11 pages. This is the final version. We added Corollary 2.6, which is an equivalent form of Theorem 2.5. In addition, we added an appendix, which provided an alternative proof of Theorem 2.2. The first author would like to express his hearty gratitude to Prof. Minghua Lin and Prof. Xiaohui Fu for detailed comments and constant encouragement
发表信息：
Operators and Matrices 15, (2021), no. 3, 1189--1199
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 (2018) 1619--1625]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.
In this short paper, we show a certain matrix trace inequality and then give a refinement of the trace inequality proven by Belmega, Lasaulce and Debbah. In addition, we give an another improvement of their trace inequality.
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.
We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{A + B}{2}} \right) +\tr f\left(\left| {\frac{A - B}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular an improvement o…
Let $A$ be a positive semidefinite $m\times m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds \[ (\mathrm{tr} A)^{mn} - \det(\mathrm{tr}_2 A)^n \ge \bigl| \det A - \det(\mathrm{tr}_1 A)^m \bigr|, \] where $\mathrm{tr}_1$ and $\mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].
In this paper we prove the Fractional Gagliardo-Nirenberg Inequality, Polya-Szego Inequality and the Sharp Fractional Sobolev Inequality, we then provide an application of such inequalities in a constraiend variational problem involving the fractional gradient and alocal non-linearity
In this note we prove that Tr (MN+ PQ)>= 0 when the following two conditions are met: (i) the matrices M, N, P, Q are structured as follows: M = A -B, N = inv(B)-inv(A), P = C-D, Q =inv (B+D)-inv(A+C), where inv(X) denotes the inverse matrix of X (ii) A, B are positive definite matrices and C, D are positive semidefinite matrices.
We prove a matrix trace inequality for completely monotone functions and for Bernstein functions. As special cases we obtain non-trivial trace inequalities for the power function x->x^q, which for certain values of q complement McCarthy's trace inequality and for others strenghten it.
We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.
Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds…
We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if ${\mathfrak A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $μ$ as a totaly order set, then \begin{align*} \int_{T} α(s) dμ(s)\int_{T}α(t)(A_t\circ B_t) dμ(t)\geq\Big{(}\int_{T}α(t) (A_tm_{r,α} B_t) dμ(t)\Big{)}\circ\Big{(}\int_{T}α(s) (A_sm_{r,1-α} B_s) dμ(s)\Big{)}, \end{align*} where $α\in[0,1]$, $r\in[-1,1]$ and $(A_t)_{t\…
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.
The goal of this review is to explain some recent results regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).
In this paper we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of operator Aczél inequality dealing with the weighted geometric mean.
We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices $A$ and $B$, by giving a lower bound on the quantity $\trace[A^r B^r A^r]^q$ in terms of $\trace[ABA]^{rq}$ for $0\le r\le 1$ and $q\ge0$, whereas the ALT inequality gives an upper bound. The bound contains certain norms of $A$ and $B$ as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (\textit{Math. Inequal. Appl.} \textbf{8}(2005) pp. 373--378) and others. Secondly, we also prove a generalisation of the ALT inequal…
Graduate School of Information Science and Technology, The University of Osaka（大阪大学情报理工学研究科）；Center for Mathematics and Data Science, Gunma University（群马大学数学与数据科学中心）；College of Science and Technology, Nihon University（日本大学理工学部）；Research and Support Center on Higher Education for the hearing and Visually Impaired, National University Corporation Tsukuba University of Technology（筑波技术大学国立大学法人听力及视障者高等教育研究与支援中心）
We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert-Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.
In this paper, we present some operator and eigenvalue inequalities involving operator monotone, doubly concave and doubly convex functions. These inequalities provide some variants of operator Aczél inequality and its reverse via generalized Kantorovich constant.
In [P. Renaud, "A matrix formulation of Grüss inequality", Linear Algebra Appl. 335 (2001), 95--100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we answer positively the Renaud's open problem.
We show that various old and new bounds involving eigenvalues of a complex n x n matrix are immediate consequences of the inequalities involving variance of real and complex numbers.
We prove that for $α\in (d-1,d]$, one has the trace inequality \begin{align*} \int_{\mathbb{R}^d} |I_αF| \;dν\leq C |F|(\mathbb{R}^d)\|ν\|_{\mathcal{M}^{d-α}(\mathbb{R}^d)} \end{align*} for all solenoidal vector measures $F$, i.e., $F\in M_b(\mathbb{R}^d,\mathbb{R}^d)$ and $\operatorname{div}F=0$. Here $I_α$ denotes the Riesz potential of order $α$ and $\mathcal M^{d-α}(\mathbb{R}^d)$ the Morrey space of $(d-α)$-dimensional measures on $\mathbb{R}^d$.
Jensen's inequality for partial traces in von Neumann algebras
Wallenberg Centre for Quantum Technology, Chalmers University of Technology（瓦伦堡量子技术中心，查尔姆斯理工大学）；Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学数学科学系）
In this paper, suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger-Moser Inequalities involving $N$-Finsler--Laplacian operator. Let $Ω\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1, N}(Ω)$ be a sequence such that anisotropic Dirichlet norm$\int_ΩF^N (\nabla u_n)dx=1$, $u_n \rightharpoonup u \not \equiv 0$ weakly in $W_0^{1, N}(Ω)$. Then for any $0 …
We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Σ\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Σ$. We provide its applications to the heat equation on $\mathbb R^d$ with an insulating cut at $Σ$ and to the Schrödinger operator with a $δ'$-interaction supported on $Σ$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.
We prove a logarithmic Sobolev trace inequality in a gaussian space and we study the trace operator in the weighted Sobolev space W^{1,p}(Ω,γ) for sufficiently regular domain. We exhibit examples to show the sharpness of the results. Applications to PDE are also considered.
We prove a Bernstein-type inequality involving the Bergman and the Hardy norms, for rational functions in the unit disc \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0
In this article we generalize the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces instead the classical Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality. We consider for simplicity only the case of whole space.
This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.
Let $Ω$ be a smooth bounded domain in $\mathbf R^2$ and $λ^{\mathsf N} (Ω)$ the first non-zero Neumann eigenvalue of the operator $-Δ$ on $Ω$. In this paper, for any $γ\in [0, λ^{\mathsf N} (Ω) )$, we establish the following improved Moser-Trudinger inequality \[ \sup_{u} \int_Ω e^{2πu^2} dx < +\infty \] for arbitrary functions $u$ in $H^1(Ω)$ satisfying $\int_Ωu dx =0$ and $\|\nabla u\|_2^2 -α\|u\|_2^2 \leqslant 1$. Furthermore, this supremum is attained by some function $u^*\in H^1(Ω)$. This strengthens the results of Chang and Yang (J. Differential Geom. 27 (1988) 259-296) and of Lu and Yan…
Consider a set represented by an inequality. An interesting phenomenon which occurs in various settings in mathematics is that the interior of this set is the subset where strict inequality holds, the boundary is the subset where equality holds, and the closure of the set is the closure of its interior. This paper discusses this phenomenon assuming the set is a Voronoi cell induced by given sites (subsets), a geometric object which appears in many fields of science and technology and has diverse applications. Simple counterexamples show that the discussed phenomenon does not hold in general, b…
We prove that for any (trace-preserving) conditional expectation $\mathcal E$ on a noncommutative $L_p$ with $p>2$, $Id-\mathcal E$ is a contraction on the positive cone $L_p^+$.
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.
We present some new inequalities related to determinant and trace for positive semidefinite block matrices by using symmetric tensor product, which are extensions of Fiedler-Markham's inequality and Thompson's inequality.
The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves $L^{n}$ norm. Let \begin{equation*} λ_{1}(Ω) = \inf_{u\in W_0^{1,n}(Ω),u\not\equiv 0} ||F(\nabla u)||_{L^n(Ω)}^n / ||u||_{L^n(Ω)}^n \end{equation*} be the first eigenvalue associated with $n$-Finsler-Laplacian. using blowing up analysis, we obtain that \begin{equation*} \sup_{u\in W_{0}^{1,n}(Ω),||F(\nabla u)||_{L^n(Ω)} = 1} \int_Ωe^{λ_n (1+α||u||_{L^n (Ω)}^n)^{\frac{1}{n-1}} |u|^{\frac{n}{n-1}}}dx \end{equation*} is finite for any $0\leq α<λ_{1}(Ω)$,and the supremum is infinite for any $α\geq λ_…
We prove a basic inequality involving anticommutators in noncommutative $L_p$-spaces. We use it to complete our study of the noncommutative Mazur maps from $L_p$ to $L_q$ showing that they are Lipschitz on balls when $0<q<p<\infty$.
In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.
We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices.
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
