Norm equalities and inequalities for three circulant operator matrices

In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered.     

Norm inequalities in operator ideals

In this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm. Among the well-known inequalities which can be proved with this technique are the Löwner-Heinz inequality, inequalities relating various operator means and the Corach-Porta-Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in C^∗-algebras, in particular to the noncommutative Lp-spaces of a semi-finite von Neumann algebra.     

具有位数3和4的双随机循环矩阵中的素元

本文研究了双随机循环矩阵中素元的分类问题．由于任一n阶双随机循环矩阵都可以唯一地表示为移位的n-1次一元多项式,从而可把双随机循环矩阵中素元的分类问题简化为解双随机循环矩阵上的一个方程．应用此原理,本文完全解决了判别具有位数3的n阶双随机循环矩阵是否为素元的问题,并给出了n阶双随机循环矩阵中一类具有位数4的素元．     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

Inequalities for two type potential operators on differential forms

The purpose of this paper is to derive some strong-type inequalities for convolution type potential operator applied on differential forms. Caccioppoli-type inequalities for integral type potential operator acting on A-harmonic tensor are also obtained.     

THE SINE TRANSFORM OPERATOR IN THE BANACH SPACE OF SYMMETRIC MATRICES AND ITS APPLICATION IN IMAGE RESTORATION

In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.     

几类矩阵范数之间的关系

1 引言 [1]中,讨论了C~(m×n)上的矩阵A的L_p范数/A/_p=(∑/a_(ij)/~p)和l_p算子范数‖A‖_p= max/AX/_p1』之间的关系,得到了下面的不等式: ‖A‖_p≤μ_p(n)/A/_p, ‖A‖_p≤μ_q(m)/A/_p, (1.1) 这里     

对角因子分块循环矩阵的对角化和谱分解

导出了对角因子分块循环矩阵的概念，把循环矩阵的对角化和谱分解推广到具有对角因子循环结构的分块矩阵中去．     

包含切比雪夫多项式的循环矩阵行列式的计算

行首加r尾r右循环矩阵和行尾加r首r左循环矩阵是两种特殊类型的矩阵,这篇论文中就是利用多项式因式分解的逆变换这一重要的技巧以及这类循环矩阵漂亮的结构和切比雪夫多项式的特殊的结构,分别讨论了第一类、第二类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式,从而给出了行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式显式表达式.这些显式表达式与切比雪夫多项式以及参数r有关.这一问题的应用背景主要在循环编码,图像处理等信息理论方面.     

关于随机矩阵Kronecker积的谱半径的不等式

研究了随机矩阵的Kronecker积的数学期望的性质,得到了随机矩阵的Kronecker积的谱半径的几个不等式.     

Double circulant matrices

Double circulant matrices are introduced and studied. By a matrix-theoretic method, the rank r of a double circulant matrix is computed, and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are posed.     

加权Morrey-Herz空间上的广义Hausdorff算子

引入了一种新的广义Hausdorff算子.它包括了许多著名的算子作为其特殊情形.我们得到了这些算子在加权Morrey-Herz空间上有界的充分必要条件,还得到了相应的新的算子范数不等式.它们是许多著名结果有意义的改进和推广.     

在两循环矩阵类中求解线性方程组反问题的快速算法

本文利用多项式最大公因式 ,给出了线性方程组的反问题在 r-循环矩阵类和对称 r-循环矩阵类中有唯一解的充要条件 ,进而得到线性方程组在 r循环矩阵类和对称 r-循环矩阵类中的反问题求唯一解的算法 .最后给出了应用该算法的数值例子 .     

循环矩阵本原指数的上界估计

本文给出了循环矩阵本原指数上界的新的估计及一种由级数较低的循环矩阵的本原指数估计级数较高的循环矩阵的本原指数的方法，解决了一类循环矩阵本原指数的计算问题．     

On the Bohr inequality of operators

This paper is focused on the operator inequalities of the Bohr type. We will give a new and transparent proof for the operator Bohr inequality through an absolute value operator identity, show some related operator inequalities by means of 2×2 (block) operator matrices, and finally we will present a generalization of the operator Bohr inequality for multiple operators.     

Some inequalities for commutators and an application to spectral variation. II

Inequalities that compare unitarily invariant norms of A - B and those of AΓ - ΓB and Γ^-1A - B Γ^-1 are obtained, where both A and B are either Hermitian or unitary or normal operators and Γ is a positive definite operator in a complex separable Hilbert space. These inequalities are then applied to derive bounds for spectral variation of diagonalisable matrices. Our new bounds improve substantially previously published bounds.     

关于矩阵Kronecker积的谱半径的不等式

本文证明了矩阵的 Kronecker积的谱半径的两个不等式     

Pascal算子矩阵在组合恒等式中的应用(英文)

定义了四种Pascal算子矩阵,给出了它们的代数性质及它们之间的关系,并且利用二项式型多项式序列、算子及哑运算得到许多组合恒等式.     

On a p-adic Hilbert-type integral operator and its applications

In this note, we deal with a $p$-adic Hilbert-type integral operator induced by a symmetric homogeneous kernel of degree $-1$ and obtain the expression of the norm of this operator. As applications, we establish some new $p$-adic Hilbert-type inequalities with best constant factors.     

Pascal算子矩阵在组合恒等式中的应用

定义了四种Pascal算子矩阵,给出了它们的代数性质及它们之间的关系,并且利用二项式型多项式序列、算子及哑运算得到许多组合恒等式.