Jensen's Operator Inequality

带等号条件的Jensen迹不等式与非交换Lamperti定理

Jensen' s trace inequality with equality condition and noncommutative Lamperti's theorem

该研究针对半有限von Neumann代数，建立带等号条件的Jensen迹不等式，解答相关问题，推导非交换Lamperti型不等式，刻画一类F-范数非交换Orlicz空间上的线性等距，得到对应非交换Lamperti定理。

广义Choi-Davis-Jensen算子不等式及其应用

Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications

针对原始Choi-Davis-Jensen不等式的推广需求，研究通过引入非线性映射、将算子凸函数拓展为紧区域上任意连续函数，借助Stone-Weierstrass定理与Kantorovich函数构建证明了广义不等式，并给出其在统计物理中的应用。

Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double p…

Operator maps of Jensen-type

Let $\mathbb{B}_J(\mathcal H)$ denote the set of self-adjoint operators acting on a Hilbert space $\mathcal{H}$ with spectra contained in an open interval $J$. A map $Φ\colon\mathbb{B}_J(\mathcal H)\to {\mathbb B}(\mathcal H)_\text{sa} $ is said to be of Jensen-type if \[ Φ(C^*AC+D^*BD)\le C^*Φ(A)C+D^*Φ(B)D \] for all $ A, B \in B_J(\mathcal H)$ and bounded linear operators $ C,D $ acting on $ \mathcal H $ with $ C^*C+D^*D=I$, where $I$ denotes the identity operator. We show that a Jensen-type map on a infinite dimensional Hilbert space is of the form $Φ(A)=f(A)$ for some operator convex funct…

有限次对角代数中的Jensen不等式

Jensen's inequality in finite subdiagonal algebras

本文将有限次对角代数中的Jensen不等式推广至更一般的递增连续函数情形，证明了迹不等式及其取等条件，并给出了关于点谱包含关系的应用。

Operator inequalities of Jensen type

We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty) \to \mathbb{R}$ is a continuous convex function with $f(0)\leq 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-δ_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$, where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $δ_f = f(0)+f(M) - 2 f(\frac{M}{2})$.

Operator extensions of Hua's inequality

We give an extension of Hua's inequality in pre-Hilbert $C^*$-modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert $C^*$-modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function.

算子Jensen不等式的广义逆及其在超复函数逼近与界代数中的应用

Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra

本研究推广了Mond-Pecaric（MP）算子不等式方法，放宽凸函数与线性映射限制，推导超复函数新不等式，应用于函数逼近与界代数构建。

The Hadamard Determinant Inequality - Extensions to Operators on a Hilbert Space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some `Fischer-type' inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction wit…

Operator inequalities related to weak 2-positivity

In this paper we introduce the notion of weak 2-positivity and present some examples. We establish some operator Cauchy--Schwarz inequalities involving the geometric mean and give some applications. In particular, we present some operator versions of Hua's inequality by using the Choi--Davis--Jensen inequality.

度量性质：从S-分歧到量子Jensen分歧

Metric Properties: From $S$-Divergence to Quantum Jensen Divergence

本文研究了从S-分歧到量子Jensen分歧的度量性质，扩展了S-分歧到迹C*-代数和有限von Neumann代数，并证明其平方根在特定框架下的度量性，同时回答了关于Hilbertianity的问题并推广了相关结果。

Monotone trace functions of several variables

We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality τ(x_1... x_n)\leτ(y_1... y_n) for a trace τon a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices.

四元数希尔伯特空间上自伴算子的对数凸函数的若干不等式

Some inequalities for log-convex functions of selfadjoint operators on quaternionic Hilbert spaces

本文针对四元数希尔伯特空间上的自伴算子，利用对数凸函数证明了若干Jensen型不等式，并通过特定对数凸函数得到算子不等式的特殊情况，丰富了该领域的不等式研究。

Grand Fujii-Fujii-Nakamoto operator inequality dealing with operator order and operator chaotic order

In this paper, we shall prove that a grand Fujii-Fujii-Nakamoto operator inequality implies operator order and operator chaotic order under different conditions.

算子矩阵数值半径不等式的改进上界及其应用

Improved Upper Bounds for Numerical Radius Inequalities of Operator Matrices and Their Applications

本研究运用多种不等式及Buzano不等式扩展形式，推导希尔伯特空间上算子矩阵数值半径的改进上界，完善经典结果，应用于量子力学等多领域的数学物理分析。

A Note Around Operator Bellman Inequality

In this paper, we shall give an extension of operator Bellman inequality. This result is estimated via Kantorovich constant.

Holder's inequality for roots of symmetric operator spaces

We prove a version of Holder's inequality with a constant for p-th roots of symmetric operator spaces of operators affiliated to a semifinite von Neumann algebra factor, and with constant equal to 1 for strongly symmetric operator spaces.

Hermite-Hadamard type inequality for operator preinvex functions

In this paper we establish a Hermite- Hadamard type inequality for operator preinvex functions and an estimate of the right hand side of a Hermite- Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved.