Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $λ(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} λ(A)*λ(B)$ for positive semidefinite matrices and $λ(|A\circ B|) \underset{w}{\prec} λ(|A|)*λ(|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $λ\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} λ(a)*λ(b)$ for $a,b\geq 0$ and $λ\big (|a\circ b|\big )\underset{w}{\prec} λ(|a|)*λ(|b|)$ for all $a$ and $b$, where $P_u$ and $λ(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $λ(|A\bullet b|)\underset{w}{\prec} λ({\mathrm{diag}}(A))*λ(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A\,\bullet\, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized Hölder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||λ(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,r\in [1,\infty]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.
In an Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, if the corresponding eigenvalue vector of x is majorized by the eigenvalue vector of y in R^n. In this article, we describe pointwise majorization inequalities of the form `T(x) majorized by S(x)', where T and S are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations…
Given a linear map $T$ on a Euclidean Jordan algebra of rank $n$, we consider the set of all nonnegative vectors $q$ in $R^n$ with decreasing components that satisfy the pointwise weak-majorization inequality $λ(|T(x)|)\underset{w}{\prec}q*λ(|x|)$, where $λ$ is the eigenvalue map and $*$ denotes the componentwise product in $R^n$. With respect to the weak-majorization ordering, we show the existence of the least vector in this set. When $T$ is a positive map, the least vector is shown to be the join (in the weak-majorization order) of eigenvalue vectors of $T(e)$ and $T^*(e)$, where $e$ is the…
This note revisits some majorization inequalities for eigenvalues, special attention is given to an elegant theorem of Hiroshima. An extension of the special case of Hiroshima's theorem is presented. Some discussion and open problems are also included.
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.
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
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lin's results imply Tung's version of Harnack's inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lin's. We also show some majorization inequalities concerning Cayley transforms. Some open pr…
We extend the notion of the determinant function $Λ$, originally introduced by T.Fack for $τ$-compact operators, to a natural algebra of $τ$-measurable operators affiliated with a semifinite von Neumann algebra which coincides with that defined by Haagerup and Schultz in the finite case and on which the determinant function is shown to be submultiplicative. Application is given to Hölder type inequalities via general Araki-Lieb-Thirring inequalities due to Kosaki and Han and to a Weyl-type theorem for uniform majorization.
Denote by $¶_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in ¶_{n_1}, \dots, D_k \in ¶_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in ¶_n$ according to $(n_1, \dots, n_k)$ so that $\Diag C = C_1\oplus \dots \oplus C_k$. We prove the following weak log majorization result: \begin{equation*} λ(C^{-1}_1D_1\oplus \cdots \oplus C^{-1}_kD_k)\prec_{w \,\log} λ(C^{-1}D), \end{equation*} where $λ(A)$ denotes the vector of eigenvalues of $A\in \Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of sing…
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all o…
Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means
High School for the Gifted, Vietnam National University Ho Chi Minh City（越南国立大学胡志明市附属资优生高中）；University of Science, Vietnam National University Ho Chi Minh City（越南国立大学胡志明市自然科学大学）；Troy University（特洛伊大学）
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).
Let $0<p<q\leq\infty$ and $α\in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,α),L_q(0,α))$ in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.
A Fan-Theobald-von Neumann system is a triple $(V,W,λ)$, where $V$ and $W$ are real inner product spaces and $λ:V \to W$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In the previous paper (arXiv:1902.06640) we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions…
We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.
We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the ma…
Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices $B$ and $C$, dominated by a third Hermitian matrix $A$. Based on…
In a Euclidean Jordan algebra $V$ of rank $n$ which carries the trace inner product, to each element $a$ we associate the eigenvalue vector $λ(a)$ in $R^n$ whose components are the eigenvalues of $a$ written in the decreasing order. For any $p\in [1,\infty]$, we define the spectral $p$-norm of $a$ to be the $p$-norm of $λ(a)$ in $R^n$. In a recent paper, based on the $K$-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on $V$ relative to the same spectral norm. In this paper, using standard complex function theo…
A well-known theorem of Korovkin asserts that if $\{T_k\}$ is a sequence of positive linear transformations on $C[a,b]$ such that $T_k(h)\rightarrow h$ (in the sup-norm on $C[a,b]$) for all $h\in \{1,ϕ,ϕ^2\}$, where $ϕ(t)=t$ on $[a,b]$, then $T_k(h)\rightarrow h$ for all $h\in C[a,b]$. In particular, if $T$ is a positive linear transformation on $C[a,b]$ such that $T(h)=h$ for all $h\in \{1,ϕ,ϕ^2\}$, then $T$ is the Identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if $T$ is a positive linear transformation on a Euclid…
We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in $L^1$). We collect a number of results for vector-valued functions on $L^1$, simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equiva…
We prove some sufficient conditions implying $l^p$ inequalities of the form $||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in certain positive real intervals. Our sufficient conditions are strictly weaker than the usual majorization relation. The conditions are expressed in terms of certain homogeneous symmetric polynomials in the entries of the vectors. These polynomials include the elementary symmetric polynomials as a special case. We also give a characterization of the majorization relation by means of symmetric polynomials.
We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result i…
Let $\mathcal{A}$ be a $C^*$-algebra and $ϕ:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we obtain a Jensen's-type inequality $f(ϕ(a)) \leq ϕ(f(a))$ where $\le$ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered i.e., operator convex, monotone convex and arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.
This paper is devoted to the study of generalized subdifferentials of spectral functions over Euclidean Jordan algebras. Spectral functions appear often in optimization problems playing the role of "regularizer", "barrier", "penalty function" and many others. We provide formulae for the regular, approximate and horizon subdifferentials of spectral functions. In addition, under local lower semicontinuity, we also furnish a formula for the Clarke subdifferential, thus extending an earlier result by Baes. As application, we compute the generalized subdifferentials of the function that maps an ele…
Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$, and $σ(X)$ the singular values with $\uparrow$ $\downarrow$ indicating its increasing or decreasing rearrangements. We wish to examine inequalities between $||A+B||_p^p+||A-B||_p^p$, $||σ_\downarrow(A)+σ_\downarrow(B)||_p^p+||σ_\downarrow(A)-σ_\downarrow(B)||_p^p$, and $||σ_\uparrow(A)+σ_\downarrow(B)||_p^p+||σ_\uparrow(A)-σ_\downarrow(B)||_p^p$ for various values of $1\leq p<\infty$. It was conjectured in [6] that a universal inequality $||σ_\downarrow(A)+σ_\downarrow(B)|…
We establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter example. We also derive several information theoretic inequalities as consequences. In particular sharp bounds are derived for the varentropy, Rényi entropies, and the concentration of information of monotone log-concave random variables. Moreover, the majorization approach utilized in the proof of the main theorem, is applied to derive analogous information theoretic results …
