评《不动点定理》

《数学研究与评论》1982年第一期上发表了颜心力同志的文章《不动点定理》,该文主要目的是企图将求算子T的不动点问题转化为寻找与T可换的算子F,使得若TF有唯一不动点,则T必有不动点(定理1)。其余结果均是定理1应用于已知结果所得到的推论。 我认为该文定理1也不是新结果,而是M.A.Krasnosel＇skii et.al.,Approximatesolution of operator equations,p.8-9中结果的改述。其证明也是类似的。这一方法主要用于证明下述事实:若算子T:X→X(距离空间)的某次迭代T~n满足具有唯一不动点性质的压缩型条件,则T~n的不动点也就是T的不动点。但这一问题容易直接得到解决。由于该文只作了转化工作而未给出任何寻找F的新方法,因此无法认为该文是有意义的。以上浅见希批评指正。     

答丁协平同志

在本刊1982年第4期上见到丁协平同志指出拙作(简称文1)《不动点定理》是他人“结果(简称文2)的改述,其证明也是类似的”。并说“这一方法主要用于证明下述事实:若算子T:X→X的某次迭代满足具有唯一不动点性质的压缩条件,则T~n的不动点也是T的不动点”。最后进一步说文1“只作了转化工作而未给出寻找F的新方法,因此无法认为该文是有意义的”。     

解析余亚正规算子的α-Weyl定理

若T有单值延伸性且T为reguloid算子,则Weyl定理对f(T)成立,其中f∈H(σ(T)),而当T~*有单值延伸性且T是reguloid算子,α-Weyl定理对f(T)成立,其中,f∈H(σ(T)),作为定理应用,我们证明了Weyl定理对解析M-亚正规算子成立,α-Weyl定理对解析余亚正规算子成立。     

上三角算子矩阵Browder定理的稳定性

令H为无限维且复可分的Hilbert空间,B(H)为H上的有界线性算子全体.若T∈B(H)满足σ_w(T)=σ_b(T),则称T有Browder定理,其中σ_ω(T)和σ_b(T)分别表示算子T的Weyl谱和Borwder谱;对任意的紧算子K∈B(H),若T+K有Browder定理,则称T满足Browder定理的稳定性.给出了2-阶上三角算子矩阵的平方满足Borwder定理的稳定性的充要条件.     

k-拟-*-A算子的谱性质及其应用Ⅱ

主要给出了k-拟-*-A算子的一些性质,若T是k-拟-*-A算子,则T有SVEP.作为此性质的应用,证明了若T是k-拟-*-A算子,则B-Weyl谱的谱映射定理成立;若T或T*是k-拟-*-A算子,则广义Browder定理对T成立.     

无界域上1-集弱压缩算子的新不动点结果

本文证明Banach空间中无界域上一类弱序列连续和1-集弱压缩算子的若干新不动点定理.我们引入原点处弱半闭算子,得到该算子的若干不动点定理.进而将著名的Leray-Schauder不动点定理、Altman定理、Roth定理和Petryshyn定理推广到弱序列连续算子和1-集弱压缩算子以及原点处弱半闭算子的情形.本文的主要结果依赖于非紧性弱原子测度的有关条件.     

关于代数拟*-n-仿正规算子的Weyl型定理(英文)

文中主要证明了:(1)若T是一个代数拟*-n-仿正规算子,则T是极.(2)若T是一个代数拟*-n-仿正规算子,则Weyl定理对f(T)成立且f∈H(σ(T)),其中f是σ(T)开邻域上的解析函数.(3)若T*是一个代数拟*-n-仿正规算子,则广义α-Weyl定理对f(T)成立,其中f∈H(σ(T)).     

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)).     

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T0+∈B(F,E)为To∈B(E,F)的一个广义逆.本文证明,每一个具有||T0+(T-T0)|| J<1的算子T∈B(E,F),B≡(I+T0+(T-T0))-1T0+是T的广义逆当且仅当(I-T0+T0)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。     

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.     

Weak* derived sets of sets of linear functionals

For a Banach space X the w^*-sequential closure operator in the adjoint space is, in general, not the topological closure operator. That is, it may happen that the w^*-sequential closure of a subspace T of X^* is not w^*-sequentially closed. The possible length of the chain of repeated w^*-sequential closures of a subspace of X^* in dependence on the dimension of X^**/X is investigated.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 607–616, April, 1978.In conclusion, the author thanks M. I. Kadets for the formulation of the problem and assistance with the article.     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.     

Stability properties of the Tikhonov regularization for nonmonotone inclusions

We study the Tikhonov regularization for perturbed inclusions of the form T(x) ' y^*{T(x) \ni y^*} where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties and y* is an element near 0. We investigate the case when T is metrically regular and strongly regular and we show the existence of both a solution x* to the perturbed inclusion and a Tikhonov sequence which converges to x*. Finally, we show that the Tikhonov sequences associated to the perturbed problem inherit the regularity properties of the inverse of T.     

Two remarks concerning the equation Πp(Χ,·)=Ip(Χ,·)

It is proved that the analog of Grothendieck's theorem is valid for a diskalgebra “up to a logarithmic factor.” Namely, if Tε? (C_A, L¹) and then π₂(T)?C (1+logn) ¦T¦. The question of whether the logarithmic factor is actually necessary remains open. It is also established that C _A ^* is a space of cotype q for any q, q > 2. The proofs are based on a theorem of Mityagin-Pelchinskii: π_p(T)?C·p·i_p(T), p?2 for any operator T acting from a disk-algebra to an arbitrary Banach space.     

On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderón-Zygmund operator in a “non-homogeneous” space ( , d, μ), where, in particular, the measure μ may be non-doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T_*, which is the supremum of the truncated operators associated with T. Specifically, for1<p<∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L^p weighted inequality holds for T_*. The main tool to deal with this problem is the theory of vector-valued inequalities for T_* and some related operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderón-Zygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C_* to characterize the existence of principal values for functions in weighted L^p.     

Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m (Ω), with m>N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs (u_e^*,y_e^*)(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast }) of (P _ε ) converges as ε goes to 0 to a pair (u ^*,y ^*) which realizes the minimum in (P), and y ^* is the solution to the original state system.     

Gauge-periodic point perturbations on the Lobachevsky plane

We study periodic point perturbations of the Shrödinger operator with a uniform magnetic field on the Lobachevsky plane. We prove that the spectrum gaps of the perturbed operator are labeled by the elements of the K₀ group of aC ^* algebra associated with the operator. In particular, if theC ^* algebra has the Kadison property, then the operator spectrum has a band structure.     

Integer Optimization on Convex Semialgebraic Sets

Let Y be a convex set in \Re ^k defined by polynomial inequalities and equations of degree at most d ≥ 2 with integer coefficients of binary length at most l . We show that if the set of optimal solutions of the integer programming problem min is not empty, then the problem has an optimal solution of binary length ld ^O(k4) . For fixed k , our bound implies a polynomial-time algorithm for computing an optimal integral solution y ^* . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming. Received August 3, 1998, and in revised form March 22, 1999.     

On real forms of JB*-triples

We introduce real JB^*-triples as real forms of (complex) JB^*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB^*-triples is an isometry if and only if it commutes with the cube mappinga→a ³={aaa}. This generalizes a result of Dang for complex JB^*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB^*-triple numerical invariants that are respected by surjective linear isometries.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.