Hermitian operators and convex functions

Convexity or concavity inequalities for Hermitian operators

Matrix versions of some basic convexity inequalities are given. Further results on the same topic are proved in the recent papers on arxiv: 1. Hermitian operators and convex functions, 2. A concavity inequality for symmetric norms, 3. Eigenvalue inequalities for convex and log-convex functions (with Jaspal Singh AUJLA).

Minimal length curves in unitary orbits of a Hermitian compact operator

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

Hermitian operators and isometries on symmetric operator spaces

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $τ$. Let $E(\mathcal{M},τ) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},τ)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},τ)$. This is the first time that the description of he…

Eigenvalues inequalities for convex and logconvex functions

Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.

广义双算子积分：有限维情形

Generalized Double Operator Integrals: Finite Dimensions

针对现有双算子积分（DOI）依赖自伴或酉假设、不适用于非厄米场景的问题，本文提出广义双算子积分（GDOIs），将其拓展至有限维任意非厄米矩阵，建立相关性质与估计，并探索其在多领域的应用。

A Laplace Principle for Hermitian Brownian Motion and Free Entropy I: the convex functional case

This paper is part of a series aiming at proving that the $\limsup$ and $\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on $[0,1]$. In the current paper, we show that microstates free entropy $χ(X_1,...,X_m)$ and non-microstate free entropy $χ^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded below, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative pol…