Bohr Inequality for Multiple Op erators

An absolute value equation is established for linear combinations of two operators. When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained. Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2] and [3].     

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.     

The Hadamard determinant inequality — Extensions to operators on a Hilbert space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some ‘Fischer-type’ inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on $[0,\infty )$. We place emphasis on documenting necessary and sufficient conditions for equality to hold.     

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.     

The converse of the Loewner–Heinz inequality via perspective

The Loewner–Heinz inequality is not only the most essential one in operator theory, but also a fundamental tool for treating operator inequalities. The aim of this paper is to investigate the converse of the Loewner–Heinz inequality in the view point of perspective and generalized perspective of operator monotone and multiplicative functions. Indeed, we give perspective inequalities equivalent to the Loewner–Heinz inequality.     

On Variance and Covariance for Bounded Linear Operators

In this paper we initiate a study of covariance and variance for two operators on a Hilbert space, proving that the c-v (covariance-variance) inequality holds, which is equivalent to the Cauchy-Schwarz inequality. As for applications of the c-v inequality we prove uniformly the Bernstein-type inequalities and equalities, and show the generalized Heinz-Kato-Furuta-type inequalities and equalities, from which a generalization and sharpening of Reid's inequality is obtained. We show that every operator can be expressed as a p-hyponormal-type, and a hyponormal-type operator. Finally, some new characterizations of the Furuta inequality are given. Received April 9, 2000, Revised July 20, 2000, Accepted August 8, 2000     

Some extensions of the operator entropy type inequalities

In this paper, we define the generalised relative operator entropy and investigate some of its properties such as subadditivity and homogeneity. As application of our result, we obtain the information inequality. In continuation, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond–Pe?ari? method.     

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.     

Maximal operator for multilinear Calderón-Zygmund singular integral operators on weighted Hardy spaces

In this paper, the maximal operator associated with multilinear Calderón-Zygmund singular integral operators will be studied by using an improved Coltlar's inequality. Moreover, weighted norm inequalities and some estimates on weighted Hardy spaces are obtained for this maximal operator.     

关于一般矩阵函数变差的不等式

该文首先展示了一般矩阵函数dχ(A)=∑[DD(X]σ∈H[DD)]χ(σ)∏[DD(]m[]i=1[DD)]aiσ(i)可作为一个酉空间张量的适当对称类的内积.然后，借助Schwarz不等式和范数证明了关于一般矩阵函数变差的三个主要不等式,而其中一个不等式是已知不等式的推广.     

Generalized Vector Variational and Quasi-Variational Inequalities with Operator Solutions

In a recent paper, Domokos and Kolumbán introduced variational inequalities with operator solutions to provide a suitable unified approach to several kinds of variational inequality and vector variational inequality in Banach spaces. Inspired by their work, in this paper, we further develop the new scheme of vector variational inequalities with operator solutions from the single-valued case into the multi-valued one. We prove the existence of solutions of generalized vector variational inequalities with operator solutions and generalized quasi-vector variational inequalities with operator solutions. Some applications to generalized vector variational inequalities and generalized quasi-vector variational inequalities in a normed space are also provided.     

黎曼流形上一类算子的低阶特征值估计

本文研究n(≥ 2)维完备黎曼流形M的有界区域Ω上算子的低阶特征值估计问题.利用Rayleigh-Ritz不等式,获得了该算子低阶特征值的万有不等式.     

三个矩阵秩不等式的多种证明

针对线性代数课程教学中三个矩阵秩的不等式的证明展开研究,对每一个不等式,分别给出相应的证明。     

Hermite正定对称矩阵迹的一些结果(英文)

本文研究了一类Hermite正定矩阵迹的不等式问题.利用文献[2-6]中的结果以及放缩法,获得了Hermite正定矩阵迹的极值定理、杨氏不等式和贝努利不等式,并且将许多初等不等式推广到Hermite正定矩阵迹的情形.     

On Bohr's Inequality

Bohr's inequality says that if is a bounded analytic function on the closed unit disc, then for 0 leq r 1/3 and that1/3 is sharp. In this paper we give an operator-theoretic proofof Bohr's inequality that is based on von Neumann's inequality.Since our proof is operator-theoretic, our methods extend toseveral complex variables and to non-commutative situations. We obtain Bohr type inequalities for the algebras of boundedanalytic functions and the multiplier algebras of reproducingkernel Hilbert spaces on various higher-dimensional domains,for the non-commutative disc algebra A_n, and for the reduced(respectively full) group C*-algebra of the free group on ngenerators. We also include an application to Banach algebras. We provethat every Banach algebra has an equivalent norm in which itsatisfies a non-unital version of von Neumann's inequality. 2000 Mathematical Subject Classification: 47A20, 47A56.     

Reduction principle for a certain class of kernel-type operators

The classical Hardy–Littlewood inequality asserts that the integral of a product of two functions is always majorized by that of their non-increasing rearrangements. One of the pivotal applications of this result is the fact that the boundedness of an integral operator acting near zero is equivalent to the boundedness of the same operator restricted to the cone of positive non-increasing functions. It is well known that an analogous inequality for integration away from zero is not true. We will show in this paper that, nevertheless, the equivalence of the two inequalities is still preserved for certain rather general class of kernel-type operators under a mild restriction and regardless of the measure of the underlying integration domain.     

A Cotlar type inequality for the multilinear singular integral operators and its applications

A Cotlar type inequality is established for the multilinear singular integral operators. As applications, some two-weight norm inequalities are obtained for the maximal operator corresponding to the multilinear singular integral operators.     

Well-posedness of a general class of elliptic mixed hemivariational–variational inequalities

In this paper, well-posedness of a general class of elliptic mixed hemivariational–variational inequalities is studied. This general class includes several classes of the previously studied elliptic mixed hemivariational–variational inequalities as special cases. Moreover, our approach of the well-posedness analysis is easily accessible, unlike those in the published papers on elliptic mixed hemivariational–variational inequalities so far. First, prior theoretical results are recalled for a class of elliptic mixed hemivariational–variational inequalities featured by the presence of a potential operator. Then the well-posedness results are extended through a Banach fixed-point argument to the same class of inequalities without the potential operator assumption. The well-posedness results are further extended to a more general class of elliptic mixed hemivariational–variational inequalities through another application of the Banach fixed-point argument. The theoretical results are illustrated in the study of a contact problem. For comparison, the contact problem is studied both as an elliptic mixed hemivariational–variational inequality and as an elliptic variational–hemivariational inequality.     

Norm inequalities for the spectral spread of Hermitian operators

In this work, we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite-dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the arithmetic–geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).     

Reverses of Schwarz inequality in inner product spaces with applications

New reverses of the Schwarz inequality in complex inner products spaces with applications for bounded linear operators are given. Some Grüss' type inequalities and their applications for numerical radius and the operator norm are provided as well.