Banach空间上的强不可约算子
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摘要
在有限维空间的矩阵论中,著名的Jordan标准形定理充分揭示了矩阵的内在结构,从Jordan标准形定理可以看出,Jordan块在矩阵论中起着基本而重要的作用.在无穷维可分Hilbert空间中,强不可约算子己被蒋春澜等证明为Jordan块的合适类似物.而且他们建立了算子在相似意义下唯一强不可约分解定理,得到了强不可约算子的谱图象和紧摄动结果以及应用K理论来寻找算子的完全相似不变量,事实上,他们已经建立了无穷维可分Hilbert空间上强不可约算子的一套系统的理论.毫无疑问,考虑建立一般Banach空间中相应的理论不仅是非常重要的,也是非常必要的.本文的主要目的就是研究一般Banach空间上的强不可约算子的包括存在性等在内的重要基本性质.全文共分六章.
第一章简要回顾了强不可约算子研究的背景以及发展概况.
第二章讨论强不可约算子的存在性,证明了当Banach空间X的共轭空间X~*w~*可分时,X上存在强不可约算子;研究强不可约算子具有的基本性质,说明强不可约算子具有非有限秩、非代数算子等性质;探讨强不可约算子与Cowen-Douglas算子的关系,证明了若X=c_0或ι_p(1≤p≤∞),则对任意1≤n≤∞,存在T∈B_n(Ω)∩(SI)(X).
第三章给出上三角算子矩阵的本性谱填洞结果,即对上三角算子矩阵M_C=(?)∈B(x×y),有σ_τ(A)∪σ_τ(B)=σ_τ,(M_C)∪W,其中W(?)σ_τ(A)∩σ_τ(B)是σ_τ(M_C)的某些洞的并,这里σ_τ对σ_b与σ_e成立,但对σ_K与σ_ι,σ_r,σ_ιe,σ_re不成立;给出M_2(A)中上三角矩阵相应的谱填洞结果;讨论算子矩阵成为强不可约算子的条件.
第四章研究遗传不可分解空间上的强不可约算子的性质.根据遗传不可分解空间特殊的算子构成,给出其上强不可约算子具有的特殊性质,例如当T是遗传不可分解空间上的强不可约算子时,kerT(?)(?);以及A'(T)/radA'(T)≈C(?)T∈(SI)此外,当算子T的点谱或压缩谱为空时,T是强不可约的.给出有限维强不可约算子的定义,说明其性质.
第五章讨论遗传不可分解空间上强不可约算子的小紧摄动问题,证明了几种特殊的单点谱算子能小紧摄动成为强不可约算子,即当具有单点谱{0}的算子T满足如下条件((1)dimkerT<∞;(2)dim(x/(?))<∞;(3)dim(kerT/(kerT∩(?))=∞;(4)dimkerT=dim(x/(?))=∞,kerT(?)(?),dim((?)/kerT)<∞)之一时,T能小紧摄动成为强不可约算子;证明有有限维Schauder分解空间上的具有单点谱的对角算子可以小紧摄动成为强不可约算子.
第六章应用K理论工具来研究算子的强不可约分解,得到了与无穷维可分Hilbert空间上结果类似的定理,即A(?)∑_(i=1)~k(?)A_i~(ni),其中A_i∈(SI),i=1,2,…,k,A_i(?)A_j(i≠j)当且仅当存在从∨(A'(A))到N~((k))上的同构h,使得h([I])=n_1e_1+n_2e_2+…+n_ke_k,其中I是A'(A)中的恒等算子,{e_i}_(i=1)~k是N~((k))的生成元,0≠n_i∈N.特别地,在遗传不可分解空间上,两个强不可约算子直和换位代数的K_0群可以用来刻画它们的相似性.
In the matrix theory of finite dimensional spaces, the famous Jordan Standard Theorem sufficiently reveals the internal structure of matrix. From the Jordan Standard Theorem, it is obvious that the Jordan block plays a fundamental and important role in matrix theory. In infinite dimensional separable Hilbert spaces, C.L.Jiang and his cooperators have proved that strongly irreducible operator is a suitable analogue of Jordan block in the set of all bounded linear operators. And they have founded the theorems concerning the unique strongly irreducible decomposition of operators in the sense of similarity, the spectral picture and compact perturbations of strongly irreducible operators. They have also used the K-theory language to find the complete similarity invariant. In fact, they have formed a theoretical system of strongly irreducible operators in infinite dimensional separable Hilbert spaces. Undoubtedly, it is very important and very essential to build the corresponding theory on general Banach spaces. The main purpose of this thesis is to research the important basic properties of strongly irreducible operators on general Banach spaces, which includes the existence of strongly irreducible operators. This thesis consists of six chapters.
Chapter 1 presents a survey of the study of strongly irreducible operators.
Chapter 2 discusses the existence of strongly irreducible operators, proves that there is a strongly irreducible operator on the Banach space X with the property that conjugate space X~* is separable under the w~* topology; researches the basic properties of strongly irreducible operators, shows that if T is a strongly irreducible operator,then T isn't of finite rank and isn't an algebraic operator. And the relation between strongly irreducible operators and Cowen-Douglas operators is considered, it is showed that if X=c_0 orι_p(1≤p<∞),then there exists T∈B_n(Ω)∩(SI)(X)for every 1≤n≤∞.
Chapter 3 gives the result of "filling in holes" of essential spectra of upper triangular operator matrices, that is, for upper triangular operator matrix M_C=(?)∈B(X×Y),σ_τ(A)∪σ_τ(B)=σ_τ(M_C)∪W,where W is the union of certain of the holes inσ_τ(M_C) which happen to be subsets ofσ_τ(A)∩σ_τ(B),σ_τcan be equal toσ_b andσ_e,and the result is not true ifσ_T is equal toσ_K andσ_ι,σ_r,σ_(ιe),σ_(re); presents the corresponding result of "filling in holes" of spectra of upper triangular matrices in M_2(A).And the conditions is discussed such that operator matrices become strongly irreducible operators.
Chapter 4 researches the properties of strongly irreducible operators on hereditarily indecomposable space. Shows the special property of strongly irreducible operators on hereditarily indecomposable space according to the special operator structure.For example, if T is a strongly irreducible operator, then ker T (?) (?); and A'(T)/radA'(T)≈C if and only if T∈(SI);moreover,ifσ_p(T)=(?) orσ_γ(T)=(?),then T∈(SI).Gives the definition of finite dimensional strongly irreducible operators and shows some properties of it.
Chapter 5 discusses the small and compact perturbations problem of strongly irreducible operators on hereditarily indecomposable space, shows that some kinds of operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation, that is, if the operator T with singleton spectra {0} satisfies one of the following conditions: (1)dim kerT <∞; (2)dim(X/(?))<∞; (3)dim(ker T/(ker T∩(?)))=∞;(4)dim ker T=dim(X/(?))=∞,ker T(?) (?),dim((?)/ ker T)<∞,then T can become strongly irreducible operators by a small and compact perturbation. Proves that diagonal operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation on the space which has a finite dimensional Schauder decomposition.
Chapter 6 uses the K-theory language to research the strongly irreducible decomposition,obtains some theorems which are similar to the conclusions on infinite dimensional separable Hilbert spaces, that is, A-∑_(i=1)~k(?)A_i~((ni)),where A_i∈(SI), i=1,2,…,k,A_i(?)A_j(i≠j) if and only if there exists a isomorphism h from∨(A'(A)) onto N~((k)) with h([I])=n_1e_1+n_2e_2+…+n_ke_k,where I is the identityoperator in A'(A),{e_i}_(i+1)~k are the generators of N~((k)),and 0≠n_i∈N.In particular, the K_0-group of commutant of direct sum of two strongly irreducible operators can been characterized the similarity of them in hereditarily indecomposable space.
第一章简要回顾了强不可约算子研究的背景以及发展概况.
第二章讨论强不可约算子的存在性,证明了当Banach空间X的共轭空间X~*w~*可分时,X上存在强不可约算子;研究强不可约算子具有的基本性质,说明强不可约算子具有非有限秩、非代数算子等性质;探讨强不可约算子与Cowen-Douglas算子的关系,证明了若X=c_0或ι_p(1≤p≤∞),则对任意1≤n≤∞,存在T∈B_n(Ω)∩(SI)(X).
第三章给出上三角算子矩阵的本性谱填洞结果,即对上三角算子矩阵M_C=(?)∈B(x×y),有σ_τ(A)∪σ_τ(B)=σ_τ,(M_C)∪W,其中W(?)σ_τ(A)∩σ_τ(B)是σ_τ(M_C)的某些洞的并,这里σ_τ对σ_b与σ_e成立,但对σ_K与σ_ι,σ_r,σ_ιe,σ_re不成立;给出M_2(A)中上三角矩阵相应的谱填洞结果;讨论算子矩阵成为强不可约算子的条件.
第四章研究遗传不可分解空间上的强不可约算子的性质.根据遗传不可分解空间特殊的算子构成,给出其上强不可约算子具有的特殊性质,例如当T是遗传不可分解空间上的强不可约算子时,kerT(?)(?);以及A'(T)/radA'(T)≈C(?)T∈(SI)此外,当算子T的点谱或压缩谱为空时,T是强不可约的.给出有限维强不可约算子的定义,说明其性质.
第五章讨论遗传不可分解空间上强不可约算子的小紧摄动问题,证明了几种特殊的单点谱算子能小紧摄动成为强不可约算子,即当具有单点谱{0}的算子T满足如下条件((1)dimkerT<∞;(2)dim(x/(?))<∞;(3)dim(kerT/(kerT∩(?))=∞;(4)dimkerT=dim(x/(?))=∞,kerT(?)(?),dim((?)/kerT)<∞)之一时,T能小紧摄动成为强不可约算子;证明有有限维Schauder分解空间上的具有单点谱的对角算子可以小紧摄动成为强不可约算子.
第六章应用K理论工具来研究算子的强不可约分解,得到了与无穷维可分Hilbert空间上结果类似的定理,即A(?)∑_(i=1)~k(?)A_i~(ni),其中A_i∈(SI),i=1,2,…,k,A_i(?)A_j(i≠j)当且仅当存在从∨(A'(A))到N~((k))上的同构h,使得h([I])=n_1e_1+n_2e_2+…+n_ke_k,其中I是A'(A)中的恒等算子,{e_i}_(i=1)~k是N~((k))的生成元,0≠n_i∈N.特别地,在遗传不可分解空间上,两个强不可约算子直和换位代数的K_0群可以用来刻画它们的相似性.
In the matrix theory of finite dimensional spaces, the famous Jordan Standard Theorem sufficiently reveals the internal structure of matrix. From the Jordan Standard Theorem, it is obvious that the Jordan block plays a fundamental and important role in matrix theory. In infinite dimensional separable Hilbert spaces, C.L.Jiang and his cooperators have proved that strongly irreducible operator is a suitable analogue of Jordan block in the set of all bounded linear operators. And they have founded the theorems concerning the unique strongly irreducible decomposition of operators in the sense of similarity, the spectral picture and compact perturbations of strongly irreducible operators. They have also used the K-theory language to find the complete similarity invariant. In fact, they have formed a theoretical system of strongly irreducible operators in infinite dimensional separable Hilbert spaces. Undoubtedly, it is very important and very essential to build the corresponding theory on general Banach spaces. The main purpose of this thesis is to research the important basic properties of strongly irreducible operators on general Banach spaces, which includes the existence of strongly irreducible operators. This thesis consists of six chapters.
Chapter 1 presents a survey of the study of strongly irreducible operators.
Chapter 2 discusses the existence of strongly irreducible operators, proves that there is a strongly irreducible operator on the Banach space X with the property that conjugate space X~* is separable under the w~* topology; researches the basic properties of strongly irreducible operators, shows that if T is a strongly irreducible operator,then T isn't of finite rank and isn't an algebraic operator. And the relation between strongly irreducible operators and Cowen-Douglas operators is considered, it is showed that if X=c_0 orι_p(1≤p<∞),then there exists T∈B_n(Ω)∩(SI)(X)for every 1≤n≤∞.
Chapter 3 gives the result of "filling in holes" of essential spectra of upper triangular operator matrices, that is, for upper triangular operator matrix M_C=(?)∈B(X×Y),σ_τ(A)∪σ_τ(B)=σ_τ(M_C)∪W,where W is the union of certain of the holes inσ_τ(M_C) which happen to be subsets ofσ_τ(A)∩σ_τ(B),σ_τcan be equal toσ_b andσ_e,and the result is not true ifσ_T is equal toσ_K andσ_ι,σ_r,σ_(ιe),σ_(re); presents the corresponding result of "filling in holes" of spectra of upper triangular matrices in M_2(A).And the conditions is discussed such that operator matrices become strongly irreducible operators.
Chapter 4 researches the properties of strongly irreducible operators on hereditarily indecomposable space. Shows the special property of strongly irreducible operators on hereditarily indecomposable space according to the special operator structure.For example, if T is a strongly irreducible operator, then ker T (?) (?); and A'(T)/radA'(T)≈C if and only if T∈(SI);moreover,ifσ_p(T)=(?) orσ_γ(T)=(?),then T∈(SI).Gives the definition of finite dimensional strongly irreducible operators and shows some properties of it.
Chapter 5 discusses the small and compact perturbations problem of strongly irreducible operators on hereditarily indecomposable space, shows that some kinds of operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation, that is, if the operator T with singleton spectra {0} satisfies one of the following conditions: (1)dim kerT <∞; (2)dim(X/(?))<∞; (3)dim(ker T/(ker T∩(?)))=∞;(4)dim ker T=dim(X/(?))=∞,ker T(?) (?),dim((?)/ ker T)<∞,then T can become strongly irreducible operators by a small and compact perturbation. Proves that diagonal operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation on the space which has a finite dimensional Schauder decomposition.
Chapter 6 uses the K-theory language to research the strongly irreducible decomposition,obtains some theorems which are similar to the conclusions on infinite dimensional separable Hilbert spaces, that is, A-∑_(i=1)~k(?)A_i~((ni)),where A_i∈(SI), i=1,2,…,k,A_i(?)A_j(i≠j) if and only if there exists a isomorphism h from∨(A'(A)) onto N~((k)) with h([I])=n_1e_1+n_2e_2+…+n_ke_k,where I is the identityoperator in A'(A),{e_i}_(i+1)~k are the generators of N~((k)),and 0≠n_i∈N.In particular, the K_0-group of commutant of direct sum of two strongly irreducible operators can been characterized the similarity of them in hereditarily indecomposable space.
引文
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