保持算子乘积谱函数的映射

设A和B为无限维复Banach空间上的标准算子代数,记ΔR(·)为下列谱函数之一σR(·),σRl(·),σRr(·),σRl(·)∩σRr(·),(a)σR(·),ησR(·),σRp(·),σRc(·),σRap(·),σRs(·),σRap(·)∩σRs(·),σRp(·)∩σRc(·),σRp(·)∪σRc(·),其中R=A或B.证明了A和B之间的每个保持算子Jordan三乘积(算子乘积)之谱函数ΔR(·)的满射φ必有形式φ=επ,其中ε是1的立方根(1的平方根)而π或者是A和B之间的代数同构,或者是代数反同构.也获得不定度规空间上的标准算子代数之间保持算子斜乘积之谱函数的映射的完全刻画.     

保Jordan正交性的映射

设A是Hilbert空间H上的*-标准算子代数,Φ是A上的满射.本文证明了Φ满足(A-B)R*+R*(A-B)=0(?)(Φ(A)-Φ(B))Φ(R)*+Φ(R)*(Φ(A)-Φ(B))=0当且仅当Φ是同构、反同构、共轭同构或共轭反同构.     

一类缺项算子矩阵的谱扰动

有界线性算子的点谱和剩余谱分别可进-步细分为两类:σ_(p1),σ_(p2)和σ_(r1),σ_(r2).设H,K为无穷维可分的Hilbert空间,本文将对于给定的A ∈B (H),B ∈B(K),给出了缺项算子M_C=(AC/OB)关于分类后所得四种谱的扰动结果.     

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.     

有界线性算子及其函数的Browder定理的判定

令H是无限维的Hilbert空间,B(H)是H上有界线性算子的全体构成的集合.称算子T∈B(H)满足Browder定理,若σ(T)σ_w(T)?π₀₀(T)或σ_w(T)=σ_b(T),其中σ(T),σ_w(T),σ_b(T)分别表示算子T的谱集、Weyl谱、Browder谱,π₀₀(T)={λ∈isoσ(T):0 相似文献   

中心化子的刻画

令X为实或复域F上的Banach空间,■为X上的标准算子代数,I是■的单位元.设Φ:■→■是可加映射.本文证明了,如果有正整数m,n,使得Φ满足条件Φ(A~(m+n+1))-A~mΦ(A)A~n∈FI对任意A成立,则存在λ∈F,使得对所有的A∈■,都有Φ(A)=λA.同样的结果对于自伴算子空间上的可加映射也成立.此外,本文还给出了中心素代数上满足条件(m+n)Φ(AB)-mAΦ(B)-nΦ(A)B∈FI的可加映射Φ的完全刻画.     

3×3阶上三角算子矩阵的点谱、剩余谱和连续谱

记■为Hilbert空间■上的上三角算子矩阵.我们借助对角元A,B和C的谱性质给出了σ_*(M_(D,E,F))=σ_*(A)∪σ_*(B)∪σ_*(C)对任意D∈B(H_2,H_1),E∈B(H_3,H_1),F∈B(H_3,H_2)均成立的充要条件,其中σ_*代表某类特定的谱,如点谱、剩余谱和连续谱等.此外,给出了一些例证.     

3×3阶上三角算子矩阵的点谱和剩余谱扰动

基于值域的稠密性和闭性,有界线性算子T的点谱和剩余谱可分别细分为σ_(p,1)(T),σ_(p,2)(T)和σ_(r,1)(T),σ_(r,2)(T).设H_1,H_2,H_3为无穷维复可分Hilbert空间,给定A∈B(H_1),B∈B(H_2),C∈B(H_3),结合分析方法与算子分块技巧给出了M_(D,E,F)的上述四种谱随D,E,F扰动的完全描述.     

Toeplitz算子的谱的某些子集的连续性

本文研究了Hilbert空间上有界线性算子的谱的某些子集的连续性,利用算子谱的精密结构的分析方法,给出了Hilbert空间H上有界线性算子T的谱σ(T)的某些子集如Φn(T),Φ(T),Φ+(T),Φ-(T),σ0p(T)等连续的充要条件.特别在Hardy空间H2(Γ)上,研究了Toeplitz算子Tφ的谱σ(Tφ)的某些子集的连续性.     

C~* -代数上完全正映射的刻画

本文给出C* -代数之间完全正映射的刻画,证明:如果A,B是有单位元的C*-代数,则映射Φ:A→B为完全正映射当且仅当存在保单位*-同态πA:A→B(K)、等距* -同态πB:B→B(H)及有界线性算子V:H→K,使得πB(Φ(1))=V*V 且■a∈A,都有πB(Φ(a))=V*π(a)V.作为推论,得到著名的Stinespring膨胀定理.     

Dense invariant sets of common cyclic vectors for Jordan operators

The purpose of this paper is to demonstrate that various collections of cyclic Jordan operators have dense invariant sets of common cyclic vectors.     

Additive maps preserving Jordan zero-products on nest algebras

Let and be nest algebras associated with the nests and on Banach Spaces. Assume that and are complemented whenever N_-=N and M_-=M. Let be a unital additive surjection. It is shown that Φ preserves Jordan zero-products in both directions, that is Φ(A)Φ(B)+Φ(B)Φ(A)=0AB+BA=0, if and only if Φ is either a ring isomorphism or a ring anti-isomorphism. Particularly, all unital additive surjective maps between Hilbert space nest algebras which preserves Jordan zero-products are characterized completely.     

Linear Maps Preserving Idempotence on Nest Algebras

In this paper, we discuss the rank-l-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear hijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.     

An algebraic invariant for Jordan automorphisms on B(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H)the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that φ is a surjective map from B(H) onto itself. If for everyλ∈ {-1, 1, 2, 3, 1/2, 1/3} and A, B ∈ B(H), A - λB ∈ I(H) (→)φ(A) - λφ(B) ∈ I(H), then φis a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that φ(A) = TAT-1 for all A ∈ B(H), or φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A - iB ∈ I(H) (→)φ(A) - iφ(B) ∈ I(H), here i is the imaginary unit, then φ is either an automorphism or an anti-automorphism.     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

An algebraic invariant for Jordan automorphisms on В(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempo-tents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ {-1,1,2,3,1/2,1/3} and A, B ∈ B(H), A - λB ∈ I(H) (?) Φ(A) - λΦ(B) ∈ I(H), then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈ I(H) (?) Φ(A) -ιΦ(B) ∈ I(H), here ι is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

Uniform primeness of the Jordan algebra of symmetric operators

In this note we establish the best possible constant for the general lower estimate for the Jacobson - McCrimmon operator on the algebra of symmetric operators acting on a Hilbert space.

     

The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

Complex symmetric operators and applications

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.

     

Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.