A system of matrix equations and its applications

We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu’s recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).     

Banach空间中极大单调算子零点的迭代构造

设E为实光滑、一致凸Banach空间,E*为其对偶空间,TE×E*为极大单调算子且T-10≠φ.本文引入了一种新迭代格式,利用Lyapunov泛函和广义投影算子等技巧,在Banach空间中证明了迭代序列弱收敛于极大单调算子T的零点的结论.     

Banach空间中极大单调算子零点的迭代逼近定理

令E为实光滑、一致凸Banach空间,E为其对偶空间．令A■ E x E为极大单调算子, A-10≠■．本文将引入新的迭代算法,并利用Lyapunov泛函, Qr算子与广义投影算子等技巧,证明了迭代序列弱收敛于极大单调算子A的零点的结论．     

鞅的(Φ_1,Φ_2)-型倒向极大不等式

设Φ_1,Φ_2是非负凸函数,证明了鞅的倒向极大算子不等式‖f‖Φ_2≤C‖f*‖Φ_1对于任意鞅f=(f_n)_n≥0成立的充分必要条件是Φ_2(?)Φ_1;鞅的极大算子均方算子的极大极小不等式‖M(f)‖Φ_2≤C_1‖m(f)‖Φ_1及‖m(f)‖Φ_2≤C_2‖M(f)‖Φ_1成立的充分必要条件是Φ_2(?)Φ_1,这里M(f)=max{f*,S(f)},m(f)=min{f*,S(f)}分别是极大算子、均方算子的极大极小函数.     

一类奇异积分算子在加权空间的界的估计

设T是由Grubb和Moore引入的一类奇异积分算子,它的核满足一种新型利普希茨正则性.T^*是由T确定的极大奇异积分算子.本文通过建立与T和T^*相关的grand极大算子的弱型端点估计,得到了算子T和T^*在加权空间的由A_p权常数表示的界的估计和弱型端点估计.     

Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件

本文研究Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件,得到定理:设C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R)分别是Ⅱk空间上第0,Ⅰ,Ⅱa,Ⅱb,Ⅲa和Ⅲb类的算子代数,则(1)C0(U),C2a(U)或C3a(U)为一致闭(弱闭)的等价条件是U是Hibert空间G上的C*-代数(W*-代数;(2)C1(U,L,R,D,V)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间,V是闭算子,L对称闭的;(3)C2b(U,R)或C3b(U,R)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间．     

一类非线性算子方程的正解问题

研究算子方程X^s+A^*X^-tA=Q的正算子解的存在性问题,通过构造有效的迭代序列,给出了算子方程X^s+A^*X^-tA=Q有正算子解的一些充分条件和必要条件,同时给出了该方程有极大解和唯一解的条件.     

交换的实W~*-代数

本文指出，交换的（复）Ｗ 代数的大部分结果对于实情形仍然成立，自然有周期为２同胚的差别．此外，不同于复情形，可分实Ｈｉｌｂｅｒｔ空间中不包含任何极小投影的交换实ＶＮ代数同构于Ｌ∞（［０，１］），Ｌ∞ｒ（［０，１］）或Ｌ∞（［０，１］）Ｌ∞ｒ（［０，１］）．     

Primitive involution rings

Summary A *-primitive involution ring Ris either a left and right primitive ring or a certain subdirect sum of a left primitive and a right primitive ring with involution exchanging the components. An example is given of a left and right primitive ring which admits no row and column finite matrix representation. We characterize *-primitive involution rings in terms of maximal *-biideals. A *-prime involution ring has a minimal left ideal if and only if it has a minimal *-biideal, and these involution rings are always *-primitive.     

The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces

The boundedness of maximal Bochner-Riesz operator Bδ* and that of maximal commutator Bδb,* generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.     

An application of Ekeland's Variational Principle to the spectrum of nonlinear homogeneous gradient operators

The Ekeland Variational Principle is used to prove the nonemptiness of the spectrum for positively homogeneous gradient mappings in Hilbert space. Under additional information on the measure of noncompactness of the operator, this leads to the existence of a maximal (or minimal) compact eigenvalue for the operator itself.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

Minimal convex functions bounded below by the duality product

It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monotone operator. Whether this converse also holds in a nonreflexive Banach space remains an open problem.

     

On Extremal Problems on a Set of Interpolation Polynomials in a Hilbert Space

We give solutions of two extremal problems, a generalization of a result of W. Porter on an interpolation polynomial in L₂(a,b) with minimal norm, which we get here in an abstract Hilbert space with a measure, and an analogue of a theorem of M. Golomb and H. Weinberger for a bounded set of operator interpolants.     

On the spectrum of well-defined restrictions and extensions for the Laplace operator

The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $\hat L$ or a Volterra well-defined extension of a minimal operator L ₀ generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $\hat L$ and of well-defined extensions of the minimal operator L ₀ generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.     

算子代数上的Jordan初等映射(英文)

给定两个环R,R’.对于满足一定条件的环R,本文证明了若M:R→R’,M*:R’→R为满射且对A,C∈R和B,D∈R’满足M（AM*（B）C+CM*（B）A）=M（A）BM（C）+M（C）BM（A）,M*（BM（A）D+DM（A）B）=M*（B）AM*（D）+M*（D）AM*（B）则M和M*是可加的;若R和R’分别包含单位I和I’,M（I）,M*（I’）可逆,则存在环同构N使得M（A）=N（A）M（I）,M*（B）=N^-1（BM（I））.特别地,若R=R’为标准算子代数或Hilbert空间套代数,则M和M*可加且存在有界可逆的线性或共轭线性算子S和T使得M（A）=SAT,M*（B）=TBS或M（A）=TA*S,M*（B）=（SBT）*对任意的A,B∈R成立.     

On the spectral theory of the Bessel operator on a finite interval and the half-line

We study the minimal and maximal operators generated by the Bessel differential expression on a finite interval and the half-line. We describe the domains of the Friedrichs and Krein extensions of the minimal operator and all nonnegative self-adjoint extensions of the minimal operator.     

Linear relations generated by an integral equation with Nevanlinna operator measure

We define families of maximal and minimal linear relations generated by an integral equation with Nevanlinna operator measure and prove their holomorphic property. We also prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We apply the obtained results for proving the constancy of deficiency indices of some integral and differential equations.