Jean-Christophe Bourin, Mitsuru Uchiyama
Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and Zhan. Our method is simpler. Several related results are presented.
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Well-known subadditivity results for positive operators (of Brown-Kosaki and Rotfeld/Ando-Zhan types) are extended to Hermitian and normal ones. Applications to Cartesian decomposition and block-matrices are given.
On an inequality of Lin, Kim and Hsieh and Strong Subadditivity
本研究给出Lin、Kim和Hsieh所提不等式的初等证明，该结果可推导冯·诺依曼熵的强次可加性。
Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.
We prove a general duality result showing that a Brascamp--Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of Brascamp--Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of th…
Subadditivity of the log-Sobolev constant on convolutions
本文证明了卷积测度对数Sobolev常数的一般次可加性不等式，并由此推出该常数在中心极限定理的标准化卷积序列上具有单调性。
We present a refinement of a known entropic inequality on the sphere, finding suitable conditions under which the uniform probability measure on the sphere behaves asymptomatically like the Gaussian measure on $\mathbb{R}^N$ with respect to the entropy.
Around Strongly Operator Convex Functions
本文研究强算子凸函数，建立其在正负半轴上的次可加性、相关算子不等式及与 Kwong 函数的关系，并刻画半直线上的严格算子单调性，给出算子差值的下界。
Some natural inequalities related to rearrangement in matrix products can also be regarded as extensions of classical inequalities for sequences or integrals. In particular, we show matrix versions of Chebyshev and Kantorovich type inequalities. The matrix approach may also provide simplified proofs and new results for classical inequalities. For instance, we show a link between Cassel's inequality and the basic rearrangement inequality for sequences of Hardy-Littlewood-Polya, and we state a reverse inequality to the Hardy-Littlewood-Polya inequality in which matrix technics are essential.
Solutions to some open problems in matrix analysis
本文解决了矩阵分析中的多个开放问题，包括加强次可加性不等式、推广互反Lie--Trotter定理、确定收缩和正分块矩阵不等式的最佳常数，并证明对称模的特征值不等式。
We present an inequality for tensor product of positive operators on Hilbert spaces by considering the tensor product of operators as words on certain alphabets (i.e., a set of letters). As applications of the operator inequality and by a multilinear approach, we show some matrix inequalities concerning induced operators and generalized matrix functions (including determinants and permanents as special cases).
New Improvements to Heron and Heinz Inequality Using Matrix Techniques
本文研究矩阵均值插值与比较，将参数$\vartheta$扩展至$\mathbb{R}^+$，引入结合Kantorovich常数的Heinz不等式新标量变体并强化算子版本，提出专为半正定矩阵迹、行列式和范数设计的改进型Young不等式。
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
For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also investigated.
In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.
A Matrix Generalization of the Hardy-Littlewood-Pólya Rearrangement Inequality and Its Applications
本文通过分析正交矩阵上的优化问题，将Hardy-Littlewood-Pólya重排不等式推广到正定矩阵与矩形矩阵，并据此推导了信号处理、机器学习领域类距离函数的新不等式。
The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that \[\left\| T \right\|\le \frac{1}{4}\left( \left\| \left| T \right|+\left| {{T}^{*}} \right|+2\mathfrak RT \right\|+\left\| \left| T \right|+\le…
We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by X. Zhan.
We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.
Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison's inequality and several operator versions of Chebyshev's inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.
The Szász inequality for matrix polynomials and functional calculus
本文探讨了具有矩阵系数或矩阵变量的多项式背景下的 Szász 不等式，证明其估计在某些情况下比 von Neumann 不等式更尖锐，并通过放宽零点位置假设改进了标量 Szász 不等式。
We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a special case. Given $n$ matrices $A_i$, $i=1,\ldots,n$, of the same size, let $Z_1$ and $Z_2$ be the block matrices $Z_1:=(A_jA_i^*)_{i,j=1}^n$ and $Z_2:=(A_j^*A_i)_{i,j=1}^n$. Then the conjectured inequality is \[ \left(||Z_1||_1-\trace Z_1\right)^2 + \left(||Z_2||_1-\trace Z_2\right)^2 \le \left(\sum_{i\neq j} ||A_i||_2 ||A_j||_2\right)^2. \] We prove this inequality for the already challenging case $n=2$ with $A_1$ equal to the identity matrix.
For any positive invertible matrix $A$ and any normal matrix $B$ in $M_{n}({\Bbb C})$, we investigate whether the inequality $ ||A\sharp (B^{*}A^{-1}B)||\geq ||B|| $ is true or not, where $\sharp$ denotes the geometric mean and $||\cdot||$ denotes the operator norm. We will solve this problem negatively. The related topics are also discussed.
We prove a version of Talagrand's concentration inequality for subordinated sub-Laplacian on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.
In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^n$ for large $n$. The main result of this paper is that the $L_p$-bounds ($p>1$) can be taken to be independent of $n$, which is a generalization of Stein and Strömberg's resut in the scalar-valued case. We also show that the weak type $(1,1)$ bound has similar behavior as Stein and Stömberg's.
Let $A$ be an $m\times m$ positive semidefinite block matrix with each block being $n$-square. We write $\mathrm{tr}_1$ and $\mathrm{tr}_2$ for the first and second partial trace, respectively. In this paper, we prove the following inequality \[ (\mathrm{tr} A)I_{mn} - (\mathrm{tr}_2 A) \otimes I_n \ge \pm \bigl( I_m\otimes (\mathrm{tr}_1 A) -A\bigr).\] This inequality is not only a generalization of Ando's result [ILAS Conference (2014)] and Lin [Canad. Math. Bull. 59 (2016) 585--591], but it also could be regarded as a complement of a recent result of Choi [Linear Multilinear Algebra 66 (201…
A note on time-uniform concentration inequality for matrix products
本文通过简单论证，将固定时间界的矩阵乘积集中性界推广到任意时间界，受Oja算法分析启发。
We characterise equality cases in matrix Hölder's inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone maps. We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.
We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued modified log-Sobolev inequality has been obtained. We also show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.
We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with ind…
A Jensen inequality for partial traces and applications to partially semiclassical limits
本文证明双体空间上厄米矩阵凸函数的矩阵不等式，以此重新证明并推广齐次势薛定谔算子特征值渐近性定理，重点研究Weyl表达式无穷大时的部分半经典极限情形。
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。
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