Analytic Functions of a General Matrix Variable

变指数矩阵加权 Lebesgue 空间中的紧致集

Precompact Sets in Matrix Weighted Lebesgue Spaces with Variable Exponent

本文研究了变指数矩阵加权 Lebesgue 空间中的紧致性，分别利用平移算子、平均算子和近似恒等给出了相应的判据，并进一步探讨了变指数矩阵加权 Sobolev 空间中的紧致性。

Analytic Functions of a Quaternionic Variable

Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show how many of those obstacles can be overcome. After a tiny bit of linear algebra we look at the beginnings of differential calculus. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*).

Analytic approximation of matrix functions in $L^p$

We consider the problem of approximation of matrix functions of class $L^p$ on the unit circle by matrix functions analytic in the unit disk in the norm of $L^p$, $2\le p<\be$. For an $m\times n$ matrix function $Φ$ in $L^p$, we consider the Hankel operator $H_Φ:H^q(C^n)\to H^2_-(C^m)$, $1/p+1/q=1/2$. It turns out that the space of $m\times n$ matrix functions in $L^p$ splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If $Φ$ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of $H_Φ$. For weird…

Analytic approximation of matrix functions and dual extremal functions

We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain special factorizations that involve thematic matrix functions.

Weighted matrix means and symmetrization procedures

Here we prove the convergence of the Ando-Li-Mathias and Bini-Meini-Poloni procedures for matrix means. Actually it is proved here that for a two-variable function which maps pairs of positive definite matrices to a positive definite matrix and is not greater than the square mean of two positive definite matrices, the Ando-Li-Mathias and Bini-Meini-Poloni procedure converges. In order to be able to set up the Bini-Meini-Poloni procedure, a weighted two-variable matrix mean is also needed. Therefore a definition of a two-variable weighted matrix mean corresponding to every symmetric matrix mean…

Matrix inequalities from a two variables functional

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).

Analytic approximation of rational matrix functions

For a rational matrix function $Φ$ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation $\AΦ$ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of $2\times2$ matrix functions. It turns out that ``generically'' $°\AΦ\le\degΦ-2$. We prove that for an arbitrary $2\times2$ rational function $Φ$, $°\AΦ\le2\degΦ-3$ whenever $\degΦ\ge2$. On the other hand, for $k\ge2$, we construct a $2\times2$ matrix function $Φ$, for which $\degΦ=k$, while $°\AΦ=2k-3$. Moreover, we conduct a detailed analysis of the situation when the…

On the multiplication operator by an independent variable in matrix Sobolev spaces

We study the operator $\mathcal{A}$ of multiplication by an independent variable in a matrix Sobolev space $W^2(M)$. In the cases of finite measures on $[a,b]$ with $(2\times 2)$ and $(3\times 3)$ real continuous matrix weights of full rank it is shown that the operator $\mathcal{A}$ is symmetrizable. Namely, there exist two symmetric operators $\mathcal{B}$ and $\mathcal{C}$ in a larger space such that $\mathcal{A} f = \mathcal{C} \mathcal{B}^{-1} f$, $f\in D(\mathcal{A})$. As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These con…

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems ove…

On the discrete spectrum of non-analytic matrix-valued Friedrichs model

We have found the sufficient conditions for the spectrum of matrix-valued Friedrichs model to be finite.

变Lebesgue空间中的矩阵加权界

Matrix-weighted bounds in variable Lebesgue spaces

本文证明了Calderón-Zygmund算子与Christ-Goldberg极大算子在矩阵加权变Lebesgue空间中的有界性，并基于方向Banach函数空间框架获得了定量外推定理。

Positive polynomials in scalar and matrix variables, the spectral theorem and optimization

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity…

相依二元随机变量的矩阵集中不等式

Matrix concentration inequalities for dependent binary random variables

本文针对满足特定独立性假设的二元随机变量建立矩阵集中不等式，并利用解耦不等式将随机子矩阵算子范数估计推广到一般强 Rayleigh 抽样方案。

On the analyticity of WLUD$^\infty$ functions of one variable and WLUD$^\infty$ functions of several variables in a complete non-Archimedean valued field

Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^k$) functions, and WLUD$^\infty$ functions at a point or on an open subset of $\mathcal{N}$. Then we show under what conditions a WLUD$^\infty$ function at a point $x_0\in\mathcal{N}$ is analytic in an interval around $x_0$, that is, it h…

Lebesgue空间中随机变量的解析构造

An Analytic Construction of Random Variables in Lebesgue Spaces

本文从泛函分析视角发展了Lebesgue空间L^p中随机变量的构造，利用Pettis定理和Riesz表示定理将Bochner积分定义为经典期望的自然推广，扩展了可测性、可积性与期望概念。

Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations

A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recen…

矩阵加权变指数Hardy空间的原子刻画及其应用

Atomic Characterization and Its Applications of Matrix-Weighted Variable Hardy Spaces

本文通过矩阵加权极大函数引入变指数Hardy空间，利用凸体值极大函数建立其等价刻画，并通过Whitney分解和凸体缩减算子获得原子刻画，进而给出对偶空间和Calderón-Zygmund算子的有界性。

Real-Variable Characterizations and Their Applications of Matrix-Weighted Triebel--Lizorkin Spaces

Let $α\in\mathbb R$, $q\in(0,\infty]$, $p\in(0,\infty)$, and $W$ be an $A_p(\mathbb{R}^n,\mathbb{C}^m)$-matrix weight. In this article, the authors characterize the matrix-weighted Triebel-Lizorkin space $\dot{F}_{p}^{α,q}(W)$ via the Peetre maximal function, the Lusin area function, and the Littlewood-Paley $g_λ^{*}$-function. As applications, the authors establish the boundedness of Fourier multipliers on matrix-weighted Triebel-Lizorkin spaces under the generalized Hörmander condition. The main novelty of these results exists in that their proofs need to fully use both the doubling property…