Hilbert空间上线性算子广义逆A(2)T,S的存在性及其表示式

设H₁和H₂是两个Hilbert空间, B(H₁,H₂)表示从H₁到H₂的所有有界线性算子的集合, T和S分别是H₁和H₂的两个闭子空间. 如果存在线性算子X ∈ B(H₂,H₁)满足XAX=X, R(X)=T, N(X)=S,则称X为线性算子$A$的具有指定像空间T和零空间S的外逆,记为A⁽²⁾_T,S. 该文进一步研究了线性算子广义逆A⁽²⁾_T,S存在的若干等价条件及其性质,建立了算子广义逆A⁽²⁾_T,S的表示形式.     

Moore-Penrose逆和Drazin逆的正则分解表示及其扰动

1引言及预备知识 设X,Y为Banach空间,B(X,Y)表示从X到Y中的有界线性算子组成的Banach空间.简记B(X,X)为B(X).对算子T∈B(X,Y),R(T)与N(T)分别表示T的值域和核空间.IP表示空间P上的恒等算子 定义1.1设T∈B(X,Y).若存在S∈B(Y,X),满足(1) TST=T;(2) STS=S,则称T广义可逆,S为T的一个广义逆,一般记为S=T+.     

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     

BANACH空间有界线性算子A_(T,S)~((2))的表示

文中R(A),N(A)分别表示算子A的值域与核空间.设A是一个n×m的复矩阵,S,T分别是Cn,Cm中的子空间,G是m × n的复矩阵.称G是A的具有指定值域T及核空间S的广义逆,若R(G)=T,N(G)=S且GAG=G.满足这样条件的G是唯一的,记为G=A(2)T,S(参见文献[7]).由文献[7]可知A(2)T,S存在的充要条件是AT+S=Cn.由于具有指定值域与核空间的广义逆是许多广义逆的统一表示形式,因此对它的研究具有普遍意义.     

保持算子乘积投影的线性映射

设B(H)是复Hilbert空间H上的有界线性算子全体且dim H≥2.本文证明了B(H)上的线性满射φ保持两个算子乘积非零投影性的充分必要条件是存在B(H)中的酉算子U以及复常数λ满足λ~2=1,使得φ(X)=λU~*XU,(?)X∈B(H).同时也得到了线性映射保持两个算子Jordan三乘积非零投影的充分必要条件.     

算子的一种更弱的相似性与互逆性

§1.定义与符号设H是可分的复Hilbert空间,B(H)表示H上全体有界算子的代数。对于A∈B(H),我们分别以R(A)、N(A)、{A}′及LatA表示它的值域、零空间、换位及不变子空间格。对于T,S∈B(H),如果有内射的稠值域的算子X,Y∈B(H),使得TX=XS,YT=SY,则说T与S是拟相似的。算子的拟相似性已经有丰富的内容。与拟相似概念有类似性的是算子互为拟仿射逆的概念[1],即:若T,S∈B(H),如果有内射的稠值域的算子X,Y∈B(H),使得TXS=X,SYT=Y,则说T与S互为拟仿射     

Banach空间中线性算子方程的约束极值解

设X,Y与Z为Banach空间L∈L(X,Z),T∈L(X,Y)为线性算子.运用线性算子的度量广义逆概念,在L(x)=y的极值解集合中,给出T(x)=h的约束极值解的精确刻画.     

保持算子束部分等距的映射

设B(H)是复Hilbert空间H上的有界线性算子全体,PI(H)表示B(H)中全体部分等距的集合.该文证明了B(H)上的满射Φ保持算子束(pencil)部分等距,即A-λB∈PI(H)Φ(A)-λΦ(B)∈PI(H)的充要条件是存在H上的两个酉算子U,V使得对于任意的X∈B(H)都有Φ(X)=UXV或存在H上的两个共轭酉算子U,V使得对于任意的X∈B(H)都有Φ(X)=UX*V.     

Banach空间中集值度量广义逆齐性单值选择的判据

本文研究了Banach空间(X,‖·‖),(Y,‖·‖)上具有闭值域的稠定闭算子T:X→Y的(集值)度量广义逆.在限定X为自反的、Y为一般的Banach空间且算子值域R(T)为空间Y中Chebyshev子空间时,证明了算子T具有非空闭凸集值的度量广义逆的存在性,运用Banach空间中广义正交分解定理,得出算子T的集值度量广义逆具有唯一齐性单值选择,并且该单值选择恰为赋等价严格凸范数的空间Xr=(X,‖·‖_r)上算子T的Moore-Penrose度量广义逆.特别地,将抽象的Banach空间X与Y具体化为有限维Banach空间l₁ⁿ=(Rn,‖·‖₁)(即n维空间Rn赋l₁范数)与有限维Hilbert空间(即m维欧式空间l₂^m=(Rm,‖·‖₂),亦即m维空间赋l₂范数),线性算子T可具体表示为m×n阶矩阵A,得到了从n维空间l₁ⁿ到m维空间l     

Banach空间中算子的秩定理

设E和F是Banach空间,B(E,F)表示映E到F的有界线性算子全体.记T+0 ∈ B(F,E)为T0 ∈ B(E,F)的一个广义逆.本文证明,每一个具有‖T+0(T-T0)‖＜1的算子T ∈ B(E,F),B≡(I+T+0(T-T0))-1T+0是T的广义逆当且仅当(I-T+0T0)N(T)=N(T0),其中N(·)表示括弧中算子的零空间.这一结果改进了Nashed和Cheng的一个有用的定理,并进一步证明Nashed和Cheng的一个引理对半-Fredholm算子有效但一般未必成立.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.