可测算子的算子不等式

设M是作用在Hilbert空间F上的,带有一个忠实的,半有限的正规迹t的半有限的von Neumann代数.设■是所有可测算子构成的集合.本文首先给出了非交换Banach函数空间上的范数不等式,其次研究了可测算子的奇异值不等式,最后讨论了t-可测算子的奇异值不等式之间的关系.     

Hilbert空间上依范连续的α-次预解算子族

主要运用Laplace-Stieltjes的方法,讨论Hilbert空间上α-次预解算子族的依范连续性与其预解式的关系,得到一个主要结果：Hilbert空间上α-次预解算子族的依范连续性与其预解式在当｜τ｜→∞时,‖（ω＋iτ）~（α-1）R（（ω＋iτ）~α,A）‖→0等价.     

τ-可测正算子生成的交换子的若干不等式

设M是具有正规忠实的半有限迹τ的von Neumann代数,‖.‖ρ是任意非交换Banach函数空间范数,‖.‖是M上的通常范数.证明了若A和B是τ-可测正算子,X∈M,则‖AX-XB‖ρ≤‖X‖‖AB‖ρ.还证明了若A,B是M中的正算子,X是τ-可测算子,则‖AX-XB‖ρ≤max(‖A‖,‖B‖)‖X‖ρ.由此得到了若A∈M是正算子,X是τ-可测正算子,则‖AX-XA‖ρ≤1/2‖A‖‖XX‖ρ.     

关于依测度拓扑收敛

给出了τ-可测算子的依测度收敛的充要条件和Hilbert空间内依测度收敛的充要条件．     

有关τ-可测算子的Young不等式与Heinz型不等式的逆向不等式

该文主要给出了有关τ-可测算子及‖·‖_2的Young不等式及Heinz型不等式的逆向不等式,并给出了有关τ-可测算子的arithmetic平均,geometric平均及Heinz型平均的奇异值不等式.     

一个算子迹的不等式

本文讨论Bellman不等式的相关问题,利用紧算子的极表示以及陈公宁的一个矩阵迹的不等式,得到算子迹的相应不等式.作为其推论,在无穷维Hilbert空间中给出了Bellman问题的一个肯定回答.     

关于Hardy-Hilbert型积分不等式

设1/p=1/q≈1：1且P〉1．通过引入一个适当的积分核函数和参数λ（λ〉-1）,创建了一种新型Hardy～Hilbert型积分不等式．证明了其常数因子（p^λ=1＋q^λ＋1）Г（λ＋1）是最佳的,其中Г（x）Г-函数．特别,当p=2时,得到了一种新的Hilbert型积分不等式．作为应用,给出了它的一种等价形式．     

Hilbert型不等式及其应用

本文通过引入两个函数u（x）和v（x），（x∈[0，＋∞））建立了一个新的Hiblert型不等式,特别,u（m）=m＋λ及v（n）=n＋λ（∈No，λ=1/2，1）时，得到了Hilbert不等式的一个改进,作为应用，给出了Fejer-Riesz不等式的推广和改进．     

关于广义Aluthge变换的谱性质的研究

设T∈H（H），T=U｜T｜是算子T的极分解，则定义T^λ=｜T｜^λU｜T｜^1-λ和T^λ（＊）=｜T＊｜^λU｜T＊｜^1-λ，（其中0〈λ〈1）分别为算子的广义Aluthge变换和广义＊-Aluthge变换．本文中主要研究了三者之间的几种谱的关系．同时，还证明了算子T满足修正的Weyl定理当且仅当弘满足修正的Weyl定理当且仅当T^λ（＊）满足修正的Weyl定理．最后证明了算子T满足a—Weyl定理当且仅当T^λ满足a—Weyl定理．     

Riemann曲面上的正常和拟正常复合算子

Riemann曲面M上的平方可测1-形式全体和解析1-形式全体均可构成Hilbert空间.本文讨论Riemann曲面上的解析映射导出的这类Hilbert空间上的复合算子,研究复合算子的正常性、拟正常性的诱导映射特征.特别地,当M有有限三角剖分时,证明了正常复合算子、拟正常复合算子、酉复合算子、等距复合算子和可逆复合算子等价.     

On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure

For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.     

Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Positivity - Let $${{\mathcal {M}}}$$ be a von Neumann algebra of operators on a Hilbert space $${\mathcal {H}}$$ and $$\tau $$ be a faithful normal semifinite trace on $$\mathcal {M}$$ . Let...     

多线性分数次积分算子一般框架下的交换子的有界性——献给陆善镇教授75华诞

对任意给定的正整数m,Z^＋×{1,...,m}的任意一个有限子集S,定义一般化的多线性分数次积分算子的交换子Iα,→b,S（f）（x）=∫（Rn）^m ∏（i,j）∈S（bi（x）-bi（yj））/（｜x-y1｜＋…＋｜x-ym｜）^mn-α∏（j=1→m）fj（yj）d→y,其中d→y=dy1…dym.此框架下的交换子包含了以往研究的各类分数次积分算子的交换子,并蕴含了多线性背景下新的交换子形式.在上述非常一般框架下,本文给出带多重A→p,q权的多线性分数次积分算子的交换子Iα,→b,S（→f）的加权强型（L^p1（ω1）×···×L^pm（ωm）,L^q（ν→ωq））估计和加权弱型端点估计.本文还得到更一般核条件下的上述结果.     

On normal τ-measurable operators affiliated with semifinite von Neumann algebras

Let τ be a faithful normal semifinite trace on the von Neumann algebra M, 1 ≥ q > 0. The following generalizations of problems 163 and 139 from the book [1] to τ-measurable operators are obtained; it is established that: 1) each τ-compact q-hyponormal operator is normal; 2) if a τ-measurable operator A is normal and, for some natural number n, the operator A ⁿ is τ-compact, then the operator A is also τ-compact. It is proved that if a τ-measurable operator A is hyponormal and the operator A ² is τ-compact, then the operator A is also τ-compact. A new property of a nonincreasing rearrangement of the product of hyponormal and cohyponormal τ-measurable operators is established. For normal τ-measurable operators A and B, it is shown that the nonincreasing rearrangements of the operators AB and BA coincide. Applications of the results obtained to F-normed symmetric spaces on (M, τ) are considered.     

On the A. M. Bikchentaev conjecture

In 1998 A. M. Bikchentaev conjectured that for positive ??-measurable operators a and b affiliated with a semifinite von Neumann algebra, the operator b ^1/2 ab ^1/2 is submajorized by the operator ab in the sense of Hardy-Littlewood. We prove this conjecture in its full generality and obtain a number of consequences for operator ideals, Golden-Thompson inequalities, and singular traces.     

Convergence in Measure and τ-Compactness of τ-Measurable Operators,Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

     

关于换位子范数的若干不等式

当f是[0,∞)上的非负算子的单调函数时,得到了对任何非交换Banach函数空间范数都有‖f (A)X - Xf (B)‖ 54‖f ( AX - XB )‖,其中A,B是τ-可测正算子, X是收缩算子.