线性有界算子序列的一致强(弱)收敛

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【一致收敛与收敛】

在数学中，一致收敛性（或称均匀收敛）是函数序列的一种收敛定义，它较逐点收敛更强，并能保持一些重要的分析性质（如连续性）。

定义：设为一集合，为一度量空间。

若对一函数序列，存在满足，对所有，存在，使得
，则称一致收敛到。

注意到，一致收敛和逐点收敛定义的区别在于，在一致收敛中仅与相关，而在逐点收敛中还与相关。

所以一致收敛必定逐点收敛，而反之则不然。

例子:
考虑区间上的函数序列，它逐点收敛到函数，
然而这并非一致收敛。

直观地想像：当愈靠近，使接近所需的便愈大。

可以依此想法循定义直接证明，也可以利用下节关于连续的性质证明，因为在此例中皆连续，而不连续。

性质：假设一致收敛到，此时有下述性质：
（1）连续性：若是集合的闭包中的一个元素，且每个都在上连续，则也在a上连续。

若对集合I的每个紧子集，每个都在上连续，则在上连续。

（2）与积分的交换：令为中的开集，或。

若每个都是黎曼
可积，则也是黎曼可积，而且。

注：在勒贝格积分的框架下能得到更广的结果。

（3）与微分的交换：令为中的开集，或。

若每个皆可微，且一致收敛到函数，则亦可微，且。

点列强收敛、弱收敛以及弱星收敛的研究

Tn T (n ) ;

（2）若 x X ,恒有 || Tn x – Tx || 0 ( n ) ,则称 {Tn } 强收敛于 T ,记作
SOT Tn T (n ) ;
（3）若 x X , f Y * ,恒有 || f (Tn x ) – f (Tx ) || 0 (n ) ,则称 {Tn } 弱收敛于 T ,记作
WOT 即 Tn T (n ) .
注 1 强收敛可能不是范数收敛. 例5 左移算子
T :l 2 l 2
( x1 , x2 , , xn , ) ( x2 , x3 , , xn , )
令 Tn T T T
n个
(n ) ,
易知:
Tn : l 2 l 2 ( x1 , x2 , , xn , ) ( xn 1 , xn 2 , ) .
2
5
Tn x 0 0 (n ) ,
SOT 从而 Tn 0 (n ) .

再证 Tn 0 ( n ) 事实上：令 ei (0, , 0,1, 0,) (i 1, 2,) ,易知
因此,当 x
x e 时,由 T 的线性可得 Tx y e ,而这里的 y t x ,即 T 是对应于阵 (t ) 的
1
n
n
n
1
1
算子.由此可知,在有限维先行空间上,如果将基选定后,线性算子与矩阵是相对应的. 设 (1 , 2 , , n ) 是一组数,那么当 x
JIANG Dong-dong
(College of Science, Shanghai University, Shanghai 200444, China)

第四章4.4-4.5 线性算子的基本定理强收敛弱收敛

T-1(k1y1+k2y2)=k1T-1y1+k2T-1y2T-1是线性算子。
定理5 （巴拿赫逆算子定理）设X, Y都是巴拿赫空间TB(X,Y)是双射，则T-1是有界线性算子。证 T是双射T-1存在且T-1是线性算子（定理4）同时，T是双射 T是开映射设 GX 是开集(T-1)-1(G)=T(G)Y是开集
u x0 r0 x r0 , Tnu Tn x0 r0Tn x
1 Tn x Tnu Tn x0 r0 1 1 2M Tn x Tnu Tn x0 Tnu Tn x0 r0 r0 r0 2M Tn sup Tn x , n 1, 2, r0 x 1
间，T: DY是线性算子，如果T的图像GT是XY的闭线性子空间，则称T为闭线性算子。
定理9 (闭线性算子的充要条件) 设X, Y都是线性赋范空间，DX 是线性子空间。T: DY是线性算子，则T是闭线性算子的充要条件是对{xn}D, 当xnxX, TxnyY时，有xD, 且Tx=y. 证 “” (x,y)GT{(xn,Txn)}GT, 使(xn,Txn)(x,y) {xn}D，使xnx, Txny xБайду номын сангаасD, 且 Tx=y (x,y)=(x,Tx)GT GT=GT T是闭线性算子 “” GT是闭集, 设{xn}D, 且xnxX, TxnyY (xn,Txn)(x,y) {(xn,Txn)}GT, GT是闭集(x,y)GT xD, 且Tx=y
对yY,有
S T Y X Y S T y sy T (Sy ) I y y y
2) T-1T=Ix, TT-1=Iy 3) 若T是线性算子，则T-1也是线性算子(将在后面证明)。

第四章4.1-4.3线性泛函与线性泛函的延拓定理(短)

n n
T 是线性算子。 {Tn }是基本列 0, N , 当 n, m N 时，Tn Tm Tn Tm Tn 为基本数列 Tn 有界，设 Tn M , ( n 1, 2,3, ) Tn x Tn x M x Tx M x（n ） T 是有界算子 T B ( X , Y )
注：1）定义中，D为算子T的定义域; M是算子T的界值；T(D)={Tx|xD}称
为算子T的值域 2）有界算子与有界函数不同。例如 f(x)=x 无界函数有界算子: |f(x)|=|x|<2|x|
3) T是连续算子 T在D上处处连续
2. 有界线性算子的性质定理1 设X、Y是线性赋范空间，DX是线性子空间，T: DY 是线性算子，则
x X
定理2 设X、Y是线性赋范空间，DX是线性子空间，T: DY是有界线性算子，则T的范数具有下列性质： (1) ||Tx||||T|| ||x||, xD (2)
T sup Tx Y sup Tx Y
x 1 xD x 1 xD
(即||T||是有界线性算子T的最小界值）（可作为范数定义）
x 1 x D
则B (X,Y)成为线性赋范空间，称之为（有界）线性算子空间。
2. 线性算子空间中的极限理论定义4 （算子序列的一致收敛与强收敛）设X、Y是两个线性赋范空间，Tn, TB(X,Y), n=1,2,…
(1) 如果||Tn-T||0, 则称算子序列{Tn}按范数收敛于T, 或称{Tn}一致收敛于T. (2) 如果xX,||Tnx -Tx||0, 则称算子序列{Tn}强收敛于T, 或称{Tn}按点收敛于T.
T su p T x T x 0 m ax

第2章有界线性算子的基本概念(1)kj

2 算子的范数及其计算
定义 2.1.2 设 X , Y 是赋范空间, T : X Y 是有界线性算子. 称
T = sup Tx .
x £1
为算子 T 的范数. 若 f 是 X 上的有界线性泛函, 则 f 的值域是标量域 K , 此时
f = sup f ( x) .
x £1
定理 2.1.3 设 T : X Y 是有界线性算子. 则
yn =
xn x - 1 (n = 1, 2, ). f ( xn ) f ( x1 )
则 f ( yn ) = 0. 因此 yn Î N ( f ) (n ³ 1). 另一方面, 由于
xn xn 1 = < 0 (n ¥), f ( xn ) f ( xn ) n
这说明
xn x 0. 因此 yn y = - 1 . 但是 f ( xn ) f ( x1 ) f ( y ) = f (x1 ) = -1 ¹ 0, f ( x1 )
T -1 y = x £
1 1 Tx = y . a a
这表明映射 T -1 : Y X 是连续的. 因此 X 与 Y 拓扑同构的充要条件是, 存在一一对应的映射 T : X Y , 使得 T 是线性的 , 并且 T 和 T -1 都是连续的. 线性泛函是线性算子的特殊情形, 因此定理 2.1.1 的结论对线性泛函当然也成立. 对线性泛函还成立如下定理. 定理 2.1.2 设 f 是赋范空间 X 上的线性泛函. 则 f 在 X 上有界的充要条件是 f 的零空间 N ( f ) 是闭集. 证明设 f 在 X 上有界, 则 f 在 X 上连续. 设 {xn } Ì N ( f ), xn x.
n¥
则 f ( xn ) = 0 ( n ³ 1 ) 于是 f ( x) = lim f ( xn ) = 0. 因此 x Î N ( f ). 这表明

Banach空间中线性算子核逆的一致有界性与收敛性[英文]

应用数学MATHEMATICA APPLICATA2021,34(1):216-223The Uniform Boundedness and Convergence for the Core Inverses of Linear Operators in Banach SpacesZHAO Yayuan(赵亚媛),CHEN Saijie(陈赛杰)ZHU Lanping(朱兰萍),HUANG Qianglian(黄强联) (School of Mathematical Sciences,Yangzhou University,Yangzhou225002,China)Abstract:The main topic of this paper is the relationship between uniform boundednessand convergence of the core inverses of linear operators in Banach spaces.Weﬁrst obtainthe equivalence of the uniform boundedness and convergence for core inverse and we givethe expression of core inverse.Secondly,we investigate the stable perturbation for the coreinverse and prove that the stable perturbation and the continuity of the core inverse areequivalent.As applications,we also give the continuity characterization for the core inverseofﬁnite rank operators and derive the suﬃcient and necessary condition for the core inverseof the perturbed operator to have the simplest possible expression.Key words:Core inverses;Uniform boundedness;Convergence;Generalized inverse;Stable perturbationCLC Number:O177.91AMS(2000)Subject Classiﬁcation:47A55;47A58Document code:A Article ID:1001-9847(2021)01-0216-081.Introduction and PreliminariesLet X,Y be Banach spaces and B(X,Y)denote the Banach space of all bounded linear operators from X into Y.We write B(X)as B(X,X).For any T∈B(X,Y),we denote the null space and the range of T by N(T)and R(T),respectively.The identity operator will be denoted by I.Recall that an operator S∈B(Y,X)is said to be a generalized inverse of T∈B(X,Y) if S satisﬁes:(1)T ST=T and(2)ST S=S.A generalized inverse of T is usually denoted by T+.While the generalized inverse may not exist and it is not unique even if it exists.In order to force its uniqueness,some further conditions have to be imposed.Let us recall deﬁnitions of three important generalized inverses.∗Received date:2020-03-02Foundation item:Supported by the National Natural Science Foundation of China(11771378, 11871064,11971419);the Yangzhou University Foundation for Young Academic Leaders(2016zqn03);the Postgraduate Research and Practice Innovation Program of Yangzhou University(XKYCX19-057) Biography:ZHAO Yayuan,female,Han,Anhui,major in functional analysis.Corresponding auther:HUANG Qianglian.No.1ZHAO Yayuan,et al.：The Uniform Boundedness and Convergence for the Core Inverses217Deﬁnition1.1Let X and Y be Hilbert spaces.An operator S∈B(Y,X)is called the Moore-Penrose inverse of T∈B(X,Y)if S satisﬁes the Penrose equations:(1)T ST=T;(2)ST S=S;(3)(T S)∗=T S and(4)(ST)∗=ST,where T∗denotes the adjoint operator of T.The Moore-Penrose inverse of T is always written by T†,which is uniquely determined if it exists.Deﬁnition1.2Let X be a Banach space.An operator S∈B(X)is said to be the group inverse of T∈B(X),always denoted by T♯,if S satisﬁes(1)T ST=T;(2)ST S=S and(5)T S=ST.The core inverse is a generalized inverse somehow between the Moore-Penrose inverse and the group inverse,which was introduced by Baksalary and Trenkler for the matrix T satisfying Rank T2=Rank T♯.In[2],Raki´c,Dinˇc i´c and Djordjevi´c extended it to the operator on a Hilbert space:Deﬁnition1.3[2]Let X be a Hilbert space.An operator S∈B(X)is said to be the core inverse of T∈B(X),denoted by T#⃝,if S satisﬁes(1)T ST=T;(2)ST S=S;(3)(T S)∗=T S;(6)ST2=T and(7)T S2=S.For the invertible operator,by the well known Banach Lemma and the identity:−T−1n=−T−1m(T m−T n)T−1n,T−1mwe can get the following theorems.Theorem1.1Let T∈B(X,Y)be invertible and T−1its inverse.If T n∈B(X,Y) satisﬁes T n→T,then there exists N∈N,such that for all n≥N,T n is invertible and=T−1[I+(T n−T)T−1]−1=[I+T−1(T n−T)]−1T−1.T−1n→T−1,which means that the mapping T→T−1is continuous. Furthermore,T−1nTheorem1.2Let T n and T∈B(X,Y)satisfy T n→T.If T n is invertible and ∥T−1n∥<+∞,then T is invertible and T−1n→T−1.supn∈N∥T−1n∥< Hence we can claim that T is invertible if and only if T n is invertible with supn∈N→T−1.It is natural to ask whether similar results hold for various +∞,in this case,T−1ngeneralized inverses.Such problems of expression,boundedness and convergence have been investigated for the generalized inverse in[3–11],the Moore-Penrose inverse in[6–8,11–15], the group inverse in[3,8,11,16–18]and the core inverse in[1,2,19].Especially,Koliha[14], ZHU,ZHU and HUANG[11]proved the equivalence between the uniform boundedness and convergence for the Moore-Penrose inverse and group inverse,respectively.In this paper,we shall investigate the same problems for the core inverse.By utilizing the stable perturbation,we obtain the equivalence of the uniform boundedness and convergence for core inverse and some expression results.For the stable perturbation of the generalized inverse,we have the following theorem which plays a crucial role in our proof.Theorem1.3[8]Let X and Y be Banach spaces and T+∈B(Y,X)be a generalized inverse of T∈B(X,Y).Assume that I+δT T+:Y→Y be bijective withδT∈B(X,Y). Then the following statements are equivalent:1)B=T+(I+δT T+)−1=(I+T+δT)−1T+is a generalized inverse of T=T+δT;2)T is a stable perturbation of T,i.e.,R(T)∩N(T+)={0};3)Y=R(T)⊕N(T+);218MATHEMATICA APPLICATA20214)Y=R(T)˙+N(T+);5)X=N(T)⊕R(T+);6)X=N(T)+R(T+);7)R(T)=T R(T+).In the next section,weﬁrst prove that T is core invertible if and only if T n is coreinvertible with supn∈N ∥T#⃝n∥<+∞and give a concrete expression of T#⃝.Secondly,we provethe equivalence between the stable perturbation and the continuity for the core inverse.As applications,we also derive the continuity characterization for the core inverse ofﬁnite rank operators and the characterization for the core inverse of the perturbed operator to have the simplest possible expression.For the null space-preserving or the dimension of null space-preserving perturbation,we give a complete answer to the problem proposed in[19].2.Main ResultsAs mentioned above,an operator T is invertible if and only if T n is invertible with supn∈N∥T−1n∥<+∞.It turns out that the same property is also enjoyed by the core inverse.Theorem2.1Let X be a Hilbert space and T n,T∈B(X)with T n→T.If the coreinverse T#⃝nexists,then the following statements are equivalent:1)supn∈N∥T#⃝n∥<+∞;2)T has the core inverse T#⃝satisfying T#⃝n→T#⃝.In this case,for all suﬃciently large n,T#⃝=W−1n T S n M−1n,where S n=T#⃝n [I+(T−T n)T#⃝n]−1,W n=T S n−T−I and M n=I−T S n−(T S n)∗.Proof It is obvious to see2)⇒1)and we only need to prove1)⇒2).By the deﬁnition of the core inverse,we can getT∗n (I−T n T#⃝n)=T∗n−T∗n(T n T#⃝n)∗=T∗n−(T n T#⃝n T n)∗=0and(I−T#⃝mT m)T m=T m−T#⃝m T2m=0.Then it follows from supn∈N∥T#⃝n∥<+∞andT#⃝m−T#⃝n=T#⃝m T n T#⃝n−T#⃝m T m T#⃝n+T#⃝m−T#⃝m T n T#⃝n+T#⃝m T m T#⃝n−T#⃝n=T#⃝m(T n−T m)T#⃝n+T#⃝m(I−T n T#⃝n)−(I−T#⃝m T m)T#⃝n=T#⃝m(T n−T m)T#⃝n+T#⃝m T m T#⃝m(I−T n T#⃝n)−(I−T#⃝m T m)T n(T#⃝n)2=T#⃝m(T n−T m)T#⃝n+T#⃝m(T m T#⃝m)∗(I−T n T#⃝n)−(I−T#⃝m T m)T n(T#⃝n)2=T#⃝m(T n−T m)T#⃝n+T#⃝m(T#⃝m)∗T∗m(I−T n T#⃝n)−T#⃝m(T#⃝m)∗T∗n(I−T n T#⃝n)−(I−T#⃝m T m)T n(T#⃝n)2+(I−T#⃝m T m)T m(T#⃝n)2=−T#⃝m(T m−T n)T#⃝n+T#⃝m(T#⃝m)∗(T∗m−T∗n)(I−T n T#⃝n)+(I−T#⃝mT m)(T m−T n)(T#⃝n)2that{T#⃝n}is a Cauchy sequence in B(X).Since B(X)is complete,we can assume T#⃝n→S∈B(X)and take the limit inﬁve equations in the deﬁnition of T#⃝n.Then S is the coreinverse of T and T#⃝n→S=T#⃝.HenceI−T#⃝nT n−T#⃝T→I−2T#⃝T.No.1ZHAO Yayuan,et al.：The Uniform Boundedness and Convergence for the Core Inverses219Noticing(I−2T#⃝T)2=I,we know that I−2T#⃝T is invertible.Thus there exists N∈N, such that for all n>N,I−T#⃝nT n−T#⃝T is invertible and∥(T−T n)T#⃝n∥≤∥T−T n∥∥T#⃝n∥≤12<1.By the Banach Lemma,I+(T−T n)T#⃝nis invertible.Therefore,R(T)=R[T(I−T#⃝nT n−T#⃝T)]=R[T(T#⃝n T n)]=T R(T#⃝n T n)=T R(T#⃝n).Since I+(T−T n)T#⃝nis invertible,from Theorem1.3,S n=T#⃝n [I+(T−T n)T#⃝n]−1is a generalized inverse of T.Observing that(T T#⃝−T−I)(T#⃝T−T#⃝−I)=(T#⃝T−T#⃝−I)(T T#⃝−T−I)=I,[I−T T#⃝−(T T#⃝)∗]2=(I−2T T#⃝)2=I,we can get that T T#⃝−T−I and I−2T T#⃝are invertible.SinceT S n−T−I→T T#⃝−T−IandI−T S n−(T S n)∗→I−T T#⃝−(T T#⃝)∗=I−2T T#⃝,we know that for all suﬃciently large n,W n=T S n−T−I and M n=I−T S n−(T S n)∗are invertible.To complete the proof,we shall show thatV n=W−1n T S n M−1nis the core inverse of T.In fact,M∗n=M n and if we set Q n=T S n,then Q n W n=−T=−Q n T,W n Q n=−T2S n=−T Q n,W n T=−T2,Q n M n=−Q n Q∗n =M n Q∗n,M n Q n=−Q∗nQ n=Q∗nM n,M n T=−Q∗nT.HenceT W−1n =−Q n,M−1nQ n=Q∗nM−1n,Q n M−1n=M−1nQ∗n.Thus,by the deﬁnition of V n,T V n=T W−1n Q n M−1n=−Q2nM−1n=−Q n M−1n,V n T=W−1n Q n M−1nT=W−1nM−1nQ∗nT=−W−1nTand soT V n T=−Q n M−1n T=−M−1nQ∗nT=T,V n T V n=−W−1n T V n=W−1nQ n M−1n=V n,(T V n)∗=(−Q n M−1n )∗=−M−1nQ∗n=−Q n M−1n=T V n,V n T2=−W−1n T T=W−1nW n T=T.Therefore,V n is a generalized inverse of T andR(V n)=R(W−1n T S n M−1n)=W−1nT R(S n)=W−1nT R(T#⃝n)=W−1n R(T)=W−1nR(T T#⃝)=W−1nR(T2)=W−1nR(W n T)=R(T)=R(T V n)=N(I−T V n).This imples(I−T V n)V n=0and so T V2n=V n.Thus,V n is the core inverse of T.The proofis complete.Theorem2.1provides a suﬃcient and necessary condition for the core invertibility of T nto imply the core invertibility of T and T#⃝n→T#⃝.Naturally,we can propose the following220MATHEMATICA APPLICATA 2021problems:Can the core invertibility of T imply the core invertibility of T n ?If T n is also coreinvertible,does the core inverse T #⃝n converge or T #⃝n →T #⃝?The following two examples showthat the answers are no in general.Example 2.1Let T = 000113−1−1−3 and T n = 1n 01n 113−1+1n−1−3+1n ,then T n →T and T is core invertible with T #⃝= 0000−1414014−14.But T n is not core invertible since Rank T n =2and Rank T 2n =1.Example 2.2Let T = 000112−1−1−2 and T n = 1n 01n 112−1+1n −1−2+1n,then T n →T ,both T and T n are core invertible with T #⃝= 0000−1212012−12and T #⃝n =13 −2n +1−n +2−n −1−6n +2−3n +4−3n −24n −12n −22n +1 .Obviously,T #⃝n is unbounded and divergent.The next theorem shows that if T is core invertible,then T n is core invertible with T #⃝n →T #⃝if and only if T n is a stable perturbation of T .Moreover,a concrete expression of T #⃝n is also obtained.Theorem 2.2Let X be a Hilbert space and T ∈B (X )be core invertible.Let T n ∈B (X )satisfy T n →T ,then the following statements are equivalent:1)For all suﬃciently large n ,T n is a stable perturbation of T ,i.e.,R (T n )∩N (T #⃝)={0};2)There exists N ∈N ,such that for all n ≥N ,T n is core invertible withT #⃝n→T #⃝.In this case,for all suﬃciently large n ,T #⃝n =K −1n T n B n G −1n ,where B n =T #⃝[I +(T n −T )T #⃝]−1,K n =T n B n −T n −I and G n =I −T n B n −(T n B n )∗.Proof 1)⇒2)It follows from Theorem 1.3that,for all suﬃciently large n ,I +(T n −T )T #⃝is invertible and B n =T #⃝[I +(T n −T )T #⃝]−1=[I +T #⃝(T n −T )]−1T #⃝is a generalized inverse of T n .Similar to the proof of Theorem 2.1,we can prove that for all suﬃciently large n ,K n =T n B n −T n −IandG n =I −T n B n −(T n B n )∗No.1ZHAO Yayuan,et al.：The Uniform Boundedness and Convergence for the Core Inverses221are invertible,K−1n T n B n G−1nis the core inverse of T n,and obviously,T#⃝n =K−1nT n B n G−1n→(T#⃝T−T#⃝−I)T T#⃝(I−2T T#⃝)=T#⃝.2)⇒1)For all suﬃciently large n,we know that both I+(T n−T)T#⃝and I−T#⃝n T n+ T#⃝T are invertible,andR(T n)=R[T n(I−T#⃝nT n+T#⃝T)]=R(T n T#⃝T)=T n R(T#⃝),By Theorem1.3,we get R(T n)∩N(T#⃝)={0}.The proof is complete.As an application,we can give a characterization that T n is core invertible with T#⃝n→T#⃝forﬁnite rank operators.Corollary 2.1Let T∈B(X)be ofﬁnite rank.If T n→T and T is core invertible, then the following statements are equivalent:1)Rank T n=Rank T for all suﬃciently large n;2)There exists N∈N,such that for all n≥N,T n is core invertible withT#⃝n→T#⃝;3)There exists N∈N,such that for all n≥N,T n is core invertible withsupn∈N∥T#⃝n∥<+∞.In this case,T#⃝n=(T n B n−T n−I)−1T n B n[I−T n B n−(T n B n)∗]−1.Proof It suﬃces to prove1)⇔2).Without loss of generality,we can assume that I+(T n−T)T#⃝is invertible.It follows from[I+(T n−T)T#⃝]T=T n T#⃝Tthat dim R(T)=dim R(T n T#⃝T)=dim T n R(T#⃝T)=dim T n R(T#⃝)=dim R(T n T#⃝).If Rank T n=Rank T,thendim R(T n)=dim R(T n T#⃝).Since R(T n T#⃝)⊆R(T n),we get R(T n)=R(T n T#⃝)=T n R(T#⃝).Using Theorem1.3,we can have R(T n)∩N(T#⃝)={0}.Hence by Theorem2.2,2)holds.Conversely,if T n is core invertiblewith T#⃝n→T#⃝,then T n T#⃝n→T T#⃝.Hence for all suﬃciently large n,I−T n T#⃝n+T T#⃝andI−T T#⃝+T n T#⃝nare invertible.ThusR(T n T#⃝n )=R[T n T#⃝n(I−T n T#⃝n+T T#⃝)]=R(T n T#⃝nT T#⃝)andR(T T#⃝)=R[T T#⃝(I−T T#⃝+T n T#⃝n )]=R(T T#⃝T n T#⃝n).Therefore,dim R(T n T#⃝n )≤dim R(T T#⃝)and dim R(T T#⃝)≤dim R(T n T#⃝n),i.e., Rank T n=Rank T.The proof is complete.It is also noteworthy that we provide a direct and brief proof in Corollary2.1and we do not use the Finite Rank Theorem[9].Next,we can give the characterization for the core inverse T#⃝nto have the simplest possible expression.Corollary2.2Let X be a Hilbert space and T∈B(X)be core invertible.If T n∈B(X) satisﬁes T n→T,then for all suﬃciently large n,T n is core invertible andT#⃝n=T#⃝[I+(T n−T)T#⃝]−1=[I+T#⃝(T n−T)]−1T#⃝if and only if T n=T T#⃝T n.222MATHEMATICA APPLICATA2021Proof Suﬃciency.If T n=T T#⃝T n,then R(T n)⊂R(T)andR(T n)∩N(T#⃝)⊆R(T)∩N(T#⃝)={0}.By Theorem1.3,B n=T#⃝[I+(T n−T)T#⃝]−1=[I+T#⃝(T n−T)]−1T#⃝is a generalized inverse of T n.Noticing B n=T T#⃝B n=B n T T#⃝andT n B n=T T#⃝T n T#⃝[I+(T n−T)T#⃝]−1=T T#⃝[I+(T n−T)T#⃝][I+(T n−T)T#⃝]−1=T T#⃝,B n T n T T#⃝=[I+T#⃝(T n−T)]−1T#⃝T n T T#⃝=[I+T#⃝(T n−T)]−1[I+T#⃝(T n−T)]T T#⃝=T T#⃝,we can get[I−T n B n−(T n B n)∗]2=(I−2T T#⃝)2=I and(T n B n−T n−I)(B n T n−B n−I)=I+T n B2n T n+B n−T n B2n−B n T n=I+T T#⃝B n T n+B n−T T#⃝B n−B n T n=I+B n T n+B n−B n−B n T n=I,(B n T n−B n−I)(T n B n−T n−I)=I+B n T2n B n+T n−B n T2n−T n B n=I+B n T n T T#⃝+T n−B n T n T T#⃝T n−T T#⃝=I+T T#⃝+T n−T T#⃝T n−T T#⃝=I+T T#⃝+T n−T n−T T#⃝=I.Hence[I−T n B n−(T n B n)∗]−1=I−2T T#⃝and(T n B n−T n−I)−1=B n T n−B n−I.Therefore, by Theorem2.2,we have=(T n B n−T n−I)−1T n B n[I−T n B n−(T n B n)∗]−1T#⃝n=(B n T n−B n−I)T T#⃝(I−2T T#⃝)=(B n T n T T#⃝−B n T T#⃝−T T#⃝)(I−2T T#⃝)=−B n+2B n T T#⃝=B n.Necessity.If B n is the core inverse of T n,then)=R(B n)=R(T#⃝)=R(T)=R(T T#⃝)=N(I−T T#⃝).R(T n)=R(T#⃝nThis means(I−T T#⃝)T n=0,i.e.,T n=T T#⃝T n.The proof is complete.Since both the null space-preserving perturbation and the dimension of null space-preserving perturbation are all stable perturbations[6−7],we can get the following corollary which gives a complete answer to the problem proposed in[19].Corollary2.3Let X be a Hilbert space.Let T∈B(X)with its core inverse T#⃝∈B(X) and T n∈B(X)with T n→T.IfN(T n)=N(T)or dim N(T n)=dim N(T)<+∞→T#⃝.holds,then for all suﬃciently large n,T n is core invertible and T#⃝nIn this case,=(T n B n−T n−I)−1T n B n[I−T n B n−(T n B n)∗]−1.T#⃝nReferences:[1]BAKSALARY O M,TRENKLER G.Core inverse of matrices[J].Linear Multilinear Algebra.,2010,58(6):681-697.No.1ZHAO Yayuan,et al.：The Uniform Boundedness and Convergence for the Core Inverses223[2]RAKI´C D S,DINˇCI´C N C,DJORDJEVI´C D S.Core inverse and core partial order of Hilbert spaceoperators[J]put.,2014,244(1):283-302.[3]CASTRO-GONZ´ALZE N,V´ELZE-CERRADA J Y.On the perturbation of the group generalizedinverse for a class of bounded operators in Banach spaces[J].J.Math.Anal.Appl.,2008,341(2): 1213-1223.[4]CHEN Guoliang,XUE Yifeng.Perturbation analysis for the operator equation T x=b in Banachspaces[J].J.Math.Anal.Appl.,1997,212(1):107-125.[5]DING Jiu.On the expression of generalized inverses of perturbed bounded linear operators[J].Mis-souri J.Math.Sci.,2003,15(1):40-47.[6]HUANG Qianglian,MA Jipu.Continuity of generalized inverses of linear operators in Banach spacesand its applications[J].Appl.Math.Mech.,2005,26(12):1657-1663.[7]HUANG Qianglian,ZHU Lanping,GENG Wanhui,et al.Perturbation and expression for innerinverses in Banach spaces and its applications[J].Linear Algebra Appl.,2012,436(9):3715-3729. [8]HUANG Qianglian,ZHU Lanping,JIANG Yueyu.On stable perturbations for outer inverses of linearoperators in Banach spaces[J].Linear Algebra Appl.,2012,437(7):1942-1954.[9]MA plete rank theorem of advanced calculus and singularities of bounded linear opera-tors[J].Front.Math.China.,2008,3(2):305-316.[10]NASHED M Z.Generalized Inverses and Applications[M].New York:Academic Press,1976.[11]ZHU Lanping,ZHU Changpeng,HUANG Qianglian.On the uniform boundedness and convergenceof generalized,Moore-Penrose and group inverses[J].Filomat.,2017,31(19):5993-6003.[12]DING Jiu.New perturbation results on pseudo-inverses of linear operators in Banach spaces[J].LinearAlgebra Appl.,2003,362(1):229-235.[13]HUANG Qianglian,MA Jipu.A note on the continuity of Moore-Penrose inverses T†[J].MathematicaApplicata.,2006,19(4):776-781.[14]KOLIHA J J.Continuity and diﬀerentiability of the Moore-Penrose inverse in C∗-algebras[J].Math-ematica Scandinavica.,2001,88(1):154-160.[15]XU Qingxiang,WEI Yimin,GU Yangyang.Sharp norm-estimations for Moore-Penrose inverses ofstable perturbations of Hilbert C*-module operators[J].SIAM J.Numer.Anal.,2010,47(6):4735-4758.[16]BEN´CTEZ J,CVETKOVI´C-ILI´C D,LIU Xiaoji.On the continuity of the group inverse in C∗-algebras[J].Banach J.Math.Anal.Appl.,2014,8(2):204-213.[17]WEI Yimin,LI Xiezhang.An improvement on the perturbation of the group inverse and obliqueprojection[J].Linear Algebra Appl.,2001,338(1):53-66.[18]WEI Yimin.On the perturbation of the group inverse and oblique projection[J]put.,1999,98(1):29-42.[19]HUANG Qianglian,CHEN Saijie,GUO Zhirong,et al.Regular factorizations and perturbation anal-ysis for the core inverse of linear operators in Hilbert spaces[J]put.Math.,2019,96(10): 1943-1956.Banach空间中线性算子核逆的一致有界性与收敛性赵亚媛,陈赛杰,朱兰萍,黄强联(扬州大学数学科学学院,江苏扬州225002)摘要：本文主要研究Banach空间中线性算子核逆的一致有界性与收敛性之间的关系.首先证明核逆的一致有界性与收敛性的等价性,给出了核逆的表达式.其次,利用稳定扰动,证明核逆的稳定扰动与连续性是等价的.作为应用,我们还给出有限秩算子核逆的连续性特征,并给出扰动算子的核逆具有最简表达式的充分必要条件.关键词:核逆;一致有界;收敛;广义逆;稳定扰动。

4 有界线性算子与线性算子的基本定理g

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第6页页 3 有界线性算子的范数定义2 是线性赋范空间， ⊂ 是线性子空间是线性子空间, 是有界线性算子, 定义设X,Y是线性赋范空间，D⊂X是线性子空间 T: D→Y是有界线性算子,则称是线性赋范空间 → 是有界线性算子 ||T||=inf { M | ||Tx||Y ≤ M||x||X, ∀x∈D} 为算子的范数为算子T的 ∈ 是线性赋范空间, 是线性子空间, 定理2 设X,Y是线性赋范空间,D⊂X是线性子空间,T: D→Y是定理是线性赋范空间 ⊂ 是线性子空间 → 是有界线性算子，的范数具有下列性质：有界线性算子，则T的范数具有下列性质：的范数具有下列性质 (1)||Tx||≤||T|| ||x||,∀x∈D(即||T||是有界线性算子的最小界值定义） ≤ 是有界线性算子T的最小界值定义 (1) ∀ ∈ ( 是有界线性算子的最小界值定义） (2) 证 ⇒ ⇒ ⇒
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一、有界线性算子的定义与性质
1 有界线性算子的定义定义1 是线性赋范空间, 是线性子空间, 定义1 设X是线性赋范空间,D⊂X是线性子空间,映射 D→Y. 是线性赋范空间 ⊂ 是线性子空间映射T: → . T(x1+x2)=Tx1+Tx2 (1)T是线性算子⇔∀x 是线性算子⇔∀ 及数 , (1) 是线性算子⇔∀ 1, x2∈D及数α∈K,有 T(αx)=αTx (2)T是连续算子⇔∀x ∈ (2) 是连续算子⇔∀ n, x∈D,n=1,2,…, xn→x, 有Txn→Tx 是连续算子⇔∀ ⇔∀x,x ⇔∀ 0∈D, x→x0, 有Tx→Tx0;⇔T在D上处处连续 → → 在上处处连续 (3)T是有界算子⇔∀x∈ (3) 是有界算子⇔∀ ∈D, ∃M>0, 使||Tx||≤M||x||X 是有界算子⇔∀ , ≤ (4)T是有界线性算子⇔ 既是有界算子既是有界算子, (4) 是有界线性算子⇔T既是有界算子,又是线性算子是有界线性算子 (5)T是连续线性算子⇔ 既是连续算子, (5) 是连续线性算子⇔T 既是连续算子,又是线性算子是连续线性算子定义中, 算子的定义域; 算子T 算子T的界值的界值;T(D)={Tx|x∈D}- 算子的值域算子T的值域注:1)定义中,D -算子T的定义域 M -算子的界值定义中 ∈ 无界函数 2）有界算子与有界函数不同,例如 f (x)=x ）有界算子与有界函数不同, 有界算子: 有界算子:|f(x)|=|x|<2|x|

泛函分析复习与总结

《泛函分析》复习与总结第一部分空间及其性质泛函分析的主要内容分为空间和算子两大部分. 空间包括泛函分析所学过的各种抽象空间, 函数空间, 向量空间等, 也包括空间的性质, 例如完备性, 紧性, 线性性质, 空间中集合的各种性质等等。

以下几点是对第一部分内容的归纳和总结。

一．空间（1）距离空间（集合+距离）！验证距离的三个条件：(,)X ρ称为是距离空间，如果对于,,x y z X ∈(i) 【非负性】(,)0x y ρ≥，并且(,)0x y ρ=当且仅当x y =【正定性】；(ii) 【对称性】(,)(,)x y y x ρρ=；(iii) 【三角不等式】(,)(,)(,)x y x y y z ρρρ≤+。

距离空间的典型代表：s 空间、S 空间、所有的赋范线性空间、所有的内积空间。

（2）赋范线性空间（线性空间 + 范数）！验证范数的三个条件：(,||||)X ⋅称为是赋范线性空间，如果X是数域K =¡（或K =£）上的线性空间，对于a K ∈和,x y X ∈，成立(i) 【非负性】||||0x ≥，并且||||0x =当且仅当0x =【正定性】； (ii) 【齐次性】||||||||||ax a x =⋅；(iii) 【三角不等式】||||||||||||x y x y +≤+。

赋范线性空间的典型代表：n ¡空间（1,2,3,n =L ）、n £空间（1,2,3,n =L ）、p l 空间（1p ≤≤∞）、([,])p L ab 空间（1p ≤≤∞）、[,]Cab 空间、[,]k C a b 空间、Banach 空间、所有的内积空间（范数是由内积导出的范数）。

（3）内积空间（线性空间 + 内积）！验证内积的四个条件：(,(,))X ⋅⋅称为是内积空间，如果X 是数域K =¡（或K =£）上的线性空间，对于a K ∈和,,x y z X ∈，成立(i) 【非负性】(,)0x x ≥，并且(,)0x x =当且仅当0x =【正定性】；(ii) 【第一变元可加性】(,)(,)(,)x y z x z x z +=+；(iii) 【第一变元齐次性】(,)(,)ax z a x z =；(iv) 【共轭对称性】(,)(,)x z z x =。

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线性有界算子序列的一致强(弱)收敛线性有界算子序列的一致强(弱)收敛，指的是在定义在线性变换
空间上的有界算子序列\{T_n\}中，存在一个定义在这个空间里的数K，使得||T_n||\leqK，并且当n\rightarrow\infty时，T_{n}以足够快
的速度向T趋近，其中||T_n||是这个序列的算子范数，T是这个空间
的有界算子。

首先要说的是，线性有界算子序列的一致强收敛，是指一个线性
变换空间上的有界算子序列，它具有线性复叱性，并且有数K使得
||T_n||\leqK，当n\rightarrow\infty时，T_n和T的定义范围趋于
一致，这个过程使得T_n不断次级收敛到T（若T是收敛点，则T也收敛到T，而T_n不断增加，最终收敛到T），使得T_n等效于T，称为
一致强收敛。

由于一致强收敛的定义具有线性复叱性，所以我们可以得出抽象
的总结：T_n的一致强收敛类似于一致收敛，但是它不是以完全一致的方式，而是以不断次级的形式收敛的，最终收敛到某个点T，即T_n等于T，称为一致强收敛。

另外，线性有界算子序列的一致弱收敛是指在定义在线性变换空
间上的有界算子序列中，存在一个定义在这个空间里的数K，使得
||T_n||\leqK，而且当n\rightarrow\infty时，T_{n}不断向T靠近，但动态幅度很小，最终没有达到等同于T，也就是T_n不能真正等效于T，但它们之间的差异趋于零，称为一致弱收敛。

总之，线性有界算子序列的一致强(弱)收敛，指的是在定义在线
性变换空间上的有界算子序列\{T_n\}中，存在一个定义在这个空间里
的数K，使得||T_n||\leqK，并且当n\rightarrow\infty时，有一致
强收敛和一致弱收敛，也就是说，T_n以不同的范围靠近T，使得T_n
逐渐收敛到T，从而减少了两者之间的偏差，使其有效的趋近于T，最
终达到稳定的状态。