线性算子的混沌性研究，强不可约意义下的极分解定理
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摘要
现代数学发展的一大趋势是各数学分支相互交叉,取长补短.本文第一部分就是将拓扑动力系统与算子理论结合起来,考察线性系统的混沌性质.拓扑动力系统与算子理论之间存在非常自然的结合点.我们强调它们之间的经典思想,概念和结论的相互借鉴,以期互相促进,共同发展.
一方面,客观物质世界的许多领域和问题(例如N-体问题)告诉我们,确定论的科学研究思想是不够的.我们还需要对随机性和不确定性进行研究.这就是所谓的混沌理论.另一方面,算子理论的一项重要任务是研究算子的结构.著名的不变子空间问题引发了人们对超循环算子的研究热潮.事实上超循环这个概念与动力系统中的传递性是完全吻合的.目前,人们对线性算子超循环性(传递性)的研究已取得不少突破性的成果:对于有界线性算子, Kitai等人给出了超循环性(传递性)的一个充分性条件—HypercyclicityCriterion, Herrero给出了复可分无穷维的Hilbert空间上超循环算子全体的闭包的谱图形刻画, Gethner, Shapiro, Salas给出了复可分无穷维的Hilbert空间上加权移位算子超循环性的等价刻画, Grosse-Erdmann考虑了一般Frechet空间上加权移位算子的超循环性, Costakis和Manoussos将拓扑动力系统中J集的概念引入到算子理论中,推广了超循环性,得到了J-类算子的概念并且考虑了与超循环算子平行的理论, Chan证明了复可分无穷维Hilbert空间H上所有超循环算子的有限线性组合在L(H)中按范数拓扑稠密.从“超循环”这个结合点出发,人们将动力系统中的混沌概念引入到算子理论中,考虑线性算子的混沌性质.目前,算子混沌理论正在发展中, Herrero证明了L(H)中存在很多的Devaney混沌算子, Grosse-Erdmann给出了加权移位算子Devaney混沌的等价刻画,侯秉喆,廖公夫,曹阳, Bermudez, Bonilla, Martinez-Gimenez和Peris分别考虑了加权移位算子的Li-Yorke混沌性,并且给出了Li-Yorke混沌的判别准则,2010年,侯秉喆,崔醭玉和曹阳考虑了Cowen-Douglas算子的分布混沌性,给出了分布混沌的一个可计算性的判别准则—范数单峰.
本文的第一部分将从整体的角度考虑复可分无穷维Hilbert空间H上的分布混沌算子和Li-Yorke混沌算子.具体地说,
第一章介绍本文研究问题的背景,以及动力系统和算子理论的一些基本概念和基本结果.
第二章首先从具体算子类出发,证明紧算子和正规算子都不可能产生混沌(分布混沌和Li-Yorke混沌),并且回顾加权移位算子和Cowen-Douglas算子的混沌性质.其次,我们借助算子逼近论的工具,用谱图形的语言刻画H上的分布混沌算子和Li-Yorke混沌算子全体在范数拓扑下的闭包和内部.结果显示,尽管分布混沌的定义从统计意义上加强了Li-Yorke混沌的定义,但是我们得到了相同的闭包和内部.我们还比较了范数单峰算子类和分布混沌算子类,得到了小扰动下分布混沌性质不变的线性算子必是范数单峰算子.再次,我们证明了上面得到的闭包和内部都是道路连通的.最后, Costakis和Manoussos在文章“J-class operators and hypercyclicity”中定义了J类算子(此类算子是超循环算子的推广),并且建议沿Herrero的思路刻画J类算子全体的闭包的谱图形.我们给出了该谱图形的刻画.
关于算子结构问题,我们可以从另一个角度看.有限维的Hilbert空间情形,线性算子表现为有限维矩阵.著名的Jordan标准型定理完全展示了矩阵的结构.定理指出矩阵的特征值和广义特征空间完全给出了矩阵的相似不变量.矩阵可以分解成Jordan块的直和(在相似的意义下).如果把Jordan块比作砖块的话,那么我们可以用这些砖块来筑起任何的“矩阵”大厦.对于无穷维Hilbert空间,我们面临同样的问题:怎么样构建类似的Jordan标准型定理,怎么样决定算子的完全相似不变量.找完全相似不变量的主要困难在于人们不清楚Jordan块在无穷维空间上的完美类似物.1968年Halmos引进了不可约算子, Voiculescu得到了著名的非交换Weyl-von Neumann定理.但是不可约性只是酉不变量,不足以显示一般算子代数和非自伴代数的结构.1970年以后,算子理论的工作者们从两个方面研究算子的结构.一方面Foias, Ringrose, Arveson, Davidson等从特殊的算子类和算子代数入手考察算子的结构问题,如Toeplitz算子,加权移位算子,拟幂零算子,三角算子,拟三角算子,三角代数,拟三角代数等.另一方面,他们用指标理论和谱图形的语言建立渐近相似不变量.其中最典型的结果莫过于Apostol, Filkow,Herrero和Voiculescu得到的相似轨道定理.这个定理用谱图形的语言给出了Hilbert空间算子的完全渐近相似不变量.另外,1970年Gilfeather和江泽坚分别地给出强不可约算子的概念.江泽坚首先认为强不可约算子可以看作Jordan块在L(H)中的类似物.算子称为强不可约的如果它的换位中没有非平凡的幂等算子.有限维情形,强不可约算子就是Jordan块(在相似意义下).在随后的20多年里,蒋春澜等人证明了强不可约算子确为Jordan块在无穷维空间中的类似物,并构建了无穷维空间中的渐近Jordan标准型定理,意义是深远的.
本文的第二部分(即第三章)将考虑如何用强不可约算子来构建极分解定理.经典的极分解定理告诉我们,任意Hilbert空间H上的算子T都可以分解成部分等距算子和正算子的乘积,即T=U|T|或者UT=|T|,其中U是部分等距算子,|T|=(T*T)~(1/2).我们将考虑如何将|T|换成强不可约算子.具体地说,我们得到:
定理3.2.1.设T∈L(H).则对任意的∈>0,存在部分等距算子U,紧算子K,||K||<∈和强不可约算子S使得T=(U+K)S.
A big trend for the development of modern mathematics is to get various branchesconnecting with each other. We will consider the chaotic properties of linear systemcoming from the natural intersection between topological dynamical system and operatortheory in the frst part of this thesis. We will emphasize the mutual infuence of classicthought, notions and conclusions in these theories in order to accelerate the developmentof them.
On the one hand, many problems in the physical world (for example, the N bodyproblem) tell us that determinism is not enough for us to solve real problems. We alsoneed to consider randomness and disorder of our world. That is the beginning of chaostheory. On the other hand, one big task in operator theory is trying to understand thestructure of operators. The most famous problem relating to it in operator theory is theinvariant subspace problem. The research of it cause the development of hypercyclic op-erators. In fact, the hypercyclicity is the same thing to transitivity in dynamical system.Up to now, there are many breakthrough progresses in the theory of hypercyclicity: Kitai,etc. gave a criterion for hypercyclicity, Herrero gave the spectral picture description of theclosure of the set of all hypercyclic operators on Hilbert space, Gethner, Shapiro, Salasgave the description of hypercyclic weighted shift operators, Grosse-Erdmann generalized the thought of them and considered the same problem on Frechet space, Costakis andManoussos generalized the hypercyclicity by borrowing a defnition from topological dy-namic system, they defned the J class operators and got related theories, Chan provedthat the closed span of all hypercyclic operators on a separable, infnitely Hilbert spaceH is L(H). Based on the junction here, people came to go one step further and beganto study the chaotic properties of linear operators. Now, the theory is ongoing. Herrerogot that there exist many Devaney chaotic operators in L(H), Grosse-Erdmann gave thedescription of Devaney chaotic weighted shift operators, Hou Bingzhe, Liao Gongfu, CaoYang, Bermudez, Bonilla, Martinez-Gimenez and Peris considered the Li-Yorke chaoticproperties for weighted shift operators and gave a sufcient condition for being Li-Yorkechaotic, In2010s, Hou Bingzhe, Cui Buyu and Cao Yang considered the distribution-al chaoticity of Cowen-Douglas operators and gave a criterion for being distributionallychaotic, called norm-unimodal.
In the frst part, we will consider the distributionally chaotic operators and Li-Yorkechaotic operators from the global point of view. Specifcally,
Chapter One. we give a brief survey on the background studied in this dissertation.Some preliminary knowledge in dynamical system and operator theory is reviewed, whichwill be used in later chapters.
Chapter Two. First we begin with particular classes of operators and obtain thatcompact operators and normal operators can not admit any chaos (distributional chaosand Li-Yorke chaos). We also review the chaotic properties for weighted shift operatorsand Cowen-Douglas operators. Second we describe the closures and interiors of the setsof all distributionally chaotic operators and Li-Yorke chaotic operators in the languageof spectral picture. It turns out that the closures and interiors are all the same thoughdistributional chaos is stronger than Li-Yorke chaos in defnition. We also compare theclasses of norm-unimodal operators and distributionally chaotic operators and get that thechaotic operators which are invariant under small perturbations must be norm-unimodaloperators. Third we prove that the closures and interiors are all arc connected. Last wegive the description of the closure of the set of all J class operators in the language ofspectral picture which Costakis and Manoussos propose to consider in their paper.
As for structure of operators, we can think of it in another way. In the matrixtheory of fnite dimensional space, the famous Jordan Standard Theorem sufciently reveals the internal structure of matrices. Jordan Theorem indicates that the eigenvaluesand the generalized eigenspaces of matrix determine the complete similarity invariantsof a matrix. Any matrix can be written in the direct sum of Jordan blocks uniquely inthe sense of similarity. If we regard Jordan block as brick, we can build up any “matrix”building using these bricks. When we consider a complex, separable, infnite dimensionalHilbert space H, we face one of the most fundamental problems in operator theory,that is how to build up a theorem in L(H) which is similar to the Jordan StandardTheorem in matrix theory, or how to determine the complete similarity invariants ofthe operators. The complexity of infnite dimensional space makes it impossible to fndgenerally similarity invariants. The main difculty behind this is that it is impossible forpeople to fnd a fundamental element in L(H), similar to Jordan’s block, so as to constructa perfect representation theorem. It is because of the introduction of the concept ofirreducible operators by Halmos in1968that Voiculescu obtained the well-known Non-commutative Weyl-von Neumann Theorem for general C*-algebra. But irreducibility isonly a unitary invariant and can not reveal the general internal structure of operatoralgebras and non-self-adjoint operators. Since the1970s, some mathematicians haveshowed their concern for the problem on Hilbert space operator structure in two aspects.In one aspect, the mathematicians such as Foias, Ringrose, Arveson, Davidson etc. havemade great eforts to study the structures of diferent classes of operators or operatoralgebras, such as Toeplitz operators, weighted shift operators, quasinilpotent operators,triangular and quasitriangular operators, triangular and quasitriangular algebras etc.In the other aspect, they have set up the approximate similarity invariants for generaloperators by introducing the index theory and spectral picture as tools. One of the mosttypical achievements, made by Apostol, Filkow, Herrero and Voiculescu is the theoremof similarity orbit of operators. This theorem suggests that the fne spectral picture isthe complete similarity invariant as far as the closure of similarity orbit of operatorsare concerned. Besides, in the1970s, Gilfeather and Jiang, Z.J. proposed the notion ofstrongly irreducible operator respectively. And Jiang, Z.J. frst thought that the stronglyirreducible operators could be viewed as the suitable replacement of Jordan block inL(H). An operator will be considered strongly irreducible if its commutant contains nonontrivial idempotent. In the theory of matrix, strongly irreducible operator is Jordanblock up to similarity. Through more than20years’ research, Jiang Chunlan etc. have shown that strongly irreducible operators are suitable replacement of Jordan blocks inL(H) defnitely and constructed the Jordan Theorem. The signifcance is profound.
In the second part, i.e. in Chapter Three, we will construct the polar decompositiontheorem by strongly irreducible operators. The classical polar decomposition theoremtell us that any operator T can be written as the multiplication of a partial isometryand a positive operator, i.e. T=U|T|or U T=|T|, where U is a partial isometryand|T|=(T*T)~(1/2). We are going to substitute|T|by a strongly irreducible operator.Specifcally,
Theorem3.2.1. For any T∈L(H) and any∈>0, there exist a partial isometryU, a compact operator K with||K||<∈and a strongly irreducible operator S such thatT=(U+K)S.
一方面,客观物质世界的许多领域和问题(例如N-体问题)告诉我们,确定论的科学研究思想是不够的.我们还需要对随机性和不确定性进行研究.这就是所谓的混沌理论.另一方面,算子理论的一项重要任务是研究算子的结构.著名的不变子空间问题引发了人们对超循环算子的研究热潮.事实上超循环这个概念与动力系统中的传递性是完全吻合的.目前,人们对线性算子超循环性(传递性)的研究已取得不少突破性的成果:对于有界线性算子, Kitai等人给出了超循环性(传递性)的一个充分性条件—HypercyclicityCriterion, Herrero给出了复可分无穷维的Hilbert空间上超循环算子全体的闭包的谱图形刻画, Gethner, Shapiro, Salas给出了复可分无穷维的Hilbert空间上加权移位算子超循环性的等价刻画, Grosse-Erdmann考虑了一般Frechet空间上加权移位算子的超循环性, Costakis和Manoussos将拓扑动力系统中J集的概念引入到算子理论中,推广了超循环性,得到了J-类算子的概念并且考虑了与超循环算子平行的理论, Chan证明了复可分无穷维Hilbert空间H上所有超循环算子的有限线性组合在L(H)中按范数拓扑稠密.从“超循环”这个结合点出发,人们将动力系统中的混沌概念引入到算子理论中,考虑线性算子的混沌性质.目前,算子混沌理论正在发展中, Herrero证明了L(H)中存在很多的Devaney混沌算子, Grosse-Erdmann给出了加权移位算子Devaney混沌的等价刻画,侯秉喆,廖公夫,曹阳, Bermudez, Bonilla, Martinez-Gimenez和Peris分别考虑了加权移位算子的Li-Yorke混沌性,并且给出了Li-Yorke混沌的判别准则,2010年,侯秉喆,崔醭玉和曹阳考虑了Cowen-Douglas算子的分布混沌性,给出了分布混沌的一个可计算性的判别准则—范数单峰.
本文的第一部分将从整体的角度考虑复可分无穷维Hilbert空间H上的分布混沌算子和Li-Yorke混沌算子.具体地说,
第一章介绍本文研究问题的背景,以及动力系统和算子理论的一些基本概念和基本结果.
第二章首先从具体算子类出发,证明紧算子和正规算子都不可能产生混沌(分布混沌和Li-Yorke混沌),并且回顾加权移位算子和Cowen-Douglas算子的混沌性质.其次,我们借助算子逼近论的工具,用谱图形的语言刻画H上的分布混沌算子和Li-Yorke混沌算子全体在范数拓扑下的闭包和内部.结果显示,尽管分布混沌的定义从统计意义上加强了Li-Yorke混沌的定义,但是我们得到了相同的闭包和内部.我们还比较了范数单峰算子类和分布混沌算子类,得到了小扰动下分布混沌性质不变的线性算子必是范数单峰算子.再次,我们证明了上面得到的闭包和内部都是道路连通的.最后, Costakis和Manoussos在文章“J-class operators and hypercyclicity”中定义了J类算子(此类算子是超循环算子的推广),并且建议沿Herrero的思路刻画J类算子全体的闭包的谱图形.我们给出了该谱图形的刻画.
关于算子结构问题,我们可以从另一个角度看.有限维的Hilbert空间情形,线性算子表现为有限维矩阵.著名的Jordan标准型定理完全展示了矩阵的结构.定理指出矩阵的特征值和广义特征空间完全给出了矩阵的相似不变量.矩阵可以分解成Jordan块的直和(在相似的意义下).如果把Jordan块比作砖块的话,那么我们可以用这些砖块来筑起任何的“矩阵”大厦.对于无穷维Hilbert空间,我们面临同样的问题:怎么样构建类似的Jordan标准型定理,怎么样决定算子的完全相似不变量.找完全相似不变量的主要困难在于人们不清楚Jordan块在无穷维空间上的完美类似物.1968年Halmos引进了不可约算子, Voiculescu得到了著名的非交换Weyl-von Neumann定理.但是不可约性只是酉不变量,不足以显示一般算子代数和非自伴代数的结构.1970年以后,算子理论的工作者们从两个方面研究算子的结构.一方面Foias, Ringrose, Arveson, Davidson等从特殊的算子类和算子代数入手考察算子的结构问题,如Toeplitz算子,加权移位算子,拟幂零算子,三角算子,拟三角算子,三角代数,拟三角代数等.另一方面,他们用指标理论和谱图形的语言建立渐近相似不变量.其中最典型的结果莫过于Apostol, Filkow,Herrero和Voiculescu得到的相似轨道定理.这个定理用谱图形的语言给出了Hilbert空间算子的完全渐近相似不变量.另外,1970年Gilfeather和江泽坚分别地给出强不可约算子的概念.江泽坚首先认为强不可约算子可以看作Jordan块在L(H)中的类似物.算子称为强不可约的如果它的换位中没有非平凡的幂等算子.有限维情形,强不可约算子就是Jordan块(在相似意义下).在随后的20多年里,蒋春澜等人证明了强不可约算子确为Jordan块在无穷维空间中的类似物,并构建了无穷维空间中的渐近Jordan标准型定理,意义是深远的.
本文的第二部分(即第三章)将考虑如何用强不可约算子来构建极分解定理.经典的极分解定理告诉我们,任意Hilbert空间H上的算子T都可以分解成部分等距算子和正算子的乘积,即T=U|T|或者UT=|T|,其中U是部分等距算子,|T|=(T*T)~(1/2).我们将考虑如何将|T|换成强不可约算子.具体地说,我们得到:
定理3.2.1.设T∈L(H).则对任意的∈>0,存在部分等距算子U,紧算子K,||K||<∈和强不可约算子S使得T=(U+K)S.
A big trend for the development of modern mathematics is to get various branchesconnecting with each other. We will consider the chaotic properties of linear systemcoming from the natural intersection between topological dynamical system and operatortheory in the frst part of this thesis. We will emphasize the mutual infuence of classicthought, notions and conclusions in these theories in order to accelerate the developmentof them.
On the one hand, many problems in the physical world (for example, the N bodyproblem) tell us that determinism is not enough for us to solve real problems. We alsoneed to consider randomness and disorder of our world. That is the beginning of chaostheory. On the other hand, one big task in operator theory is trying to understand thestructure of operators. The most famous problem relating to it in operator theory is theinvariant subspace problem. The research of it cause the development of hypercyclic op-erators. In fact, the hypercyclicity is the same thing to transitivity in dynamical system.Up to now, there are many breakthrough progresses in the theory of hypercyclicity: Kitai,etc. gave a criterion for hypercyclicity, Herrero gave the spectral picture description of theclosure of the set of all hypercyclic operators on Hilbert space, Gethner, Shapiro, Salasgave the description of hypercyclic weighted shift operators, Grosse-Erdmann generalized the thought of them and considered the same problem on Frechet space, Costakis andManoussos generalized the hypercyclicity by borrowing a defnition from topological dy-namic system, they defned the J class operators and got related theories, Chan provedthat the closed span of all hypercyclic operators on a separable, infnitely Hilbert spaceH is L(H). Based on the junction here, people came to go one step further and beganto study the chaotic properties of linear operators. Now, the theory is ongoing. Herrerogot that there exist many Devaney chaotic operators in L(H), Grosse-Erdmann gave thedescription of Devaney chaotic weighted shift operators, Hou Bingzhe, Liao Gongfu, CaoYang, Bermudez, Bonilla, Martinez-Gimenez and Peris considered the Li-Yorke chaoticproperties for weighted shift operators and gave a sufcient condition for being Li-Yorkechaotic, In2010s, Hou Bingzhe, Cui Buyu and Cao Yang considered the distribution-al chaoticity of Cowen-Douglas operators and gave a criterion for being distributionallychaotic, called norm-unimodal.
In the frst part, we will consider the distributionally chaotic operators and Li-Yorkechaotic operators from the global point of view. Specifcally,
Chapter One. we give a brief survey on the background studied in this dissertation.Some preliminary knowledge in dynamical system and operator theory is reviewed, whichwill be used in later chapters.
Chapter Two. First we begin with particular classes of operators and obtain thatcompact operators and normal operators can not admit any chaos (distributional chaosand Li-Yorke chaos). We also review the chaotic properties for weighted shift operatorsand Cowen-Douglas operators. Second we describe the closures and interiors of the setsof all distributionally chaotic operators and Li-Yorke chaotic operators in the languageof spectral picture. It turns out that the closures and interiors are all the same thoughdistributional chaos is stronger than Li-Yorke chaos in defnition. We also compare theclasses of norm-unimodal operators and distributionally chaotic operators and get that thechaotic operators which are invariant under small perturbations must be norm-unimodaloperators. Third we prove that the closures and interiors are all arc connected. Last wegive the description of the closure of the set of all J class operators in the language ofspectral picture which Costakis and Manoussos propose to consider in their paper.
As for structure of operators, we can think of it in another way. In the matrixtheory of fnite dimensional space, the famous Jordan Standard Theorem sufciently reveals the internal structure of matrices. Jordan Theorem indicates that the eigenvaluesand the generalized eigenspaces of matrix determine the complete similarity invariantsof a matrix. Any matrix can be written in the direct sum of Jordan blocks uniquely inthe sense of similarity. If we regard Jordan block as brick, we can build up any “matrix”building using these bricks. When we consider a complex, separable, infnite dimensionalHilbert space H, we face one of the most fundamental problems in operator theory,that is how to build up a theorem in L(H) which is similar to the Jordan StandardTheorem in matrix theory, or how to determine the complete similarity invariants ofthe operators. The complexity of infnite dimensional space makes it impossible to fndgenerally similarity invariants. The main difculty behind this is that it is impossible forpeople to fnd a fundamental element in L(H), similar to Jordan’s block, so as to constructa perfect representation theorem. It is because of the introduction of the concept ofirreducible operators by Halmos in1968that Voiculescu obtained the well-known Non-commutative Weyl-von Neumann Theorem for general C*-algebra. But irreducibility isonly a unitary invariant and can not reveal the general internal structure of operatoralgebras and non-self-adjoint operators. Since the1970s, some mathematicians haveshowed their concern for the problem on Hilbert space operator structure in two aspects.In one aspect, the mathematicians such as Foias, Ringrose, Arveson, Davidson etc. havemade great eforts to study the structures of diferent classes of operators or operatoralgebras, such as Toeplitz operators, weighted shift operators, quasinilpotent operators,triangular and quasitriangular operators, triangular and quasitriangular algebras etc.In the other aspect, they have set up the approximate similarity invariants for generaloperators by introducing the index theory and spectral picture as tools. One of the mosttypical achievements, made by Apostol, Filkow, Herrero and Voiculescu is the theoremof similarity orbit of operators. This theorem suggests that the fne spectral picture isthe complete similarity invariant as far as the closure of similarity orbit of operatorsare concerned. Besides, in the1970s, Gilfeather and Jiang, Z.J. proposed the notion ofstrongly irreducible operator respectively. And Jiang, Z.J. frst thought that the stronglyirreducible operators could be viewed as the suitable replacement of Jordan block inL(H). An operator will be considered strongly irreducible if its commutant contains nonontrivial idempotent. In the theory of matrix, strongly irreducible operator is Jordanblock up to similarity. Through more than20years’ research, Jiang Chunlan etc. have shown that strongly irreducible operators are suitable replacement of Jordan blocks inL(H) defnitely and constructed the Jordan Theorem. The signifcance is profound.
In the second part, i.e. in Chapter Three, we will construct the polar decompositiontheorem by strongly irreducible operators. The classical polar decomposition theoremtell us that any operator T can be written as the multiplication of a partial isometryand a positive operator, i.e. T=U|T|or U T=|T|, where U is a partial isometryand|T|=(T*T)~(1/2). We are going to substitute|T|by a strongly irreducible operator.Specifcally,
Theorem3.2.1. For any T∈L(H) and any∈>0, there exist a partial isometryU, a compact operator K with||K||<∈and a strongly irreducible operator S such thatT=(U+K)S.
引文
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[51] HOU B Z, CUI P Y, CAO Y. Chaos for Cowen-Douglas operators [J]. Proc. Amer.Math. Soc,2010,138:929–936.
[52] HOU B Z, LIAO G F, CAO Y. Dynamics of shift operators [J]. Houston Journal ofMathematics,2012,38(4):1225–1239.
[53] HUANG W, YE X D. Devaneys’s chaos and2-scattering imply Li-Yorke chaos [J].Topol. Appl,2002,117:259–272.
[54] JI Y Q. Quasitriangular+small compact=strongly irreducible [J]. Trans. Amer.Math. Soc,1999,351(11):4657–4673.
[55] JI Y Q, JIANG C L. Small compact perturbation of strongly irreducible operators[J]. Integr. Equ. Oper. Theory,2002,43:417–449.
[56] JI Y Q, JIANG C L, WANG Z Y. Essentially normal+small compact=stronglyirreducible [J]. Chinese Ann. Math1996,24:41–72.
[57] JI Y Q, JIANG C L, WANG Z Y. The unitary orbit of strongly irreducible operatorsin the nest algebra with well-ordered set [J]. Mich, Math. J,1997,44:85–98.
[58] JI Y Q, JIANG C L, WANG Z Y. The strongly irreducible operators in nest algebras[J]. Integral Equations Operator Theory,1997,28(1):28–44.
[59] JI Y Q, JIANG C L, WANG Z Y. The closure of the unitary orbit of the stronglyirreducible operators in continuous nest algebra [J]. Acta Math. Sinica (N. S.),1998,14:605–612.
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[61] JI Y Q, LI J X, SUN S L. The essential spectrum and Banach reducibility of operatorweighted shifts [J]. Acta Math. Sinca,2003,46(3):413–424.
[62] JI Y Q, LI J X, YANG Y H. A characterization of bilateral operator weighted shiftsbeing Cowen-Douglas operators [J]. Proc. Amer. Math. Soc,2001,129(11):3025–3210.
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