关于代数拟*-n-仿正规算子的Weyl型定理(英文)

文中主要证明了:(1)若T是一个代数拟*-n-仿正规算子,则T是极.(2)若T是一个代数拟*-n-仿正规算子,则Weyl定理对f(T)成立且f∈H(σ(T)),其中f是σ(T)开邻域上的解析函数.(3)若T*是一个代数拟*-n-仿正规算子,则广义α-Weyl定理对f(T)成立,其中f∈H(σ(T)).     

(n,k)-拟仿正规算子的广义Weyl型定理和谱的连续性

若T或T*是某可分Hilbert空间上的(n,k)-拟仿正规算子,则f(T)满足广义Weyl定理;进一步地,若T*是完全(n,k)-拟仿正规算子,则f(T)满足广义a-Weyl定理,其中f∈H(σ(T))满足在其定义域的每一个连通分支上是非常值的.最后,证明谱在(n,k)-拟仿正规算子类上是连续的.     

解析余亚正规算子的α-Weyl定理

若T有单值延伸性且T为reguloid算子,则Weyl定理对f(T)成立,其中f∈H(σ(T)),而当T~*有单值延伸性且T是reguloid算子,α-Weyl定理对f(T)成立,其中,f∈H(σ(T)),作为定理应用,我们证明了Weyl定理对解析M-亚正规算子成立,α-Weyl定理对解析余亚正规算子成立。     

拟-k-仿正规算子的一个注解

设T是复希尔伯特空间H上的有界线性算子,若对任意的x∈H,T满足||T~(k+2)x||||Tx||~k≥||T~2x||~(k+1),则称T为拟-k-仿正规算子,其中k为正整数.该文给出了拟-k-仿正规算子的一些性质,如拟-k-仿正规算子是极,作为此性质的应用,证明了拟-k-仿正规算子满足Weyl定理.     

包含广义逆的算子乘积的谱

给定三个算子A,B,C∈ B(H),其中算子B的值域R(B)是闭的,利用算子矩阵分块技巧给出了∪σ(AB(1))C)=C的充分必要条件,其中σ(D)是算子D ∈B(H)的B(1) ∈B{1}谱,B{1}={X∈B(H):B×B=B}.     

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.     

关于导算子的若干问题

<正> 设X_1,X_2是Banach空间.对A∈B(X_1),B∈B(X_2),定义广义导算子:δ_(AB)|T→AT-TB,T∈B(X_2,X_1). 当X_1=X_2=X时,设A=B,则称δ_(AA)(≡δ_A)为内导算子,简称导算子. 本文分两部分.前一部分讨论几个有关导算子值域的未解决问题;后一部分讨论刻     

反对角算子矩阵及其平方的(ω)性质的摄动

设H为复的无限维可分的Hilbert空间,B(H)为H上的有界线性算子的全体.若σ_a(T)\σ_(ea)(T)=π_(00)(T),则称T∈B(H)满足(ω)性质,其中σ_a(T)和σ_(ea)(T)分别表示算子T的逼近点谱和本质逼近点谱,π_(00)(T)={λ∈isoσ(T):0dimN(T-λI)∞}.T∈B(H)称为满足(ω)性质的摄动,若对任意的紧算子K,T+K满足(ω)性质.本文证明了反对角算子矩阵及其平方具有(ω)性质的摄动的等价性.     

算子的拟仿射变换与拟仿射逆

设H是可分的复Hilbert空间,B(H)是H上全体有界线性算子的代数。以后把B(H)的元简单地叫做算子。对于算子T∈B(H),用R(T)、N(T)、σ(T)及LatT分别表示其值域、零空间、谱及不变子空间的格。算子X∈B(H)叫做拟仿射,如果它满足N(X)=N(X~*)={0}。若T、S、X∈B(H),X是拟仿射,TX=XS,则S叫做T的拟仿射变换。与此类似的一个概念是:若TXS=X,X是拟仿射,则T(S)叫做S(T)的左(右)拟仿射逆([1])。在§1中,找到了有左(右)拟仿射逆的算子是可逆的一些     

C2（H）上广义导算子

本文主要给出了 C_2(H)上广义导算子δ_(A,B)、τ_(A,B)为弱正规算子、Jordon 类算子的充要条件。     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$

Let $\mathcal{B}(\mathcal{H})$ be the $C^∗$-algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. It is proved that an additive surjective map $φ$ on $\mathcal{B}(\mathcal{H})$ preserving the star partial order in both directions if and only if one of the following assertions holds. (1) There exist a nonzero complex number $α$ and two unitary operators $\boldsymbol{U}$and$\boldsymbol{V}$ on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$. (2) There exist a nonzero $α$ and two anti-unitary operators$\boldsymbol{U}$and$\boldsymbol{V}$on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$.     

On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma \to \Gamma$, $\sigma_{\varphi}:X^{\Gamma} \to X^{\Gamma}$with $\sigma_{\varphi}((x_{\alpha})_{\alpha \in \Gamma})=(x_{\varphi(\alpha)})_{\alpha \in \Gamma}$ (for $(x_{\alpha})_{\alpha \in \Gamma} \in X^{\Gamma}$) is a generalized shift. In this text for $\mathcal{S} = \{\sigma_{\varphi}:\varphi \in \Gamma^{\Gamma}\}$ and $\mathcal{H}=\{\sigma_{\varphi}:\Gamma \xrightarrow{\varphi} \Gamma$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$. Regarding proximal relation we prove: $$P(\mathcal{S}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \exists \beta \in \Gamma (x_{\beta} = y_{\beta})\}$$and $P(\mathcal{H}, X^{\Gamma} ) \subseteq \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\beta \in \Gamma : x_{\beta} = y_{\beta}\}$ is infinite$\}$ $\cup\{($ $x,x) : x \in \mathcal{X}\}$. Moreover, for infinite $\Gamma$, both transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$ are regionally proximal, i.e., $Q(\mathcal{S}, X^{\Gamma}) = Q(\mathcal{H}, X^{\Gamma} ) = X^{\Gamma} \times X^{\Gamma}$, also for sydetically proximal relation we have $L(\mathcal{H}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\gamma ∈ \Gamma :$ $x_{\gamma} \neq y_{\gamma}\}$ is finite$\}$.     

THE FUNCTION {\tf d}$_{{\boldkey k}}$({\tf n}) AT CONSECUTIVE INTEGERS

Let $d_{k}(n)$ denote the $k$-fold iterated divisor function $(k\geq 2)$. It is proved that for sufficiently large $x$, $d_{k}(n)=d_{k}(n+1)$ holds for $\gg x(\log\log x)^{-3}$ integers $n\leq x$.     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

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”.     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.