保持算子截断的可加映射

Let H be a complex Hilbert space and B(H)the algebra of all bounded linear operators on H.An operator A is called the truncation of B in B(H)if A=P_ABP_A^*,where PA and P_A^*denote projections onto the closures of R(A)and R(A^*),respectively.In this paper,we determine the structures of all additive surjective maps on B(H)preserving the truncation of operators in both directions.     

ON PGLAR PRODUCT OPERATOR

Let H be a Hilbert space, and let A be a linear bounded operator on H. For \(\lambda \in \rho (A)\), the \({U_\lambda } = {(A - \lambda )^{ * - 1}}(A - \lambda )\) is called polar.Produot operator. In this paper, we discuss the properties of \({U_\lambda }\) and the relation between \({U_\lambda }\) and A. We obtain tbe following results. Definition. Let B be a linear bounded operator on H, suppose \(0 \in \rho (B)\). For every \(x,y \in H\), we definite \([x,y] = (Bx,y)(H,B)\)(or (H, [·,·]) is called a non- degenerate bilinear space (it is obvious that if B=B*,then (H,B)is a space with an indefinite metric; and that if B>0, then (H,B) is a Hilbert Space. If an operator U(A) satisfies \[[Ux,Uy] = [x,y]([Ax,y] = [x,Ay]),x,y \in H\] then the operator U(A) is called a wvitary (self adjoint) on (H,B). Theorem I . Suppose A is a linear bounded operator on H, (1) If \(0 \in \rho (A)\), then \(U = {A^{ * - 1}}A\) is a unitary operator on (H,A) or (H, A*), and \(\sigma (U) = \frac{1}{{\sigma (U)}}\). (2) If there is a complex number \(\alpha \), such that \({\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then a)\(0 \in \rho (A)\), and the operator \(U = {A^{ * - 1}}A\) is a unitary on Hilbert space \((H,{\mathop{\rm Im}\nolimits} A) and 1 \in \rho (U)\);b) there exist two Hilbert spaces \((H,{v_1}),(H,{v_2})\), such that A, A* are all the unitary operator from (H,v1) onto (H,v2), and there are two spectral measures \(\{ E_\lambda ^i,\lambda \in [\alpha ,\chi ] \subset (0,2\pi )\} ,i = 1,2,\), such that \(AE_\Delta ^1H \subset E_\Delta ^2H,{A^ * }E_\Delta ^1H \subset E_\Delta ^2H\) for any \(\Delta = (\lambda ,u] \subset (0,2\pi ]\). (3) If \(0 \in \rho (A) \cap \rho ({\mathop{\rm Im}\nolimits} A)\) then the operator \(U = {A^{ * - 1}}A\) is a unitary on \((H,{\mathop{\rm Im}\nolimits} A)\). with an indefinite met He, and \(1 \in \rho (U)\). (4) For any complex number \(\lambda = r{e^{i\theta }},\left| \lambda \right| > \left\| A \right\|\), then \({U_\lambda }\) must be a unitary operator on the Hilbert space \(\left( {H,{\mathop{\rm Im}\nolimits} (\frac{1}{{i\lambda }}A + iI)} \right),and - {e^{i2\theta }} \in \rho ({U_\lambda })\) Theorem 2. (1) A is a normal operator iff there exists a complex number \(\lambda \), \(\lambda \in \rho (A)\), such that \(\frac{{\partial {U_\lambda }}}{{\partial \lambda }}{U_\lambda } = {U_\lambda }\frac{{\partial {U_\lambda }}}{{\partial \lambda }}\),where \(\frac{\partial }{{\partial \lambda }}\) is the directional derivative. (2) If there exists a complex number \(\alpha ,{\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then A is a normal operator iff \(U = {A^{ * - 1}}A\) is also. (3) If A is a hyperiwrmal or a subnormal, then for every \(\lambda \in \rho (A),\sigma ({U_\lambda })\) lies on the circle. 3, 4期 关于极?积算子炉一U 499 Theorem 3? Let A be a linear bounded operator on Hm Suppose 0￡p(J.)? (1) If U=А*~гА is a unitary operator in a certain non-degenerate bilinear space (H} 1 J |Л|>1 入 J Ш=1 solmble iff (l)cr(ü) =-=i=-, (2) the operotor Ui= f XdE\ is unitarity equivalent to cr(C7) J |л|>1 the operator ül_1 == ШЕ1, 、 J 1Л|>1 If the conditions (1),(2) are satified, then we have (3) the subspace JJi?^2 of H reduces any solution of the equation A*"1 A — U, and 1 AI (Ягея*)1 = ^Uo where Ъ is any in vertible self-adjoint on (i?i?J?2) L, and bv Uo (Uo = Jiai i ЫЕ1 )}⑷41Я1фя, = , Ла = A1ü2, if the operator V is to realize TJt and Ut^unitary equivalence^ then Ai = VSf where S is any invertible on Нг and SvUi. " Corollary. Swpp)se operator U is a normal on a certain Hilbert space(Hy v) (^the U is similar to a certain normal operator on IT), if U satisfies the conditions (1)、(2) of Theorem^ on ?S,v) ? Then the general form of the solution of А*~гА = U is A = vA'y where A' is same as A in Theorem^. Theorem 5. Let U be a linear bounded operator on H. Suppose O?p(J.)and p(ü) is a simply connected region, then the equation А*~гЛ = U is a solvable iff there exists a certain space {H,v) with a indefinite metric, such that U is a unitary operator on (Hf 丨）. If is a unitary operator on ?H,v),then there exists a particular solution of I X+$ А*~гА = U: Ar = 2e v [ (?7 —X)_1+X], where eie ? p (JJ), and the general form of the solution is Аж АУ, wliere V is any in vertible self-adjoint on (H, -u)and VvU,     

Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH : B H(I) → B(?2), β : B H(I) → B(H), respectively, so that B H(I) becomes a unital C*-algebra under each kind of multiplication and involution. It is proved that the two C*-algebras(B H(I), ?, ?) and(B H(I), ·, *) are *-isomorphic. It is also proved that the set F H(I) of all frames for H is a unital multiplicative semi-group and the set R H(I) of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set K H(I) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

Analytic Operator Rings and Analytic J-Self Adjoint Operator Ring

Let A be a linear bounded operator in Hilbert space H with polar respresentation A =J(A^*A)^{1/2} where J^2=I, J^* = J. we use \pho_J(A) to denote the set of all complex \lambda, such that for any $\lambda \in \rho_J(A)$ there exist an bounded inverce R_J(A,\lambda) of (A—\lambda J)and \sigma_J(A)to complement of \rho_J(A). Let S be a closed Cauchy domain, S\supset \sigma_J(A) and f (z) an analytic function on S. We define $f(A)=\frac{1}{2\pi i}\oint\limits_{2s} {f(\zeta ){R_J}(A,\zeta )d\zeta }$, the set of all such f(A)is denoted M. If f(z) be analytic on S and symmetrical for real axis then f(A) is J-self adjoint. The set of all such f(A)is denoted M'. Let A\otimes B = AJB for A, B \in M(or M'). We have Theorem, the ring of functions analytic on S (or analytic symmetrical for real axis on S) is a algebra homomorphism of M (or M'). The constant function 1 or z corresponds to operator J or A^* respectively. Let $M_J={JB|B \in M}$ and $M'_J={JB|B \in M'}$ If the spectrum of (A^*A)^{1/2} is detached, we have Theorem. M_J has common non-trivial reducing subspace and it is true for M_J.     

Problem with Critical Sobolev Exponent and with Weight

The authors consider the problem: -div(p▽u) = uq-1 λu, u > 0 inΩ, u = 0 on (?)Ω, whereΩis a bounded domain in Rn, n≥3, p :Ω→R is a given positive weight such that p∈H1 (Ω)∩C(Ω),λis a real constant and q = 2n/n-2, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrōdinger Equations

The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=2m/2m-1∈（1,2].The following five problems are studied： （I） A sharp asymptotic behaviour of solutions as t → ＋∞ is governed by a discrete spectrum and a countable set Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same spectrum σ（B^＊）=σ（B）.Applications of these spectral results also include： （Ⅲ） a unique continuation theorem, and （IV） boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜u｜^nu）,in R^N×R＋,where n〉0.For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n → 0^＋ and to use spectral theory of the pair {B,B^＊}.     

POWERS OF AN INVERTIBLE ω-HYPONORMAL OPERATOR

§1　IntroductionIn whatfollows,H means a complex Hilbertspace.A bounded linear operator T on His said to be positive(in symbol:T≥0 ) if(Tx,x)≥0 for any x∈H.Also an operator T isstrictly positive(in symbol:T>0 ) if T is positive and invertible.If A and B are invertiblepositive operators,itis well known that A≥B implies log A≥log B.However[1 ] ,log A≥log B does notnecessarily imply A≥B.Let T be a bounded linear operator and p≥0 .T is said to be a p-hyponormal operatorif(T* T)…     

不定酉不变线性映射的刻戈及相关结果

Let A and B be unital C*-algebras, and let J ∈ A, L ∈ B be Hermitian invertible elements. For every T ∈ A and S ∈ B,define TJ(?)=J-1T*J and SL(?) =L-1S*L. Then in such a way we endow the C*-algebras A and B with indefinite structures. We characterize firstly the Jordan (J, L)-(?)-homomorphisms on C*-algebras. As applications, we further classify the bounded linear maps ?:A→B preserving (J, L)-unitary elements. When A = B(H) and B = B(K), where H and K are infinite dimensional and complete indefinite inner product spaces on real or complex fields, we prove that indefinite-unitary preserving bounded linear surjections are of the form T →UVTV-1((?)T ∈ B(H)) or T→UVT(?)V-1 ((?)T ∈ B(H)), where U ∈ B(K) is indefinite unitary and, V : H→K is generalized indefinite unitary in the first form and generalized indefinite anti-unitary in the second one. Some results on indefinite orthogonality preserving additive maps are also given.     

非游荡算子的性质*

本文研究一类具有混纯性质的线性算子:非游荡算子,该类算子仅在无穷维线性空间中.我们给出非游荡算子紧集上的超循环分解.     

Nilpotent operators similar to irreducible operators

We give a necessary and sufficient condition for the nilpotent operators to be similiar to irreducible operators, and give an answer to D. A. Herrero Conjectures for nilpotent operators.     

Disjointly homogeneous Banach lattices and compact products of operators

The notion of disjointly homogeneous Banach lattice is introduced. In these spaces every two disjoint sequences share equivalent subsequences. It is proved that on this class of Banach lattices the product of a regular AM-compact and a regular disjointly strictly singular operators is always a compact operator.     

冯·诺依曼代数的可测算子的性质

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.     

广义弱亚正规算子的正规性

主要研究了T是广义弱亚正规算子时,T~t是拟正规算子的充要条件是T是拟正规算子.并且举例说明了存在非次正规的广义弱亚正规算子T使得T~t是次正规的.     

Operator Method of Solving Nonlinear Differential Equations

We discuss the possibility of applying linear structures for solving nonlinear differential equations.     

Stable and stabilizable means involving linear operator arguments

In this paper we extend the stability and stabilizability concepts from the case that the variables are positive real numbers to the case that the variables are positive linear operators. Since the algebra of bounded linear operators is not commutative, such extension does not appear to be obvious. As applications, iterative operator algorithms, converging to some operator means, are discussed.     

一类解析算子函数空间上的复合算子

本文用算子函数论的方法，研究了解析算子函数的Banach空间X，X0上的复合算子．给出此复合算子为有界的条件，并刻划了此复合算子在X0上为紧的特征．     

完全超约化算子的几个性质

本文讨论了完全超约化算子的性质，共得到三个结果，主要结果是给出了完全超约化算子有一个超约的特征向量的充分条件。