2×2阶上三角型算子矩阵的Moore-Penrose谱

设$H_{1}$和$H_{2}$是无穷维可分Hilbert空间. 用$M_{C}$表示$H_{1}\oplusH_{2}$上的2$\times$2阶上三角型算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\\end{array}\right)$. 对给定的算子$A\in{\mathcal{B}}(H_{1})$和$B\in{\mathcal{B}}(H_{2})$,描述了集合$\bigcap\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$与$\bigcup\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$,其中$\sigma_{M}(\cdot)$表示Moore-Penrose谱.     

具有一对零态射的Morita Context环(II)

设$(A,B,V,W,\psi,\phi)$是一个Morita Context,具有一对零态射$\psi=0$, $\phi=0$, $C =\left ( \begin{array} {cc}A & V \\W & B \end{array}\right)$是对应的Morita Context环.本文给出了$C$与$A,B,V,W$之间关于环的$\pi$-正则性、semiclean性、Mophic性和环的Exchgange性、Potent性、GM性的关系.     

A Hypothesis Testing Problem in the Linear Model (in English)

本文讨论了多元线性模型中的一个假设检验问题。假定 $\[{E(Y) = A\theta + B\eta }\]$ $Y的各行独立、正太、同协差阵V$ 现在要检验假设H_0:存在矩阵C使$\theta= C\eta$ 是否成立。首先可将问题化为法式的形式，对法式分两种情况进行讨论： （一）$[V = {\sigma ^2}I,{\sigma ^2}\]$未知，此时可求出 \theta,C，\sigma ^2的最大似然估计（当 H^0成立时）是 $[\left\{ {\begin{array}{*{20}{c}} {\hat \theta = {{({I_p} + \hat C'\hat C)}^{ - 1}}({y_1} + \hat C'{y_2})}\{\hat C = - {{({{T'}_{22}})}^{ - 1}}{{T'}_{12}}}\{{{\hat \sigma }^2} = \frac{1}{{nk}}(\sum\limits_{j = p + 1}^{p + q} {\lambda _j^* + \sum\limits_{j = 1}^k {{d_j})} } } \end{array}} \right.\]$ 其中y_1,y_2是法式 $[E\left( {\begin{array}{*{20}{c}} {{y_1}}\{{y_2}}\{{y_3}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} \theta \\eta \0 \end{array}} \right)\begin{array}{*{20}{c}} p\q\{n - (p + q)} \end{array}\]$ 中的资料阵y_1,y_2,d_1,\cdots，d_k是y^'_3y_3的全部特征根，$[\lambda _1^* \ge \cdots \lambda _{p + q}^*\]$是$[\left( {\begin{array}{*{20}{c}} {{y_1}}\{{y_2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{y'}_1}}&{{{y'}_2}} \end{array}} \right)\]$的全部特征根，相应特征向量依$\lambda^*_i$的大小顺序从左到右排成矩阵T，T的分块子阵是T_ij，即 $[T = \left( {\begin{array}{*{20}{c}} {{T_{11}}}&{{T_{12}}}\{{T_{21}}}&{{T_{22}}} \end{array}} \right)\begin{array}{*{20}{c}} p\q \end{array}\]$ 对H_0的广义似然比检验是 $[\Lambda = \sum\limits_{j = p + 1}^k {{\lambda _j}/\sum\limits_{j = 1}^k {{d_j}} } \]$ $=lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k$是$y_1^'y_1+y_2^'y_2$的全部特征根。 （二）一般情形V未知，此时 \theta,C的估计量同前，可求出 $[\hat V = \frac{1}{n}({y_2}^\prime {T_{22}}{T_{22}}^\prime {y_2} + {y_2}^\prime {y_2})\]$ H_0相应的Lawley不变检验是 $[\sum\limits_{j = p + 1}^k {{\beta _j}} \ge {\alpha _1}\]$ 其中 $\beta_1 \geq \beta_2 \geq \cdots \beta_k$是$y'_1y_1+y'_2y_2$的相应于$y'_sy_s$的全部特征根。 有关$\Lambda \$的以及$[\sum\limits_{j = p + 1}^k {{\beta _j}} \]$的极限分布将在另外的文章中讨论。     

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     

在Banach空间中推广的 Roper-Suffridge算子(III)

假定 $X$ 是具有范数$\|\cdot\|$的复 Banach 空间, $n$ 是一个满足 $\dim X\geq n\geq2$的正整数. 本文考虑由下式定义的推广的Roper-Suffridge算子 $\Phi_{n,\beta_2, \gamma_2, \ldots , \beta_{n+1}, \gamma_{n+1}}(f)$: \begin{equation} \begin{array}{lll} \Phi _{n, \beta_2, \gamma_2, \ldots, \beta_{n+1},\gamma_{n+1}}(f)(x) &;\hspace{-3mm}=&;\hspace{-3mm}\dl\he{j=1}{n}\bigg(\frac{f(x^*_1(x))}{x^*_1(x)})\bigg)^{\beta_j}(f''(x^*_1(x))^{\gamma_j}x^*_j(x) x_j\\ &;&;+\bigg(\dl\frac{f(x^*_1(x))}{x^*_1(x)}\bigg)^{\beta_{n+1}}(f''(x^*_1(x)))^{\gamma_{n+1}}\bigg(x-\dl\he{j=1}{n}x^*_j(x) x_j\bigg),\nonumber \end{array} \end{equation} 其中 $x\in\Omega_{p_1, p_2, \ldots, p_{n+1}}$, $\beta_1=1, \gamma_1=0$ 和 \begin{equation} \begin{array}{lll} \Omega_{p_1, p_2, \ldots, p_{n+1}}=\bigg\{x\in X: \dl\he{j=1}{n}| x^*_j(x)|^{p_j}+\bigg\|x-\dl\he{j=1}{n}x^*_j(x)x_j\bigg\|^{p_{n+1}}<1\bigg\},\nonumber \end{array} \end{equation} 这里 $p_j>1 \,( j=1, 2,\ldots, n+1$), 线性无关族 $\{x_1, x_2, \ldots, x_n \}\subset X $ 与 $\{x^*_1, x^*_2, \ldots, x^*_n \}\subset X^* $ 满足 $x^*_j(x_j)=\|x_j\|=1 (j=1, 2, \ldots, n)$ 和 $x^*_j(x_k)=0 \, (j\neq k)$, 我们选取幂函数的单值分支满足 $(\frac{f(\xi)}{\xi})^{\beta_j}|_{\xi=0}= 1$ 和 $(f''(\xi))^{\gamma_j}|_{\xi=0}=1, \, j=2, \ldots , n+1$. 本文将证明: 对某些合适的常数$\beta_j, \gamma_j$, 算子$\Phi_{n,\beta_2, \gamma_2, \ldots, \beta_{n+1}, \gamma_{n+1}}(f)$ 在$\Omega_{p_1, p_2, \ldots , p_{n+1}}$上保持$\alpha$阶的殆$\beta$型螺形映照和 $\alpha$阶的$\beta$型螺形映照.     

素*-代数上的非线性混合双斜Jordan三重导子

设$\mathcal{A}$是一个包含非平凡投影的单位素*-代数.本文证明了一个映射$\Phi:\mathcal{A}\rightarrow\mathcal{A}$满足对任意$A,B,C\in\mathcal{A}$有$\Phi([A,B]_{\diamond}\circC)=[\Phi(A),B]_{\diamond}\circC+[A,\Phi(B)]_{\diamond}\circC+[A,B]_{\diamond}\circ\Phi(C)$当且仅当$\Phi$是一个可加的*-导子, 其中$A\circ B=A^{*}B+B^{*}A$和$[A,B]_{\diamond}=A^{*}B-B^{*}A$.     

关于$3\times 3$阶三角矩阵环上模的研究

文章对$3\times 3$阶三角矩阵环$$\Gamma = \left(\begin{array}{ccc}T & 0 & 0 \\M & U & 0\\{N \otimes _U M} & N & V \\\end{array}\right)$$上的模作了研究,其中T,U,V均是环, M,N分别是U-T, V-U双模.通过用一个五元组$(A,B,C;f,g)$来描述一个左$\Gamma$-模 (其中$A \in \mod T, B\in {\rm mod} U, C \in {\rm mod} V$, $f:M \otimes _T A \to B \in {\rm mod} U, g:N \otimes _U B \to C \in {\rm mod} V$), 文章分别刻画了$\Gamma$上的一致模、空的模、有限嵌入模,并且确定了${ }_\Gamma (A \oplus B \oplus C)$的根和基座．     

SECOND INITIAL-BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR HYPERBOLIC-PARABOLIC COUPLED SYSTEMS

On a rectangular domain \[R(\delta ) = \{ 0 \leqslant t \leqslant \delta ,0 \leqslant x \leqslant 1\} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\] We consider the second initial-boundary value problem for the quasi-linear hyperbolic- parabolic coupled system \[{\begin{array}{*{20}{c}} {\sum\limits_{j = 1}^n {{\zeta _{ij}}(t,x,u,v)(\frac{{\partial {u_j}}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial {u_j}}}{{\partial x}})} } \\ { = {\zeta _l}(t,x,u,v)(\frac{{\partial v}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial v}}{{\partial x}})} \\ { + {\mu _l}(t,x,u,v,{v_x}),(l = 1,...,n)} \\ {\frac{{\partial v}}{{\partial t}} - a(t,x,u,v,{v_x})\frac{{{\partial ^2}v}}{{\partial {x^2}}} = b(t,x,u,v,{v_x})} \end{array}}\] without loss of generatity,the initial conditions may be written as \[t = 0,{u_j} = 0,(j = 1,...,n),v = 0\] and we can suppose that \[\left\{ {\begin{array}{*{20}{c}} {a(0,x,0,0,0) \equiv 1} \\ {b(0,x,0,0,0) \equiv 0} \\ {{\zeta _{ij}}(0,x,0,0) \equiv {\delta _{lj}} = \left\{ {\begin{array}{*{20}{c}} {1,if{\kern 1pt} {\kern 1pt} {\kern 1pt} l = j} \\ {0,if{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \ne j} \end{array}} \right.} \end{array}} \right.\] The boundary conditions are as follows: \[\begin{gathered} on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}} {{u_{\bar r}} = {G_{\bar r}}(t,u,v),(\bar r = 1,...,h;h \leqslant n)} \\ {\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);} \end{array}} \right. \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}} {{u_{\hat s}} = {{\hat G}_{\hat s}}(t,u,v),(\hat s = m + 1,...,n;m \geqslant 0)} \\ {\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right. \hfill \\ \end{gathered} \] Uf = Q-f(t> u, x), (r = 1> k^n), We assume that the following conditions are satisfied: (1) the orientability condition \[\begin{gathered} {\lambda _{\bar r}}(0,1,0,0,0) < 0,{\lambda _s}(0,1,0,0,0) > 0,\left( {\begin{array}{*{20}{c}} {\bar r = 1,...,h} \\ {s = h + 1,...,n} \end{array}} \right) \hfill \ {\lambda _{\bar r}}(0,0,0,0,0) < 0,{\lambda _{\hat s}}(0,0,0,0,0) > 0,\left( {\begin{array}{*{20}{c}} {\hat r = 1,...,m} \\ {\hat s = m + 1,...,n} \end{array}} \right) \hfill \\ \end{gathered} \] (2) the compatibility condition \[\begin{gathered} \frac{{\partial {G_{\bar r}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,1,0,0,0) = {\mu _{\bar r}}(0,1,0,0,0) \hfill \ \frac{{\partial {{\hat G}_{\hat s}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,0,0,0,0) = {\mu _{\hat s}}(0,0,0,0,0) \hfill \ (\bar r = 1,...,h;\hat s = m + 1,...,n);{F_ \pm }(0,0,0) = 0 \hfill \\ \end{gathered} \] (3) the condition of characterizing number \[\begin{gathered} \sum\limits_{j = 1}^n {\left| {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1 \hfill \ \sum\limits_{j = 1}^n {\left| {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1(\bar r = 1,...,h,\hat s = m + 1,...,n \hfill \\ \end{gathered} \] (4)The smoothness condition: the coefficients of the system and the boundary conditions are suitably smooth. By means of certain a priori estimations for the solution of the heat equation and the linear hyperbolic system, using an iteration method and Leray-Schauder fixed point theorem, we have proved Theorem 1. Under the preceding hypotheses, for the second initial-boundary value problem (2)—(4), (6), (7), there exists uniquely a classical solution on R(8) where \[\delta \]>0 is suitably small. Theorem 2. In theorem the 1,condition of characterizing number (13) may be ameliorated as the following solvable condition; \[\left\{ {\begin{array}{*{20}{c}} {\det |({\delta _{\bar rr'}} - \frac{{\partial {G_{\bar r}}}}{{\partial {u_{r'}}}}(0,0,0)| \ne 0,(\bar r,r' = 1,...,h)} \\ {\det |({\delta _{\hat s\hat s'}} - \frac{{\partial {G_{\hat s}}}}{{\partial {u_{\hat s'}}}}(0,0,0)| \ne 0,(\hat s,\hat s' = m + 1,...,n)} \end{array}} \right.\] i.e,the boundary condition (6),(7)may be written as \[\begin{gathered} on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}} {{u_{\bar r}} = {H_{\bar r}}(t,{u_s},v),} \\ {\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);} \end{array}} \right.\left( {\begin{array}{*{20}{c}} {\bar r = 1,...,h} \\ {s = h + 1,...,n} \end{array}} \right) \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}} {{u_{\hat s}} = {H_{\hat s}}(t,{u_{\hat r}},v){\kern 1pt} ,} \\ {\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right.\left( {\begin{array}{*{20}{c}} {\hat r = 1,...,m} \\ {\hat s = m + 1,...,n} \end{array}} \right) \hfill \\ \end{gathered} \]     

Approximate Controllability of Nonlinear Delay Integro Differential Evolution Equations with Random Impulses

The basic concept of this research is to analyse the approximate controllability (AC) of a nonlinear delay integrodifferential evolution system (NDIDES) with random impulse of the type \begin{align*}&z''(\zeta)=\mathfrak{A}(\zeta)z(\zeta)+(\mathfrak{B}x)(\zeta)+\int_{0}^{\zeta}\mathcal{H}(\zeta, s,z(\beta(s))), \ \sigma_{q} <\zeta < \sigma_{q+1}, \ \zeta\in [\zeta_{0}, \mathcal{T}], \\ &z(\sigma_{q})=a_{q}(\tau_{q})z(\sigma^{-}_{q}), ~~q = 1,2,\ldots,\\ &z_{\zeta_{0}}=\upsilon,\end{align*} by assuming that the linear system is approximately controllable. The existence and uniqueness of the mild solution to above system have been determined by using the Banach contraction principle and trajectory accessible sets. We generalize the results for NDIDES with and without fixed-type impulsive moments.     

ON DISCRETE PHENOMENA IN THE MIXED PROBLEM

On discrete phenomena in uniqueness of the initial value problem, F. Treves studied an interesting example and proved that the Oauohy problem \[\left\{ \begin{array}{l} {L_p}u = {u_{xx}} - {x^2}{u_{tt}} + p{u_t} = 0,t \ge 0;\u(x,0) = {u_t}(x,0) = 0, \end{array} \right.\] has non-triyial solutions if and only if p = 3, 5, …. Wang Guang-ymg and others proved that the Oauohy problem \[\left\{ \begin{array}{l} {L_p}u = 0,t \ge 0;\u(x,0) = {\varphi _1}(x);{u_t}(x,0) = {\varphi _2}(x), \end{array} \right.\] and Goursat problem \[\left\{ \begin{array}{l} {L_p}u = 0,t \ge \frac{{{x^2}}}{2};\u(x,\frac{{{x^2}}}{2}) = {\varphi _3}(x), \end{array} \right.\] both have a unique solution if and only if p≠1, 3, 5, …. In this paper, we discuss in detail the equation Lvu = 0 for discrete phenomena. We prove that solution of the mixed problem \[\left\{ \begin{array}{l} {L_p}u = 0,x \ge 0,t \ge 0,\u(x,0) = \varphi (x),\{u_t}(x,0) = \psi (x),\u(0,t) = 0 \end{array} \right.\] is not only existent but also unique, for р≠3, 7, 11，…，neither existence nor uniqueness could be proved in this problem, for p = 3, 7, 11，….，more precisely, only under some compatibility condition can the solution exist for the equation \({L_p}u = 0\).     

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.     

Extensions of hereditary algebras

Let be a triangular matrix algebra, uhere k is an algebraically closed field, B is the path algebra of an oriented Dynkin diagram of type E₆ or E₇ or E₈ and M is a finite dimensional k-B-bimodule. The aim of this paper is to determine the representation type of A for any orientation of the Dynkin diagram and for any indecomposable B-module M. This classification is obtained by comparing the representation types of the algebras and using the theory of tilting modules.     

Weak amenability of triangular Banach algebras

Let and be unital Banach algebras, and let be a Banach -module. Then becomes a triangular Banach algebra when equipped with the Banach space norm . A Banach algebra is said to be -weakly amenable if all derivations from into its dual space are inner. In this paper we investigate Arens regularity and -weak amenability of a triangular Banach algebra in relation to that of the algebras , and their action on the module .

     

形式三角矩阵环的PS性质和CESS性质

设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\     

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.     

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

Drazin谱和算子矩阵的Weyl定理

A∈B(H)称为是一个Drazin可逆的算子,若A有有限的升标和降标．用σ_D(A)={λ∈C：A-λI不是Drazin可逆的)表示Drazin谱集．本文证明了对于Hilbert空间上的一个2×2上三角算子矩阵M_C=■,从σ_D(A)∪σ_D(G)到σ_D(M_C)的道路需要从前面子集中移动σ_D(A)∩σ_D(B)中一定的开子集,即有等式：σ_D(A)∪σ_D(B)=σ_D(M_C)∪G,其中G为σ_D(M_C)中一定空洞的并,并且为σ_D(A)∪σ_D(B)的子集．2×2算子矩阵不一定满足Weyl定理,利用Drazin谱,我们研究了2×2上三角算子矩阵的Weyl定理,Browder定理,a-Weyl定理和a-Browder定理．