mono-graph,1978. Bull.London. Math. Soc.29(1997)338-344.
[24]定光桂.巴拿赫空间引论,科学出版社,1983.
[25]李炳仁.Banach代数,科学出版社,1997.
[26]邓生华.Banach代数中理想概念的推广,西南师范大学学报,1999.
[27]A.Pietsch.Inessential operators in Banach spaces.Integral Equations and Operator Theory,l(1978):589-591.
[28]Yu.V.Turovskii and V.S.Shulman,Radicals in Banach Algebras and Some Problems. in the Theory of Radical Banach Algebras, Functional Analysis and Its Applica-tions,Vol.35,No.4, pp.312-314,2001.
[29]A. Pietro and M.Gonzalez. Essentially incomparable B.S and Fredholm theory.Proc. R. Ir. Acad.39A:1(1993):49-59.
[30]M.Schechter.Riesz operators and Fredholm perturbations.Bull.AMS.74(1968):1139-1144.
[31]P.Aiena.On Riesz and Inessential Operators,Math.Z.201,521-528(1989).
[32]W. T. Gowers,B.Maurey,Banach spaces with small spaces of operators,Math. Ann. 307,543-568(1997).
[33]陈东阳,钟怀杰.本性不可比的Banach空间,应用泛函分析学报,第3卷,.第3期,2001年9月.
[34]M.Gonzalez.The perturbation classes problem in Fredholm theory,Journal of Func-tional Analysis 200 (2003)65-70.
[35]P. Aiena and M.Gonzalez.On the operators which are invertible modulo an operator ideal,Bull.Austral.Math.Soc.Vol.64(2001) [233-243].
[36]陈剑岚,钟怀杰.超投影空问与严格余奇异算子,福建师范大学学报,1997,13(4)：19-21.
[37]J.Zemanek.A note on the radical of a Banach algebra,manuscripta math.20,191-196 (1977).
[38]钟怀杰.关于非平凡的黎斯算子,福建师范大学学报,1993,9(3)：10-14.
[39]林丽琼,钟怀杰.关于弱自反严格奇异算子,福建师范大学学报,第3卷,第34期,2006年6月.
[40]林丽琼,钟怀杰.关于弱自反严格余奇异算子,福州大学学报,第35卷,第1期,2007年2月.
[41]Manuel Gonzalez.The perturbation classes problem in Fredholm theory,Journal of Functional Analysis 200 (2003)65-70.
[42]P.Aiena and M.Gonzalez.On inessential and improjective operators. Studia Math.l31(1998),271-287.
[43]钟怀杰,陈东晓,陈剑岚.某些G-M型Banach空间及其上算子代数的K群.中国科学,2003,A33(4)：338-353(英文版2004,47(3)：372-392).
[44]P. Aiena,M. Gonzalez,Incomparable Banach spaces and operator semigroups. Arch. Math.79(2002)372-378.
[45]P. Aiena,An internal characterization of inessential operators,Proc. Amer. Math. Soc. 102(1988),625-626.
[46]T. Alvarez and M. Gonzalez. Some examples of tauberian operators,Proc. Amer. Math. Soc.111(1991),1023-1027.
[47]M.Basallote.Representabilidad finita por cocientes y operators,Doctoral Thesis,Univ. Sevilla,1998.
[48]W. T. Gowers and B. Maurey. The unconditional basic sequence problem[J]. JAMS,1993,6(3):851-874.
[49]Pietro Aiena.Fredholm and local spectral theory with applications to multipli-ers [M].Kluwer Academic Publishers,2004.
[50]Rudin W.Functional analysis[M].2nd ed.New York:McGraw-Hill,1991.
[51]陈焕艮.关于Jacobson根的特征.广西师范大学学报,第11卷,第2期,1993.
[52][苏]O.A.德洛兹德,B.B.基里钦柯著,刘绍学.张英伯译.有限维代数[M].北京师范大学出版社1984.
[53]S.R.Caradus and W.E.Pfaffenberger and Bertram Yood.Calkin Algebras and Alge-bras of Operators on Banach Spaces.Marcel Dekker,INC. New York 1974.
[54]Nathan Jacobson. Basic Algebra.W.H.FREEMAN AND COMPANY.San Fran-cisco.1980.
[55]张长耀.Banach空间上强不可约算子的初步探讨.福建师范大学硕士学位论文2008.