上三角算子矩阵的几类固零谱

设M_C表示Hilbert空间H_1⊕H_2上的上三角算子矩阵M_C=(ACOB),用∩_*表示∩_(C∈B(H_2,H_1))σ_*(M_C),其中*表示某类谱,称满足等式∩_*=σ_*(M_0)的谱为固零谱,本文集中给出上三角算子矩阵的三类固零谱,并举例说明谱等式σ_*(M_0)=σ_*(A)∪σ_*(B)对这三类固零谱失效.     

2×2上三角算子矩阵的Drazin谱

设MC=[A C 0 B]是从Hilbert空间H⊕K到H⊕K中的2×2上三角算子矩阵.该文主要研究MC的Drazin可逆性和MC的Drazin谱.此外,对给定算子A∈B(H)和B∈B(K),将给出在一定条件下所有上三角算子矩阵Mc的Drazin谱的交∩C∈B(K,K)σD(MC)的具体表达式.     

2×2-上三角算子矩阵谱的Fredholm扰动

设H和K是复无穷维可分Hilbert空间,A∈B(H),B∈B(K),C∈B(K,H)且M_C=(ACOB).本文给出了上三角算子矩阵M_C的Weyl谱、本性谱、谱、左谱、右谱、下半本性谱、下半Weyl谱和上半Weyl谱的Fredholm扰动的完全刻画.     

上三角算子矩阵Browder定理的稳定性

令H为无限维且复可分的Hilbert空间,B(H)为H上的有界线性算子全体.若T∈B(H)满足σ_w(T)=σ_b(T),则称T有Browder定理,其中σ_ω(T)和σ_b(T)分别表示算子T的Weyl谱和Borwder谱;对任意的紧算子K∈B(H),若T+K有Browder定理,则称T满足Browder定理的稳定性.给出了2-阶上三角算子矩阵的平方满足Borwder定理的稳定性的充要条件.     

2×2上三角算子矩阵的Drazin谱

设MG=[ O B^A C]是从Hilbert空间H＋K到H＋K中的2×2上三角算子矩阵．该文主要研究MC的Drazin可逆性和Mc的Drazin谱．此外,对给定算子A∈B（H）和B∈B（K）,将给出在一定条件下所有上三角算子矩阵Mc的Drazin谱的交∩C∈B（K,H）σD（Mc）的具体表达式。     

上三角算子矩阵Browder定理稳定性的判定

用σ(T)和σ_w)分别表示算子T的谱与weyl谱,π_(00)(T)={λ∈isoσ(T),0dimN(T-λI)∞},若σ(T)\σ_w(T)■π_(00)(T)成立,则就认为T满足Browder定理.主要研究了2×2上三角算子矩阵的Browder定理在紧摄动下的稳定性,并给出了判定稳定性的等价条件.     

拓扑一致降标与Weyl定理的摄动

若σ(T)\σ_ω(T)■π_(00)(T),则称算子T满足Browder定理,其中σ(T)和σ_ω(T)分别表示算子T的谱和Weyl谱,且π_(00)(T)={λ∈isoσ(T);0dim N(T-λI)∞}.若σ(T)σ_ω(T)=π_(00)(T),则称T满足Weyl定理.该文利用拓扑一致降标域的特征,研究了Browder定理在紧摄动下的稳定性,并且给出了Browder定理的紧摄动具有稳定性的算子的特征.     

3×3阶上三角算子矩阵的点谱、剩余谱和连续谱

记■为Hilbert空间■上的上三角算子矩阵.我们借助对角元A,B和C的谱性质给出了σ_*(M_(D,E,F))=σ_*(A)∪σ_*(B)∪σ_*(C)对任意D∈B(H_2,H_1),E∈B(H_3,H_1),F∈B(H_3,H_2)均成立的充要条件,其中σ_*代表某类特定的谱,如点谱、剩余谱和连续谱等.此外,给出了一些例证.     

上三角算子矩阵的局部谱性质及其应用

本文研究了Banach空间中上三角算子矩阵■∈L(X⊕Y)的局部谱性质,其中A∈L(X),B∈L(Y),C∈L(Y,X),X,Y是无穷维复Banach空间,L(X,Y)表示X到Y的所有有界线性算子.首先考察了MC的单值扩张性,借助于向量值解析函数和解析核等工具给出了集合S(M_C)={λ∈C:M_C在λ没有单值扩张性}的刻画,并得到对任意C∈L((Y,X)等式S(M_C)=S(A)∪S(B)都成立的条件.进一步,研究了M_C的单值扩张性扰动,得到了对于给定A∈L(X),B∈L(Y),等式S(M_C)=S(A)∪S(B)成立时C所需的条件.同时,举例说明了这些条件的合理性.最后,把所得结果运用到上三角算子矩阵的谱和局部谱上,得到了σ(M_C)=σ(A)∪σ(B)和σ_MC(x⊕0)=σ_A(x)成立的条件,并给出了M_C局部谱子空间的一个刻画.     

有界线性算子及其函数的Browder定理的判定

令H是无限维的Hilbert空间,B(H)是H上有界线性算子的全体构成的集合.称算子T∈B(H)满足Browder定理,若σ(T)σ_w(T)?π₀₀(T)或σ_w(T)=σ_b(T),其中σ(T),σ_w(T),σ_b(T)分别表示算子T的谱集、Weyl谱、Browder谱,π₀₀(T)={λ∈isoσ(T):0 相似文献   

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (H) preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.     

算子AB和BA的Drazin可逆性

给定Hilbert空间${\cal H}$上的有界线性算子$A$和$B$, 本文证明了$AB$和$BA$的Drazin可逆性是等价的. 作为应用, 我们证明了$\sigma_D(AB)=\sigma_D(BA)$和$\sigma_D(A)=\sigma_D(\widetilde{A})$,这里$\sigma_D(M)$和$\widetilde{M}$分别表示算子$M$的Drazin谱和Aluthge变换.     

以仿正规算子为参数的初等算子的性质

若对x∈H,‖Tx‖~2≤‖T~2x‖‖x‖,则称T是仿正规算子.d_(AB)表示δ_(AB)或△_(AB),其中δ_(AB)和△_(AB)分别表示Banach空间B(H)上的广义导算子和初等算子,其定义为δ_(AB)X=AX-XB,△_(AB)X=AXB-X,X∈B(H).若A和B~*是仿正规算子,则可证d_(AB)是polaroid算子,f∈H(σ(d_(AB))),f(d_(AB))满足广义Weyl定理,f(d_(AB)~*)满足广义a-Weyl定理,其中H(σ(d_(AB)))表示在σ(d_(AB))的某邻域上解析的函数全体.     

一类缺项算子矩阵的四类点谱的扰动

有界线性算子的点谱可进一步细分为4类,分别为$\sigma_{p1}$, $\sigma_{p2}$, $\sigma_{p3}$ 和$\sigma_{p4}$.设 $H, K$为无穷维可分的Hilbert空间,用$M_C$表示$2\times 2$上三角算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\ \end{array} \right)$,对于给定的 $A\in B(H),~B\in B(K)$,描述了集合$\bigcap\limits_{C\in B(K,H)}\sigma_{p1}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p2}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p3}(M_C)$和$\bigcap\limits_{C\in B(K,H)}\sigma_{p4}(M_C)$.     

Weyl spectra of operator matrices

In this paper it is shown that if is a upper triangular operator matrix acting on the Hilbert space and if denotes the ``Weyl spectrum", then the passage from to is accomplished by removing certain open subsets of from the former, that is, there is equality where is the union of certain of the holes in which happen to be subsets of .

     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.