Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

On matrix approximation problems with Ky Fank norms

LetA be a realm xn matrix whose elements depend onl free parameters forming the vectorx. Then a class of approximation problems can be defined by the requirement thatx be chosen to minimize A(x), for a given matrix normon m ×n matrices. For example, it may be required to approximate a given matrix by a particular type of matrix, or by a linear combination of matrices. In the derivation of effective algorithms for such problems, a prerequisite is the provision of appropriate conditions satisfied by a solution, and the subdifferential of the matrix norm plays a crucial role in this. Therefore a characterization of the subdifferential is important, and this is considered for a class of orthogonally invariant norms known as Ky Fank norms, which include as special cases the spectral norm and the trace norm. The results lead to a consideration of efficient algorithms.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Ky Fan's N-matrices and linear complementarity problems

We consider the linear complementarity problem (LCP),w=Az + q, w0,z0,w ^T z=0, when all the off-diagonal entries ofA are nonpositive (the class of Z-matrices), all the proper principal minors ofA are positive and the determinant ofA is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that ifA is a Z-matrix, thenA is an F-matrix if and only if LCP(q, A) has exactly two solutions for anyq0,q0, and has at most two solutions for any otherq. Research supported by AFOSR-89-0512.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Isometries of Lipschitz type function spaces

In this article, we describe isometries over the Lipschitz spaces under certain conditions. Indeed, we provide a unified proof for the main results of 3 and 5 in a more general setting. Finally, we extend our results for some other functions spaces like the space of vector‐valued little Lipschitz maps and pointwise Lipschitz maps.     

Isometries of Weighted Spaces of Harmonic Functions

We investigate the isometries between weighted spaces of harmonic functions. We show that, under some mild conditions, every isometry is a composition operator. Our research shows that the structure of isometries of weighted spaces of harmonic functions is, in general, simpler than that observed for weighted spaces of holomorphic functions. Supported by MEC and FEDER Project MTM 2005-08210.     

Apollonian等距与Mobius变换

设D是R²中至少包含三个边界点的单连通区域, 对任意x, y∈ D, aD(x, y)表示D中关于x, y两点的Apollonian度量．1998年A. F. Beardon猜测: 若f: D→ D是Apollonian等距映射,则f必是D上的Mobius变换.在该文中作者对D是圆的情况肯定并证明了A. F. Beardon的上述猜想     

Isometries between Banach spaces of Lipschitz functions

The Banach spaces Lip ^a (S, Δ), lip ^a (S, Δ), Lip ^a (S, Δ;s ₀) and lip ^a (S, Δ;s ₀) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces. This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Dr. Y. Benyamini.     

Isometries between completely regular vector-valued function spaces

Periodica Mathematica Hungarica - In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of $$C_0(X,E)$$ and $$C_0(Y,F)$$ where X...     

Isometries between projection lattices of von Neumann algebras

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ^?-isomorphisms. In particular, we prove that two von Neumann algebras without type ${\mathrm{I}}_1$ direct summands are Jordan ^?-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. ?emrl, which is a generalization of Wigner's theorem.     

单位球面间等距映射的线性延拓

本文研究实赋范空间E和F的单位球面S_1(E)和S_1(F)之间的等距映射线性延拓问题。得到:若T:S_1(E)→S_1(F)是一个满等距映射,且对于(?)x,y∈S_1(E),有‖T(x)-|λ|T(y)‖≤‖x-|λ|y‖,(?)λ∈R,则T可延拓为全空间上的实线性算子。     

矩阵空间上保弱伴随矩阵的线性映射

为了刻画矩阵空间上保弱伴随矩阵的线性映射f,引入了保弱伴随矩阵的概念,以矩阵的弱伴随矩阵为不变量,得到了当n≥3时数域F上从线性矩阵空间Mn×n（F）到Mm×m（F）的保弱伴随矩阵的线性映射f的形式．     

一般化凸空间上的Ky Fan型重合点定理及其应用

我们应用已有的一般化凸空间上的KKM型定理得到K y Fan型重合点定理,然后作为应用给出截口定理和择一性问题.我们的主要结果对文[1-5]中的相应的结果进行了改进和一般化.     

Minimax inequalities of Ky Fan

In this paper, we found a new result by relaxing the condition of [1, Theorem 3]. As its direct consequence, we have obtained some new minimax inequalities of Ky Fan and minimax theorems.