四阶正则微分算子耦合自共轭边界条件的基本标准型

本文利用新的方法给出了4阶正则微分算子耦合自共轭边界条件的基本标准型,新标准型中的4个分块小矩阵为对称矩阵,且其行列式的模为1.这与2阶微分算子耦合边界条件的标准型极为类似,这为给出一般的高阶微分算子自共轭边界条件标准型提供了新的思路.     

希尔伯特空间上的李雅普诺夫定理

我们用■(C)记n维欧氏空间 C~n 上的 n×n 阶矩阵全体,其中自共轭矩阵全体记为■_n.关于矩阵的 Lyapunov 定理和 Stein 定理通常分别叙述成Lyapunov 定理.对矩阵 A∈■(C),存在一自共轭矩阵 x∈■_n,且 X>0(正定),使 AX XA~*>0的充要条件是 A 的特征值完全落在复平面的右半开平面内.这里 A~*表示 A 的转置共轭矩阵,而右半开平面是指不包含虚轴的右半平面.     

具有转移条件的四阶微分算子自共轭边界条件的统一规范型

研究了具有转移条件的四阶正则微分算子自共轭边界条件的统一规范型.在标准型的基础上通过对自共轭边界条件矩阵左乘非奇异矩阵和右乘辛矩阵给出了四阶微分算子自共轭边界条件的统一规范型.结果表明具有转移条件的四阶自共轭微分算子的边界条件的统一规范型不仅与边界条件矩阵的秩有关,而且与转移条件矩阵的行列式有关.     

正则四阶微分算子的左定边界条件

本文讨论了-类四阶微分算子的左定边界条件,利用自共轭扩张的正定性来研究左定问题.通过自共轭微分算子的系数、区间端点以及边界条件给出了问题左定性的充要条件,并相应地得到了所有四阶自共轭微分算子的左定边值矩阵的情形.     

Hilbert C~*-模上可共轭算子的并联和

本文研究了Hilbert C~*-模上可共轭算子的并联和,推广了矩阵和Hilbert空间上有界线性算子的一些相关结果.通过举例说明:存在一个Hilbert C~*-模H,以及H上的两个可共轭的正算子A和B,使得算子方程A~(1/2)=(A+B)~(1/2)X, X∈■(H)无解,其中■(H)为H上的可共轭算子全体.     

与薛定谔算子相关的Riesz算子在BMO型空间上的有界性

假设薛定谔算子L=-Δ+V中的非负位势函数V属于逆H(o|")lder函数类RH_s(s> n/2).本文我们证明了Riesz算子T_α=L~(-α)V~α(0 <α相似文献   

一类2n阶具有转移条件的对称微分算子的特征值问题

研究了具有边界条件及转移条件的2n阶对称微分算子的特征值问题.首先构建了新的Hilbert空间使得所研究的微分算子在新的Hilbert空间中是自共轭的.然后利用微分算子谱分析经典方法,得到了λ是边值问题的特征值的充要条件,并给出了边值问题特征值的某些特点.     

拟-*-A(n)算子的谱性质

我们证明了以下结论:(1)若T是拟-*-A(n)算子,则T是似正规算子.(2)若E是拟-*-A(n)算子T的非零孤立谱点λ的Riesz幂等算子,则E是自共轭的且满足R(E)=N(T-λ)=N(T-λ)*.(3)若T或T^*是代数拟-*-A(n)算子,则f(T)满足Weyl定理.(4)若T*是代数拟-*-A(n)算子,则f(T)满足Weyl定理.(4)若T^*是代数拟-*-A(n)算子,则f(T)满足α-Weyl定理,其中f∈H(σ(T)).     

针对某类小输入控制系统的Hamilton QR算法

上标*表示矩阵的共轭转置,象文[1]、[2],我们记(2)的矩阵H=HAM(A,G,F)。 由于H与—H~·相似(JHJ~(-1)=—H~·),因此H的特征值是成(λ,—λ)对出现的。解线性二次最优控制问题所需要的是求H相应于左半平面特征值所对应的不变子空间X。 A.Laub利用QR算法计算X,他把H当作一般的2n阶矩阵而忽略了H的特殊结     

2n阶微分算子乘积自伴的充分必要条件

给定Hilbert空间L2[a,∞)上两个由2n阶对称微分算式生成的微分算子Li(i=1,2),该文给出了乘积算子L2L1是自伴算子的一个充分必要条件．     

Two Suficient and Necesary Conditions for Quasidisks

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线性时滞微分方程的点态退化(英文)

本文讨论了线性时滞微分方程的点态退化问题.借助于矩阵的有关知识,我们给出了判定n阶时滞微分方程点态退化的充分必要条件及代数判据.     

Invertibility Conditions for Second-Order Differential Operators in the Space of Continuous Bounded Functions

We prove the invertibility of second-order differential operators with constant operator coefficients acting on the Banach space of bounded continuous functions on the real line under the condition that they are uniformly injective (in particular, left invertible) or surjective (in particular, right invertible). We show that if these operators are considered on the space of periodic functions, then the unilateral invertibility does not imply the invertibility of such operators. We obtain criteria for the injectivity, surjectivity, and invertibility of differential operators on the space of periodic functions.     

Analyticity of Semigroups Generated by a Class of Differential Operators with Matrix Coefficients and Interface

In this work we are interested in studying a class of mixed differential operators with matrix coefficients and the matching interface conditions, the basic question treated in this paper is that such mixed operators generate an analytic semigroup on an appropriate Hilbert space X.     

单位球的Bergman空间上Schatten类加权复合算子

本文研究了单位球的Bergman空间上Schatten类加权复合算子,得到了这种加权复合算子属于Schatten-Von Neumann.理想S_p的几个充要条件.作为推论给出了Wφ,φ是一个Hilbert- Schmidt算子的充要条件是∫_(Bn)(|ψ(ω)|~2)/((1-|φ(ω)|~2)~(n 1)dV(ω)<∞..     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

On Domination Conditions for Systems of Minimal Differential Operators Acting in the Space L ∞ $$\mathbb{R}^n $$

In this paper, we consider the linear space of minimal differential operators with constant coefficients which are dominated by systems of minimal operators acting in spaces with uniform norm. In terms of domination conditions, we prove the quasiellipticity test for a given operator; this criterion generalizes a similar result of De Leeuw and Mirkil to elliptic operators.     

两类四阶微分算子积的自伴性研究

利用算子理论及矩阵运算方法,讨论了由两类不同的对称微分算式D~((4))+D~((2))+q_1(t)和D~((4))+q_2(t)(D=d/dt,t∈I=[a,b])生成的微分算子的积算子的自伴性,获得了积算子是自伴算子的充分必要条件.     

一般迁移方程相应余项的弱紧性及其应用

在L¹空间,研究了板几何中一类具周期边界条件的各向异性、连续能量、均匀介质的迁移方程.通过构造算子,利用比较算子方法,证明了该迁移算子A相应的迁移半群V(t)(t≥0)的Dyson-Phillips展开式的伽阶余项R_n(t)(n≥1)的弱紧性,得到了半群V(t)与U(t)(streaming算子B产生)本质谱相同,本质谱型一致;迁移算子A的谱在区域Γ中由有限个具有限代数重数的离散本征值组成;迁移方程解的渐近稳定性.     

On Iterations of the Green Integrals and their Applications to Elliptic Differential Complexes

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operators P of order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equation Pu = f in the Sobolev space W^p,²(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of a P-Neumann problem for elliptic differential operators is proved.