具有不可约性算子类与G—M型空间上算子结构的研究
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摘要
Banach空间上的算子结构问题是泛函分析Banach空间理论与算子代数理论共同关注的主要问题之一.本文在特殊的Banach空间—G-M型空间上,利用G-M型空间的特殊结构,借助各种具有不可约性的算子作为工具.应用算子代数K理论的语言来研究算子结构,体现了空间结构和算子结构研究的互动作用.相对于单纯空间理论,或者单纯算子代数理论的研究,其指导思想独特和新颖.
关于Banach空间上算子结构的研究,江泽坚先生认为强不可约算子可以视为Jordan块在无限维空间上有界线性算子中的合适类似物,并希望能在无限维空间算子结构的研究中,建立起类似于Jordan标准形定理的相应系列结果.在可分Hilbert空间上,D.A.Herrero, S.Power和蒋春澜等人完全证实了这一思想.本文的工作可以视为是在新的一类空间—G-M型空间上继续实践江先生的思想.
至今为止,国内外学者只研究强不可约算子类或以强不可约算子类作为工具研究算子结构,本文拓广为以强不可约算子类为中心的一系列具有不可约性的算子类,并以其为工具研究空间结构和算子结构,这也是本文的主要特色之一
本文有以下三个方面的主要工作.
第一方面工作是给出了一系列具有不可约性算子类的定义,包括有限维不可约算子、无限维不可约算子、(NCI)算子、(NFI)算子、Bn算子和B算子.本文讨论这几种算子类的存在性,还详细讨论它们和强不可约算子类、Cowen-Douglas算子类之间的关系及其性质,例如(拟)相似不变性和共轭不变性等.
第二方面工作是讨论具有不可约性算子类的小紧摄动问题,主要是讨论有限维不可约算子：强不可约算子与(NFI)算子的小紧摄动问题.本文证明了可分Banach空间上的每个谱单点算子均可小紧摄动成为有限维不可约算子.对于强不可约算子的小紧摄动,在可分的不可分解的Σcdc型空间上,每个谱连通算子均可小紧摄动成为强不可约算子,进而在可分的不可分解的Σcdc型空间上,谱连通算子集是强不可约算子集的闭包.对于(NFI)算子的小紧摄动,本文证明了几种特殊情形的谱为单点集{0}的算子可小紧摄动成为(NFI)算子.在此基础上,建立了某些有Schauder基的Banach空间上算子的近似Jordan标准形.
第三方面工作是利用K理论语言来研究算子的相似不变量,主要是利用(序)KKo群给出Σcdc型空间X上两个强不可约算子以及(ΣSI)(X)(X上可分解为有限个强不可约算子的直和的算子集)中两个算子相似的充要条件.
此外,本文还研究Banach空间X上算子代数B(X)的K0群,主要研究是否存在Banach空间X使得K0(B(X))=Z2|问题,这是A.Zsak提出的一个猜想.本文首先给出K0(B(X))=Z2的充分条件.进而讨论G-M型空间XGM4上幂等算子的性质,为最终得到Banach空间X满足K0(B(X))=Z2提供一些思想方法.
The research of the operator structure on Banach spaces is one of the main issues both for the theory of Banach spaces and for the theory of operator algebras in functional analysis. This paper study the operator structure on the special Banach spaces—G-M type spaces by using the special structure of G-M type spaces and K-theory for operator algebras, with a variety of operators of irreducibility as tools. It reflects the interaction between the research of the structure of Banach spaces and the research of operator structure. Compared to the pure theory of Banach spaces or the pure theory of operator algebras, the idea is unique and innovative.
Concerning the research of the operator structure on Banach spaces, Z.J.Jiang thought that the strongly irreducible operators can be considered as the approximate replacement of Jordan blocks on infinite dimensional spaces and hoped that a series of results similar to the Jordan Standard Theorem can be set up with this replacement on the research of the operator structure on infinite dimensional Banach spaces. On separable Hilbert spaces. D.A.Herrero, S.Power and C.L.Jiang had fully confirmed this idea. The work of this paper can be seen to put into practice the idea of Z.J.Jiang on a new class of Banach spaces—G-M type spaces.
So far, the domestic and foreign scholars only studied the class of strongly irre-ducible operators or only used the class-of strongly irreducible operators to study the operator structure. This paper expands to a series of operators with irreducibility with the class of strongly irreducible operators as the center. And it uses such classes of operators as tools to study the space structure and the operator structure. This is one of the main feature of this paper.
This paper has three main works.
Firstly, it gives the concepts of a series of operators with irreducibility, including finite dimensional irreducible operators, infinite dimensional irreducible operators, (NCI) operators, (NFI) operators, Bn operators and B operators. Then it discusses the existence of these classes of operators and discusses in detail the relationships among strongly irreducible operators, Cowen-Douglas operators and them. This paper also discusses some properties of these classes of operators, such as the (quasi)similar invariance and the conjugation invariance.
Secondly, this paper discusses the small and compact perturbation problem of operators with irreducibility. It mainly discusses the small and compact perturbation problem of finite dimensional irreducible operators, strongly irreducible operators and (NFI) operators. This paper proves that every operator with a singleton spectrum on separable Banach spaces is a small compact perturbation of a finite dimensional irreducible operator. For the small and compact perturbation problem of strongly irreducible operators, it shows that every operator with a connected spectrum on separable indecomposableΣcdc spaces is a small compact perturbation of a strongly irreducible operator. It further shows that the set{T∈B(X)σa(T) is connected } is the norm-closure of the set of strongly irreducible operators on separable inde-composableΣcdc spaces. For the small and compact perturbation problem of (NFI) operators, this paper shows that an operator with a singleton spectrum{0} can be-come an (NFI) operator by a small and compact perturbation in some cases. Based on these results, this paper establishes the approximate Jordan forms of operators on some kinds of Banach spaces with Schauder bases.
Thirdly, this paper studies the similar invariants of operators by using the lan-guage of K-theory. It mainly uses the (order) K-groups to give the sufficient and necessary conditions for the similarity of two strongly irreducible operators and the similarity of two operators in (ΣSI)(X) (where (ESI)(X) denotes the class of op-erators which can be decomposed into the direct sum of finitely strongly irreducible operators on X) onΣcdc spaces.
In addition, this paper studies the K0-groups of the operator algebras B(X) on Banach spaces X. It mainly discusses the following problem:does there exist a Banach space X with K0(B(X))=Z2, which is a guess raised by A.Zsak. This paper gives a sufficient condition for K0(B(X))=Z2. Then it discusses the properties of the idempotents on G-M type space XGM4, which provides some ideals and methods to get a Banach space X with K0(B(X))=Z2.
关于Banach空间上算子结构的研究,江泽坚先生认为强不可约算子可以视为Jordan块在无限维空间上有界线性算子中的合适类似物,并希望能在无限维空间算子结构的研究中,建立起类似于Jordan标准形定理的相应系列结果.在可分Hilbert空间上,D.A.Herrero, S.Power和蒋春澜等人完全证实了这一思想.本文的工作可以视为是在新的一类空间—G-M型空间上继续实践江先生的思想.
至今为止,国内外学者只研究强不可约算子类或以强不可约算子类作为工具研究算子结构,本文拓广为以强不可约算子类为中心的一系列具有不可约性的算子类,并以其为工具研究空间结构和算子结构,这也是本文的主要特色之一
本文有以下三个方面的主要工作.
第一方面工作是给出了一系列具有不可约性算子类的定义,包括有限维不可约算子、无限维不可约算子、(NCI)算子、(NFI)算子、Bn算子和B算子.本文讨论这几种算子类的存在性,还详细讨论它们和强不可约算子类、Cowen-Douglas算子类之间的关系及其性质,例如(拟)相似不变性和共轭不变性等.
第二方面工作是讨论具有不可约性算子类的小紧摄动问题,主要是讨论有限维不可约算子：强不可约算子与(NFI)算子的小紧摄动问题.本文证明了可分Banach空间上的每个谱单点算子均可小紧摄动成为有限维不可约算子.对于强不可约算子的小紧摄动,在可分的不可分解的Σcdc型空间上,每个谱连通算子均可小紧摄动成为强不可约算子,进而在可分的不可分解的Σcdc型空间上,谱连通算子集是强不可约算子集的闭包.对于(NFI)算子的小紧摄动,本文证明了几种特殊情形的谱为单点集{0}的算子可小紧摄动成为(NFI)算子.在此基础上,建立了某些有Schauder基的Banach空间上算子的近似Jordan标准形.
第三方面工作是利用K理论语言来研究算子的相似不变量,主要是利用(序)KKo群给出Σcdc型空间X上两个强不可约算子以及(ΣSI)(X)(X上可分解为有限个强不可约算子的直和的算子集)中两个算子相似的充要条件.
此外,本文还研究Banach空间X上算子代数B(X)的K0群,主要研究是否存在Banach空间X使得K0(B(X))=Z2|问题,这是A.Zsak提出的一个猜想.本文首先给出K0(B(X))=Z2的充分条件.进而讨论G-M型空间XGM4上幂等算子的性质,为最终得到Banach空间X满足K0(B(X))=Z2提供一些思想方法.
The research of the operator structure on Banach spaces is one of the main issues both for the theory of Banach spaces and for the theory of operator algebras in functional analysis. This paper study the operator structure on the special Banach spaces—G-M type spaces by using the special structure of G-M type spaces and K-theory for operator algebras, with a variety of operators of irreducibility as tools. It reflects the interaction between the research of the structure of Banach spaces and the research of operator structure. Compared to the pure theory of Banach spaces or the pure theory of operator algebras, the idea is unique and innovative.
Concerning the research of the operator structure on Banach spaces, Z.J.Jiang thought that the strongly irreducible operators can be considered as the approximate replacement of Jordan blocks on infinite dimensional spaces and hoped that a series of results similar to the Jordan Standard Theorem can be set up with this replacement on the research of the operator structure on infinite dimensional Banach spaces. On separable Hilbert spaces. D.A.Herrero, S.Power and C.L.Jiang had fully confirmed this idea. The work of this paper can be seen to put into practice the idea of Z.J.Jiang on a new class of Banach spaces—G-M type spaces.
So far, the domestic and foreign scholars only studied the class of strongly irre-ducible operators or only used the class-of strongly irreducible operators to study the operator structure. This paper expands to a series of operators with irreducibility with the class of strongly irreducible operators as the center. And it uses such classes of operators as tools to study the space structure and the operator structure. This is one of the main feature of this paper.
This paper has three main works.
Firstly, it gives the concepts of a series of operators with irreducibility, including finite dimensional irreducible operators, infinite dimensional irreducible operators, (NCI) operators, (NFI) operators, Bn operators and B operators. Then it discusses the existence of these classes of operators and discusses in detail the relationships among strongly irreducible operators, Cowen-Douglas operators and them. This paper also discusses some properties of these classes of operators, such as the (quasi)similar invariance and the conjugation invariance.
Secondly, this paper discusses the small and compact perturbation problem of operators with irreducibility. It mainly discusses the small and compact perturbation problem of finite dimensional irreducible operators, strongly irreducible operators and (NFI) operators. This paper proves that every operator with a singleton spectrum on separable Banach spaces is a small compact perturbation of a finite dimensional irreducible operator. For the small and compact perturbation problem of strongly irreducible operators, it shows that every operator with a connected spectrum on separable indecomposableΣcdc spaces is a small compact perturbation of a strongly irreducible operator. It further shows that the set{T∈B(X)σa(T) is connected } is the norm-closure of the set of strongly irreducible operators on separable inde-composableΣcdc spaces. For the small and compact perturbation problem of (NFI) operators, this paper shows that an operator with a singleton spectrum{0} can be-come an (NFI) operator by a small and compact perturbation in some cases. Based on these results, this paper establishes the approximate Jordan forms of operators on some kinds of Banach spaces with Schauder bases.
Thirdly, this paper studies the similar invariants of operators by using the lan-guage of K-theory. It mainly uses the (order) K-groups to give the sufficient and necessary conditions for the similarity of two strongly irreducible operators and the similarity of two operators in (ΣSI)(X) (where (ESI)(X) denotes the class of op-erators which can be decomposed into the direct sum of finitely strongly irreducible operators on X) onΣcdc spaces.
In addition, this paper studies the K0-groups of the operator algebras B(X) on Banach spaces X. It mainly discusses the following problem:does there exist a Banach space X with K0(B(X))=Z2, which is a guess raised by A.Zsak. This paper gives a sufficient condition for K0(B(X))=Z2. Then it discusses the properties of the idempotents on G-M type space XGM4, which provides some ideals and methods to get a Banach space X with K0(B(X))=Z2.
引文
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