Problems and Conjectures in Matrix and Operator Inequalities
K. M. R. Audenaert, F. Kittaneh
我们提出了一个宝库中的矩阵和算子不等式的开放问题，一个功能分析的性质，并具有不同程度的硬度。
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
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Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.
In this article, we show multiple inequalities for the singular values of the difference of matrix means. The obtained results refine and complement some well established results in the literature. Although we target singular values inequalities, we will show several matrix means inequalities, as well.
In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath [13] in a more general setting and also relate to recent work of Gressman [8]. In particular, we establish optimisers for these determinant inequalities. We then use these inequalities to establish our main theorem which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ in [5].
Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.
We study an elementary inequality supporting the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such new Schatten p-norm estimates and new majorization
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus' inequalities. Our results are stronger and more general than the existing ones.
Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented.
Fe{j}ér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For matrix Levin-Stečkin type, we present more rigorous results involving the partial Loewner ordering for Hermitian matrices.
Direct and inverse conversion formulas associated with Khabibullin's conjecture for integral inequalities
Khabibullin猜想讨论了一类非负连续函数$q(t)$的两个线性积分不等式.这两个不等式中的第一个中的积分将$q(t)$转换为另一元函数$g(t)$。该积分产生直接转换公式。逆转换公式是指将$q(t)$表示回$g(t)$的公式。推导了这种逆转换公式。
Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed version of this result. Our approach will tackle concave function properties and some delicate manipulations of matrices and inner products.
We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
A counterexample to Khabibullin's conjecture for integral inequalities
Khabibullin关于积分不等式的猜想在其陈述中有两个数值参数$n$和$α$，其中$n$是正整数，$α$是正实数.这个猜想已经在$n>0$和$0<α\leq 1/2$ .但是，对于$α>1/2$，它并不总是有效的。本文构造了$n=2$和$α=2$的一个反例。然后将Khabibullin猜想以一种适合于所有$α>0$的方式重新表述.
Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces, and then provide a unified extension of the recent inequalities due to Lin [Electron. J. Linear Algebra 27 (2014) 821-826], Zhang et al. [Linear Algebra Appl. 498 (2016) 99-105] and [Electron. J. Linear Algebra 27 (2014) 332-341] and a result of Choi [Linear Algebra Appl. 532 (2017) 1-7]. Moreover, we demonstrate the application of a positive semidefinite $3\times 3$ block matrix, which motivates us to give a…
We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian matrix.
Inequalities involving upper bounds for certain matrix operators
本文考虑了从序列空间$l_p(v)$（或$d(v,p)$）到序列空间$l_p(w)$（或$d(w,p)$）的Hausdorff矩阵算子的上界问题.本文还讨论了从$d(v,1)$到$d(w,1)$，从$e(w,\infty)$到$e(v,\infty)$的矩阵算子的上界问题，并推导出了最近在\cite{5}和\cite{6}中考虑的与\cite{10}类似的Cesaro，Copson和Hilbert矩阵算子的上界.
The Horn conjecture for compact selfadjoint operators
我们确定了紧自伴算子A，B，C.具有A=B+C+...当所有这些算子都是正的时，已知特征值服从某些不等式，这些不等式从A=B+C的有限维情况推广了Horn不等式。我们找到适当的扩展霍恩不等式，并表明，他们，以及他们的反向类似物，提供了一个完整的表征。我们的结果也使我们能够讨论更一般的情况下，只有一些运营商的特征值指定。一个特殊的情况是要求B+C+...最多为r的秩为正。
Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea and technical tool is a two variables log-convex norm functional (Theorem 1.2).
The main goal of this article is to present new types of inequalities refining and reversing inequalities of the harmonic mean of scalars and matrices. Furthermore, implementing the spectral decomposition of positive matrices, we present a new type of inequalities treating certain harmonic matrix perturbation.
On singular value inequalities for matrix means
对于半正定$n\times n$矩阵$A$和$B$，证明了奇异值不等式$(2+t)s_{j}(A^{r}B^{2-r}+A^{2-r}B^{r})\leq 2s_{j}(A^{2}+tAB+B^{2})$对$r=\frac{1}{2}, 1, \frac{3}{2}$和所有$-2<t\leq 2$成立.
In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{î}rtoaje in the following two works: "{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented.
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
In 2015, I. Soprunov and A. Zvavitch have shown how to use the Bernstein-Khovanskii-Kushnirenko theorem to derive non-negativity of a certain bilinear form $F_Δ$, defined on (pairs of) convex bodies. Together with C. Saroglou, they proved non-negativity of $F_K$ characterizes simplices, among all polytopes. It is conjectured the characterization further holds among all convex bodies. Towards this conjecture, several necessary conditions on $K$ (for non-negativity of $F_K$), were derived. We give a new necessary condition, expressed with isoperimetric ratios, which provides a further step towar…
Utilizing the notion of positive multilinear mappings, we give some matrix inequalities. In particular, Choi--Davis--Jensen and Kantorovich type inequalities including positive multilinear mappings are presented.
A Note on Matrix Concentration Inequalities via the Method of Exchangeable Pairs
本文的目的是通过发展Mackey等人的方法证明矩阵值函数的有界差分不等式的一个改进版本：矩阵浓度不等式通过交换对的方法。在此过程中，我们证明了新的迹不等式的矩阵指数。
On matrix inequalities between the power means: counterexamples
我们证明了矩阵幂平均不等式$((A^p+B^p)/2)^{1/p}\le((A^q+B^q)/2)^{1/q}$对每对矩阵$A,B>0$成立的实参数$(p,q)$的已知充分条件确实是最佳可能的.通过构造$2\times2$反例进行证明。本文还阐明了$(p,q)$上$Φ(A^p)^{1/p}\leΦ(A^q)^{1/q}$对每个单位正线性映射$Φ$和$A>0$成立的最佳可能条件。
Matrix versions of some classical inequalities
一些与矩阵乘积重排有关的自然不等式也可以看作是经典的数列或积分不等式的推广。特别是，我们显示矩阵版本的Chebyshev和Kantorovich型不等式。矩阵方法还可以为经典不等式提供简化的证明和新的结果。例如，我们证明了Cassel不等式与Hardy-Littlewood-Polya序列的基本重排不等式之间的联系，并给出了Hardy-Littlewood-Polya不等式的一个逆不等式，其中矩阵技巧是必不可少的.
We are interested in the norm of the inclusion between the standard weighted Bergman spaces $A^2_α$ and $A^p_{\frac{p}{2} (α+ 2)-2}$, $p > 2$, which is conjectured to be contractive by O.F. Brevig, J. Ortega-Cerdà, K. Seip and J. Zhao (in the limit case $α= -1$) and E. Lieb and J.P. Solovej (for $α> -1$). We provide some partial answers and improvements of some recent results.
We propose a new, self-contained, approach to H. Raufi's extension of Prekopa's theorem for matrix-valued log-concave functions. Along the way, new related inequalities are established, in particular a Brascamp-Lieb variance inequality for matrix weights.
Matrix subadditivity inequalities and block-matrices
给出了矩阵（或算子）、分块矩阵、凹函数和范数的次可加性的几个结果和定理。
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $Ω\subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial Ω$. This reduces the study of the Neumann Poincaré constant on $Ω$ to that of the cone and Lebesgue measures on $\partial Ω$; these may be bounded via the curvature of $\partial Ω$. A second reduction is obtained to the class of harmonic functions on $Ω$. We also study the relation between the Poincaré constant of a log-concave measure $μ$ and its associated K. Ball body $K_μ$. In particular, we obtain a simple proof…
Deriving Matrix Concentration Inequalities from Kernel Couplings
本文给出了随机矩阵的谱范数离均值的指数尾界和多项式矩不等式。这一论点取决于矩阵扩展斯坦的方法交换对浓度的措施，介绍了查特吉。Mackey等人最近的工作使用这些技术来分析具有加性结构的随机矩阵，而本文中的增强覆盖了更广泛的一类矩阵值随机元。特别是，这些想法导致一个有界差分不等式，适用于随机矩阵构造的弱相关随机变量。证明需要新的迹不等式，可能是独立的利益。
Matrix Hermite-Hadamard type inequalities
给出了几个Hermite-Hadamard型矩阵不等式和算子不等式。首先建立了矩阵上单调凸函数的优化形式。然后利用Mond-Pecaric方法得到了凸函数的算子形式。我们还介绍了一些应用。最后得到了算子凸函数、正线性映射和算子作用在希尔伯特空间上的一个Hermite-Hadamard不等式。
Slicing inequalities for subspaces of $L_p.$
我们证明了超平面猜想适用于$k$ -相交体的类，并以任意测度代替体积。
We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.
A fractional Muckenhoupt-Wheeden theorem and its consequences
在20世纪70年代，Muckenhoupt和Wheeden提出了几个将Hardy-Littlewood极大算子的两个权范数不等式与奇异积分的不等式联系起来的定理。使用最近的$A_2$猜想的证明技术，我们证明了一个相关的对连接的里斯位势和分数极大算子的图解。结果证明了里斯位势的一个和两个权不等式。
Fractional generalizations of Young and Brunn-Minkowski inequalities
本文对两个以上函数卷积的情形，给出了具有锐常数卷积的Young不等式的推广，并在一定的参数范围内证明了它的正确性。这个猜想将为最近Barron和Madiman的熵幂不等式，以及Brunn-Minkowski不等式的（简化）推广提供一个统一的证明。它表明，广义Brunn-Minkowski猜想是真实的凸集;这一应用的随机集的大数定律描述。
Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes
本文研究了凸体满足Soprunov和Zvavitch提出的混合体积Bezout不等式的边界结构必要条件，并给出了三维情形的新解。
We systematically study how properties of abstract operator systems help classifying linear matrix inequality definitions of sets. Our main focus is on polyhedral cones, the 3-dimensional Lorentz cone, where we can completely describe all defining linear matrix inequalities, and on the cone of positive semidefinite matrices. Here we use results on isometries between matrix algebras to describe linear matrix inequality definitions of relatively small size. We conversely use the theory of operator systems to characterize special such isometries.
In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein's homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous groups. In this environment, the theory of Hardy type inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. Particularly, we discuss the Badiale-Tarantello conjecture and a conjecture on the geometric Hardy inequality in a half-space of the Heisenberg group with a shar…
Müntz spaces and Remez inequalities
解决了两个相对长期存在的关于Müntz多项式的问题。中心工具是非稠密Müntz空间的有界Remez型不等式。
Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections
本文证明了Hollenbeck-Verbitsky猜想，确定了圆环上解析与共解析投影Lp范数不等式的最佳常数，并推广到实直线、半空间乘子及解析鞅情形。
Given two density matrices $ρ$ and $σ$, there are a number of different expressions that reduce to the $α$-Rényi relative entropy of $ρ$ with respect to $σ$ in the classical case; i.e., when $ρ$ and $σ$ commute. Only those expressions for which the Data Processing Inequality (DPI) is valid are of potential interest as quantum divergences in quantum information theory. Audenaert and Datta have made a conjecture on the validity of the DPI for an interesting family of quantum generalizations of the $α$ - Rényi relative entropies, the $α-z$ - Rényi relative entropies. They and others have contribu…
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>…
Oppenheim--Schur inequalities for causal products
本文研究了因果卷积设置中正半定矩阵的Oppenheim-Schur型不等式，扩展了经典Schur和Oppenheim不等式，并提出了统一的因果矩阵乘积不等式框架。
We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group and then by Lieb and Solovej for SU(2) and by Kulikov for SU(1,1) and the affine group. In this paper, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber--Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1,1) cases. Finally, we prove a family of reverse H…
Isoperimetric inequalities in high-dimensional convex sets
本讲义笔记围绕Bourgain切片问题及Kannan、Lovasz与Simonovits提出的KLS等周猜想的近期研究进展展开阐述。
From Matrix to Operator Inequalities
推广了Loewner证明矩阵单调函数是算子单调函数的方法.有界算子上的关系x \leq y是我们定义剩余有限维的C*-关系的模型。我们的主要结果是一个关于有界算子关系的元定理。如果我们能证明存在剩余有限维关系，并验证一个技术条件，那么这样一个定理将由它对矩阵的限制而推出。应用程序示出关于指数的规范，规范和“积极的”非交换 *-多项式。
Universality and sharp matrix concentration inequalities
该研究提出非渐近矩阵普适性原理，结合高斯理论得到适用于非齐次、元素有依赖的广泛随机矩阵模型的精确浓度不等式，并在多个随机矩阵相关场景中验证应用。
We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovász (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by Saloff-Coste (Colloq. Math. 2010), a theorem by Milman and Pisier from conv…
Matrix Chaos Inequalities and Chaos of Combinatorial Type
本文针对元素为独立随机变量多项式的矩阵混沌模型，提出了基于系数展平的通用矩阵集中不等式，并识别了展平参数可机械计算的组合型混沌，为图矩阵等应用提供了统一处理与改进的界限。
Sharp Matrix Empirical Bernstein Inequalities
提出了两个针对具有有界特征值的对称随机矩阵的锐利闭式经验伯恩斯坦不等式，其偏差能紧凑适应未知方差，分别适用于独立矩阵样本均值和停时下鞅依赖的均值估计量。
Matrix concentration inequalities for dependent binary random variables
本文针对满足特定独立性假设的二元随机变量建立矩阵集中不等式，并利用解耦不等式将随机子矩阵算子范数估计推广到一般强 Rayleigh 抽样方案。
The foundational inequalities of D.L. Burkholder and some of their ramifications
本文综述了D.L. Burkholder到$L^p$ -界的奇异积分算子，集中在希尔伯特变换，一阶和二阶里斯变换，Beurling-Ahlfors算子和其他由随机积分变换的投影（条件期望）得到的乘子。虽然鞅不等式可以用来证明更广泛的一类Calderón-Zygmund奇异积分的有界性，本文的目的是显示结果，给出最佳或接近最佳的范围内的规范，因此我们的限制上述运营商。连接伯克霍尔德的基础工作尖锐的鞅不等式的其他领域的数学的结果本身或技术来证明它们已成为相当大的兴趣在最近几年，进行了讨论。1952年，C.B. Morrey对秩一凸和拟凸函数与变分法中的问题和1982年T。Iwaniec关于Beurling-Ahlfors算子的$L^p$ -范数与拟共形映射理论中的问题。开放式的问题，问题，并列出了整个文件和丰富的参考资料。
$S$-nodes, factorisation of spectral matrix functions and corresponding inequalities
本文通过分解和Arov-Krein不等式结果，推导出与S节点相关的插值问题中的重要不等式。
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