置换空间P_XX_n的范数k-粗性

利用Banach空间理论的方法,主要研究了k-粗性和k-强粗性从Banach空间Xn到置换空间PXXn上的提升问题,证明了这两种k-粗性都可以在置换空间PXXn上得到提升.     

向量范数的几何刻画

从向量范数的定义出发,通过研究范数为1的超曲面的几何性质,给出了范数等价的几何刻画.利用该等价刻画验证了p范数(1≤p≤∞)满足的条件以及p范数关于p递减两个事实.最后进一步讨论了矩阵诱导范数的有界性,得出了在n阶方阵中只有纯量矩阵的诱导范数有界的结论.     

关于矩阵2-范数的一个定理

证明了矩阵2-范数的一个定理.该定理指出了一类实方阵的2-范数恒不大于1.在未来的研究中,这个定理可以用来分析图像处理中的某些算法.     

关于2R（w2R）范数的一个注记

证明了自反Banach空间X上的等价w2R范数全体构成一个剩余集;同时证明了X闭子空间上等价的w2R范数均可延拓为X上等价的w2R范数.特别地,当X是可分时,上述w2R范数可替换为2R范数.     

一个范数构造定理的反例及修正

1 引 言 近年来,范数在矩阵计算[2]、扰动理论[4]等领域中的应用越来越广泛.有关范数不等式的运用在一些证明中也越来越常见. 在文[3]中,Horn和Johnson引用了这样的一个范数定理: 命题 V为域F(C或R)上的向量空间,若‖·‖a1,…,‖·‖am为V上的向量范数,‖·‖β是Rm上的向量范数,则函数f:V|→R     

基于半范数的风险测度

由于一致性风险测度公理中的平移不变性公理存在不合理性,故可将该该公理从一致性风险测度公理中去掉.将半范数的概念加以扩展,增加单调性要求,则去掉平移不变性公理之后一致性风险测度公理与半范数的要求就完全相同,这样风险度量从本质上讲就是定义在某空间上的半范数.本文发现F ishburn的风险测度是满足正齐次性、次可加性、单调性要求的.从这个意义讲,F ishburn的风险测度是一个比较科学的风险度量方法.     

Musielak-Orlicz序列空间中的S-点

本文给出了Musielak-Orlicz序列空间中S-点的充要判别准则.作为推论,得到Musielak-Orlicz序列空间具有S-性质的充分必要条件.     

关于PN空间上线性算子的概率范数

本文提出PN空间上线性算子的概率范数的新定义,并用它对算子有界性进行刻划,还讨论了算子空间的完备性.     

Hilbert空间中广义算子半群范数连续的特征

研究了Hilbert空间上最终范数连续广义算子半群的特征条件,利用半群的生成元的预解式,给出了Hilbert空间上广义算子半群范数连续的三个特征条件.     

单位圆盘加权Bergman空间上Toeplitz算子的本性范数

对于D上的Carleson测度μ而言,本文研究在加权Bergman空间Aα~2（D）上具有符号μ的Toeplitz算子Tμ的一些特殊的性质.近几年,在加权Bergman空间Aα~2（D）上的Toeplitz算子的有界性和紧性已经被广泛研究.为了了解Toeplitz算子Tμ的一些其他性质,本文需要估算出单位圆盘的加权Bergman空间上Toeplitz算子的本性范数的界限.     

Principal powers of matrices with positive definite real part

We study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.     

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25 :362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A ? Ã‖₂ ? ‖A ? Ã‖_F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14 :588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.     

Block-matrix generalizations of infinite-dimensional schur products and schur multipliers

We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.     

An extension of invertibility of Hammerstein-type operators

My aim is to show that some properties, proved to be true for the square matrices, are true for some not necessarily linear operators on a linear space, in particular, for Hammerstein-type operators.     

Optimizing matrix stability

Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.

     

Rings and groups of matrices with a nonstandard product

We define a new operation of multiplication on the set of square matrices. We determine when this multiplication is associative and when the set of matrices with this multiplication and the ordinary addition of matrices constitutes a ring. Furthermore, we determine when the nonstandard product admits the identity element and which elements are invertible. We study the relation between the nonstandard product and the affine transformations of a vector space. Using these results, we prove that the Mikha?lichenko group, which is a group of matrices with the nonstandard product, is isomorphic to a subgroup of matrices of a greater size with the ordinary product.     

On orthostochastic, unistochastic and qustochastic matrices

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size n?4.     

Rectangular random matrices, related convolution

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore, we can define a “rectangular-free convolution”, which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values. This convolution is linearized by cumulants and by an analytic integral transform, that we called the “rectangular R-transform”.     

再生核希尔伯特空间连续线性泛函的范数及其应用北大核心

考虑了再生核希尔伯特空间连续线性泛函范数的表示,得到了用其范数平方等于该线性泛函连续两次作于再生核的简明表示.对于常见的Sobolev-Hibert空间而言,其再生核则可用截幂函数来表示,从而得到Sobolev-Hibert空间上连续线性泛函范数的简洁表示,以新视角解释和简化了文献中的现有结果.