On Berezin Number Inequalities for Operator Matrices

In this article, we refine certain earlier existing bounds for Berezin number of operator matrices. We also prove some new Berezin number inequalities for general n × n operator matrices.Further, we establish several upper bounds for Berezin number and generalized Euclidean Berezin number for off-diagonal operator matrices. Finally, some interesting examples are discussed.     

Development of the Berezin Number Inequalities

We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.     

任意多个Hermitian矩阵的Khatri-Rao乘积的一些不等式

最后S.Liu[2]和笔[4]得到了两个Hermite矩阵的Khatri-Rao乘积的一些不等式。我们以两种方式来推广这些结果。首先，将结论推广到任意有限个Hermite矩阵的Khatri-Rao乘积；其次，给出了相应不等式的等式成立的充分必要条件。     

几个算子不等式的最佳常数

该文给出了单位圆上解析函数中φ(z),|φ(z)<1,对于 Hilbert空间 H上任一真压缩算子 A,算子φ(A)的范数估计,得出 sup b~2(φ)=2, sup b~2(φ)=1. b(φ)=sup(||φ(A)||(1-||A||~2) ||φ(A)||~2)~(1/2),b(φ)=sup{||φ(A)||(1-||A||~2) inf ||φ(A)x||~2}~(1/2),这里A取遍H上一切真压缩算子,并且若φ(z)=sum from σ=0 to n(0/σ),|φ(z)|,|φ(z)|<1,则||φ(A)|| ||φ(A)||≤2n,其中φ(z)=z~nφ(z~(-1)),且2是与n,φ(z)及A无关常数中的最小者.     

四元数矩阵的范数与迹的一些不等式

本文获得了四元数矩阵的范数和迹的一些不等式,改进了近期的某些结果．     

The Berezin and Garding Inequalities

Let ? be a convex function on ?, let ?(σ) be a pseudodifferential operator with symbol σ, let Λ_σ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λ_σ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that \(\sum {_{\lambda \in \Lambda _\sigma } m(\lambda )} \varphi (\lambda ) \leqslant \operatorname{Tr} L(\varphi (\sigma )) + R\), where R is an error term of the same order as the remainder term in the Garding inequality.     

关于算子迹的几个不等式的等价性

本文给出算子迹的三个不等式,利用柯西推理方法证明其等价性,且推广了一些文献的结果.     

有关实正定方阵行列式的几个不等式

本文得到实正定方阵行列式的几个不等式，改进了近期研究的一些结果．     

Some Generalization for an Operator Which Preserving Inequalities Between Polynomials

For a polynomial p(z) of degree n which has no zeros in |z| 1, Dewan et al.,(K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363(2010), 38–41) established zp′(z) +nβ2p(z) ≤n2{( β2 + 1+β2)max|z|=1|p(z)|-( 1+β2- β2)min|z|=1|p(z)|},for any complex number β with |β|≤ 1 and |z| = 1. In this paper we consider the operator B, which carries a polynomial p(z) into B[ p(z)] := λ0p(z) + λ1(nz2)p′(z)1!+ λ2(nz2)2 p′′(z)2!,where λ0, λ1, and λ2are such that all the zeros of u(z) = λ0+c(n,1)λ1z+c(n,2)λ2z2lie in the half plane |z| ≤ |z-n/2|. By using the operator B, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.     

复正定矩阵的行列式不等式

本文指出了文献[3-6]中的一些不正确的结论,并给出了复正定矩阵的行列式不等式.     

算子矩阵的亚循环性

讨论了亚循环算子和具有拓扑一致降标的算子之间的关系,同时研究了算子矩阵的亚循环性和超循环性.     

亚正定矩阵Schur补的几个不等式

本文给出了亚正定方阵Ｓｃｈｕｒ 补的几个不等式,推广了Ｆａｎｋｙ的结果．     

Determinantal Inequalities for Accretive-Dissipative Matrices

A matrix is said to be accretive-dissipative if, in its Hermitian decomposition , both matrices B and C are positive definite. Further, if B= I _n, then A is called a Buckley matrix. The following extension of the classical Fischer inequality for Hermitian positive-definite matrices is proved. Let be an accretive-dissipative matrix, k and l be the orders of A ₁₁ and A ₂₂, respectively, and let m = min{k,l}. Then For Buckley matrices, the stronger bound is obtained. Bibliography: 5 titles.     

亚正定阵的几个开问题及一些不等式

亚正定阵的几个开问题及一些不等式谢清明（湘潭大学数学系，湖南４１１１０５）关键词亚正定阵，ｋ－局部对称阵，Ｓｃｈｕｒ补．     

Lq Inequalities and Operator Preserving Inequalities

Let Pn be the class of polynomials of degree at most nRather and Shah [15]proved that if P∈Pnand P(z)≠0 in|z| 1, then for every R 0 and 0≤q ∞,|B[ P( Rz)]|q ≤|RnB[zn] + λ0 |q|1 + zn|q|P(z)|q,where B is a Bn-operator.In this paper, we prove some generalization of this result which in particular yields some known polynomial inequalities as specialWe also consider an operator Dαwhich maps a polynomial P(z) into DαP(z) := n P(z) +(α-z) P′(z) and obtain extensions and generalizations of a number of well-known Lq inequalities     

关于Hodge-Dirac微分算子的Caccioppoli不等式

利用广义Hlder不等式、满足Dirac-调和方程的微分形式的弱逆Hlder不等式等重要引理,在已有的关于Hodge-Dirac微分算子的Caccioppoli-型不等式的基础上,首先给出局部的双权Caccioppoli-型积分不等式的参数形式.进一步,作为上述局部结论的应用,在有界域Ω上给出了相应的全局加权积分不等式.结论中的四个参数λ_1,λ_2,λ_3,α使得到的结论更具灵活性,若赋予参数以适当的值可以得到了其它经典权函数的加权积分不等式.     

亚正定矩阵的Bergstrom型不等式

首先证明亚正定矩阵的一个偏序,利用该偏序得到了亚正定矩阵的一些Bergstrom型不等式,推广了近期关于亚正定矩阵行列式不等式的一些结果.