Polynomial inequalities for non-commuting operators

Orthogonal polynomials in several non-commuting variables. II

In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.

Relations on noncommutative variables and associated orthogonal polynomials

This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on some dilation theoretic techniques that are also described in some details.

Curvature Inequalities and Extremal Operators

A curvature inequality is established for contractive commuting tuples of operators in the Cowen-Douglas class of rank n. Properties of the extremal operators, that is, the operators which achieve equality, are investigated. Specifically, a substantial part of a well known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.

Global holomorphic functions in several non-commuting variables II

We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.

Bernstein Type Inequalities for Restrictions of Polynomials to Complex Submanifolds of ${\mathbf {\mathbb C^N}}$

The paper studies Bernstein type inequalities for restrictions of holomorphic polynomials to graphs $Γ_f\subset\mathbb C^{n+m}$ of holomorphic maps $f:\mathbb C^n\rightarrow\mathbb C^m$. We establish general properties of exponents in such inequalities and describe some classes of graphs admitting Bernstein type inequalities of optimal exponents and of exponents of polynomial growth.

Bernstein and Markov-type inequalities for polynomials on $L_{p}(μ)$ spaces

In this work, we discuss generalizations of the classical Bernstein and Markov type inequalities for polynomials and we present some new inequalities for the $k$th Fréchet derivative of homogeneous polynomials on real and complex $L_{p}(μ)$ spaces. We also give applications to homogeneous polynomials and symmetric multilinear mappings in $L_{p}(μ)$ spaces. Finally, Bernstein's inequality for homogeneous polynomials on both real and complex Hilbert spaces has been discussed.

Non-linear Plank Problems and polynomial inequalities

We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works when the number of polynomials is large.

关于一些不等式、由此产生的不确定性关系和相关性的注释。1. 一般的数学形式主义

Notes on some inequalities, resulting uncertainty relations and correlations

本文分析Schwarz不等式及其推广，以及由Jensen不等式导出的不等式，用于量子理论中推导两个非对易可观测量的Heisenberg-Robertson和Schroedinger-Robertson不确定性关系及其推广。

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain~$C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based…

Estimations of zeros of a polynomial using numerical radius inequalities

We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new estimations for zeros of that polynomial. We also show with numerical examples that our new estimations improve on the existing estimations.

Polynomial inequalities on the $π/4$-circle sector

A number of sharp inequalities are proved for the space ${\mathcal P}\left(^2D\left(\fracπ{4}\right)\right)$ of 2-homogeneous polynomials on ${\mathbb R}^2$ endowed with the supremum norm on the sector $D\left(\fracπ{4}\right):=\left\{e^{iθ}:θ\in \left[0,\fracπ{4}\right]\right\}$. Among the main results we can find sharp Bernstein and Markov inequalities and the calculation of the polarization constant and the unconditional constant of the canonical basis of the space ${\mathcal P}\left(^2D\left(\fracπ{4}\right)\right)$.

Dilations of commuting $C_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality

Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz…

Algebraic pairs of isometries

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial.

A note on Sobolev type inequalities on graphs with polynomial volume growth

We prove a scale of generalized $L^p$-Poincaré inequalities and Sobolev type inequalities on graphs with polynomial volume growth. They are optimal on Vicsek graphs.

Operator Algebras of Functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are …

Some functional inequalities on polynomial volume growth Lie groups

We study in this article some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended without the use of the Littlewood-Paley decomposition to this general framework.

关于保持多项式间不等式的算子

On operators preserving inequalities between polynomials

该综述探讨了Bernstein型多项式不等式极值性质的算子方面，指出将多项式映射到多项式且具零保持性的线性算子自然保持Bernstein不等式。

Variation inequalities for Riesz transforms and Poisson semigroups associated with Laguerre polynomial expansions

In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.

Bergman空间上的交换斜Toeplitz算子

Commuting Slant Toeplitz Operators on the Bergman Space

本文研究了开单位圆盘Bergman空间上的斜Toeplitz算子，证明了两个此类算子可交换的充要条件是两者的有界解析函数相等。

Joint similarity for commuting families of power bounded matrices

An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is not jointly similar to a pair of commuting contractions. We show that this phenomenon does not occur in finite dimensions. More precisely, we show that a finite family of power bounded commuting matrices is always jointly similar to a family of contractions. In fact, the result can be extended to infinite families satisfying certain uniformity conditions. Our …

Bergmann与Hardy空间中多项式的Marcinkiewicz-Zygmund不等式

Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy Spaces

本文研究无限维解析函数Hilbert空间中采样序列与多项式子空间Marcinkiewicz-Zygmund不等式的关系，重点比较单变量Hardy空间与Bergman空间中截然不同的行为。

Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$

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.

关于双曲多项式 的Schur-Horn型不等式

Schur--Horn type inequalities for hyperbolic polynomials

本文为双曲多项式建立了Schur-Horn型不等式，并解决了关于Hadamard型不等式的猜想，通过群对称性原理证明了PSD稳定的lpm多项式Hadamard-Fischer不等式。

Polynomial capacities, Poincare' type inequalities and Spectral synthesis in Sobolev space

Contains a further development of the V.G. Maz'ya concept of polynomial capacities.A different different approach is given together with new results. The primary application is for Hardy inequalities. This is treated in a later report. Here is given an application which consists of a reformulation of the Spectral synthesis question in terms of polynomial capacities. However so far this does not give a new proof, though a similar but different question for the nonnegative cone in Sobobolev space of order 2 is resolved.

Bide - Side Exponential and Moment Inequalities for Tails of Distributions of Polynomial Martingales

In this paper non-asymptotic exponential estimates are derived for the tail distribution of polynomial martingale differences in terms unconditional tails distributions of summands. Applications are considered in the theory of polynomials on independent random variables, to the theory of U-statistics, multiply martingale series and in the theory of weak compactness measures on the Banach spaces. Partially supported by the Israel Ministry of Absorbtion Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.

Fock空间中多项式的Marcinkiewicz-Zygmund不等式

Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

本研究探究加权L²空间中多项式的Marcinkiewicz-Zygmund族与Fock空间整函数采样定理的关系及相关对偶关系，最终得到借助Gabor框架描述Hermite函数张成的信号子空间的方法。

Decoupling inequalities with exponential constants

Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring decoupling inequalities with good constants depending only exponentially on the degree of the polynomial. Assuming the Banach space has finite cotype we achieve this for classical decoupling inequalities that compare the polynomials with their associate…

Caffarelli-Kohn-Nirenberg inequalities on Lie groups of polynomial growth

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli-Kohn-Nirenberg type, where the weights involved are powers of the Carnot-Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak $L^p$ spaces is crucial in our proofs and we formulate these inequalities within the framework of $L^{p,q}$ Lorentz spaces (a scale of (quasi)-Banach spaces which extend the more classical $L^p$ Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy-Sobolev inequalities.

非交换算子球的冯·诺依曼型不等式

Von Neumann Type Inequalities for NC Operator Balls

该研究针对非交换算子球中的交换矩阵元组建立有限维冯·诺依曼型不等式，给出相关常数估计，对行与列球乘积得无条件估计，对一般I型Cartan域仅得条件性结果。