Non-Archimedean Normal Operators

Non-Archimedean Group Algebras with Baer Reductions

Within the concept of a non-Archimedean operator algebra with the Baer reduction (A. N. Kochubei, On some classes of non-Archimedean operator algebras, Contemporary Math. 596 (2013), 133--148), we consider algebras of operators on Banach spaces over non-Archimedean fields generated by regular representations of discrete groups.

关于单参数算子半群到阿基米德向量格完备化的扩张

On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices

本文研究了阿基米德向量格上单参数算子半群到其完备化的扩张，证明了在正半群类中到ru-完备化扩张的存在性与唯一性，并通过证明性质(R)向量格上的扩张定理，去除了相关结论中对ru-完备性的要求。

Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform

In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^α$, $α>0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^α$ that the appropriate change of variables reduces equations with $D^α$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary dif…

Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces

A main objective of the present paper is to develop the theory of hypercyclicity and supercyclicity of linear operators on topological vector space over non-Archimedean valued fields. We show that there does not exist any hypercyclic operator on finite dimensional spaces. Moreover, we give sufficient and necessary conditions of hypercyclicity (resp. supercyclicity) of linear operators on separable $F$-spaces. It is proven that a linear operator $T$ on topological vector space $X$ is hypercyclic (supercyclic) if it satisfies Hypercyclic (resp. Supercyclic) Criterion. We consider backward shifts…

非阿基米德算子值域的Borel复杂度

The Borel complexity of non-Archimedean operator ranges

该研究分类了波兰非阿基米德非平凡赋值域上可分巴拿赫空间非闭算子值域的Borel复杂度类，还补充了弗雷歇空间对应的额外复杂度类。

算子束在非阿基米德Banach空间上的条件伪谱

Condition pseudospectrum of operator pencils on non-archimedean Banach spaces

本文刻画了非阿基米德Banach空间上算子束的条件伪谱及其与通常谱的关系，并利用非阿基米德完全连续算子给出了本质谱的特征描述。

非阿基米德希尔伯特空间上无界对角算子的有限秩扰动的谱与本质谱分析

Spectral and Essential Spectral Analysis of Finite-Rank Perturbations of Unbounded Diagonal Operators on Non-Archimedean Hilbert Spaces

研究非阿基米德希尔伯特空间上无界对角算子经有限秩扰动后的谱、点谱和本质谱，利用Fredholm算子理论描述其结构。

关于原子阿基米德向量格上运算代数的正自同构

On positive automorphisms of algebras of operators on atomic Archimedean vector lattices

研究原子阿基米德向量格上运算代数的正自同构性质，证明其必为空间自同构，并利用Kakutani定理证明有限维子空间的序闭性。

Orthosymmetric spaces over an Archimedean vector lattice

We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.

非阿基米德开尔文变换

Non-Archimedean Kelvin Transformation

本文在非阿基米德局部域上引入并研究开尔文变换的类比，将其与作用于K^n上实值或复值函数的Vladimirov-Taibleson算子联系起来。

Taibleson算子在非阿基米德局部域上的函数 calculus 与半线性演算方程

Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

本文研究了Taibleson算子在非阿基米德局部域上的有界H^∞(Σ_θ)函数calculus及半线性演算方程的解的存在性。

半范数序向量空间的 Archimedean 序单位化

The Archimedean order unitization of seminormed ordered vector spaces

本文提出半范数预序向量空间的 Archimedean 序单位化构造，证明其泛性质，并用于阐明半范数正规性及完善 Werner 偏单位化的正锥描述与完全同构嵌入条件。

Matricial Archimedean order unit spaces and quantum correlations

We introduce a matricial analogue of an Archimedean order unit space, which we call a $k$-AOU space. We develop the category of $k$-AOU spaces and $k$-positive maps and exhibit functors from this category to the category of operator systems and completely positive maps. We also demonstrate the existence of injective envelopes and C*-envelopes in the category of $k$-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional $k$-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjectu…

Non-Archimedean Duality: Algebras, Groups, and Multipliers

We develop a duality theory for multiplier Banach-Hopf algebras over a non-Archimedean field K. As examples, we consider algebras corresponding to discrete groups and zero-dimensional locally compact groups with K-valued Haar measure, as well as algebras of operators generated by regular representations of discrete groups.

Spectral theory of $p-adic$ Hermite operator

We give the definition of $p-adic$ Hermite operator and set up the $p-adic$ spectral measure. We compare the Archimedean case with non-Archimedean case. The structure of Hermite conjugate in $C^{*}$-Algebra corresponds to three canonical structures of $p-adic$ ultrametric Banach algebra: 1. mod $p$ reduction 2. Frobenius map 3. Teichmüller lift. There is a nature connection between Galois theory and Hermite operator spectral decomposition. The Galois group $\mathrm{Gal}(\bar{\mathbb{F}}_p|\mathbb{F}_p)$ generate the $p-adic$ spectral measure. We point out some relationships with $p-adic$ quant…

Non-Archimedean Pseudo-Differential Operators With Bessel Potentials

In this article, we study a class of non-archimedean pseudo-differential operators associated via Fourier transform to the Bessel potentials. These operators (which we will denote as $J^{α},$ $α>n$) are of the form (J^{α})(x)=\mathcal{F}_{ξ\rightarrow x}^{-1}\left[ (\max\{1,||ξ||_{p}\})^{-α}\widehat{φ}(ξ)\right] ,\text{ } φ\in \mathcal{D}\mathbb{Q}_{p}^{n}),\text{ } x\in\mathbb{Q}_{p}^{n}. We show that the fundamental solution $Z(x,t)$ of the $p-$adic heat equation naturally associated to these operators satisfies $Z(x,t)<= 0,x\in\mathbb{Q} _{p}^{n},t>0. So this equation describes the cooling …

非阿基米德有理映射的鲁厄ζ函数

Ruelle's zeta function for non-Archimedean rational maps

本文研究了非阿基米德有理映射的鲁厄ζ函数及其相关性质，证明了其解析性并探讨了朱利亚集的形状。

非阿基米德赋值域上留数域为形式实域的谱定理

A spectral theorem for a non-Archimedean valued field whose residue field is formally real

本文证明非阿基米德赋值域上自伴紧致算子的谱定理，该定理仅依赖于留数类域，解决了留数类域为形式实域的情形。

t-余范数对t-范数的弱支配性研究

On weak dominance of t-conorms over t-norms

本文完整刻画了连续t-余范数弱支配连续t-范数的条件，通过归结为阿基米德分量并用加法生成元表达，统一覆盖严格与幂零情形。

Non-Archimedean Welch Bounds and Non-Archimedean Zauner Conjecture

Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2, \quad \forall λ_j \in \mathbb{K}, 1\leq j \leq n, \forall n \in \mathbb{N}. \end{align*} For $d\in \mathbb{N}$, let $\mathbb{K}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{K}^d)$ be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If $\{τ_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle τ_j, τ_j\rangle =1$ for all $1\l…

A non-archimedean Algebra and the Schwartz impossibility theorem

In the 1950s L. Schwartz proved his famous impossibility result: for every k in N there does not exist a differential algebra (A,+,*,D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibnitz rule (namely D(u*v)=Du*v+u*Dv) and * is an extension of the pointwise product on the continuous functions. In this paper we prove that, by changing the requests, it is possible to avoid the impossibility result of Schwartz. Namely we prove that it is possible to construct an algebra of functions (A,+,*,D) such that (1) …

p-adic Heisenberg-Robertson-Schrodinger 和 p-adic Maccone-Pati 不确定性原理

Non-Archimedean Heisenberg-Robertson-Schrodinger and Non-Archimedean Maccone-Pati Uncertainty Principles

本文提出了p-adic环境下Heisenberg-Robertson-Schrodinger和Maccone-Pati不确定性原理，通过定义新的不确定性度量，推导了两个新的不确定性不等式，扩展了p-adic量子力学中的不确定性理论。

Spectral analysis for a class of bounded linear operators

We study the Spectral Analysis for a class of bounded linear operators T = D + F in a non Archimedean Hilbert space E, where D is a diagonal linear operator and where F is a finite rank linear operator. In this study of the Spectral Analysis, we use extensively the Theory of Fredholm Operators to deduce some of our main results.

On Archimedean Zeta Functions and Newton Polyhedra

Let $f$ be a polynomial function over the complex numbers and let $ϕ$ be a smooth function over $\mathbb{C}$ with compact support. When $f$ is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to $f$ and $ϕ$. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to $f$. More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, …

The Super Operator System Structures and their applications in Quantum Entanglement Theory

An operator system $\cl S$ with unit $e$, can be viewed as an Archimedean order unit space $(\cl S,\cl S^+,e)$. Using this Archimedean order unit space, for a fixed $k\in \bb N$ we construct a super k-minimal operator system OMIN$_k(\cl S)$ and a super k-maximal operator system OMAX$_k(\cl S)$, which are the general versions of the minimal operator system OMIN$(\cl S)$ and the maximal operator system OMAX$(\cl S)$ introduced recently, such that for $k=1$ we obtain the equality, respectively. We develop some of the key properties of these super operator systems and make some progress on charact…

保持不相交性的多重线性算子的Arens扩张

Arens extensions of disjointness preserving multilinear operators

本文研究保持不相交性的多重线性算子的Arens扩张，通过反例说明其一般不保持不相交性，并在有限格秩、Archimedean限制或Banach格条件下给出保持不相交性的充分条件。

序向量空间之间算子集合的集体序有界性

Collective order boundedness of sets of operators between ordered vector spaces

本文证明了序向量空间间算子集合的集体序有界性定理，并探讨了其在交换算子半群中的若干应用。

Non-Archimedean and p-adic Functional Welch Bounds

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2$ for all $ λ_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{…

Hardy inequalities on constant-order noncommutative Vilenkin groups

In this note we extend several integral inequalities to the context of noncommutative Vilenkin groups. We prove some sharp weak and strong type estimates for the Hardy operator and the Hardy-Littlewood-P{ó}lya operator on constant-order noncommutative Vilenkin groups. In particular for graded $\K$-Lie groups, where $\K$ is a non-archimedean local field, we additionally provide some functional inequalities, like the Hardy-Littlewood-Sobolev unequality and the Stein-Weiss inequality, linking some classes of homogeneous pseudo-differential operators, like the Vladimirov-Taibleson operator and the…

Spectral measures over c-algebras of operators defined in $c_0$

The main goal of this work is to introduce an analogous in the non-archimedean context of the Gelfand spaces of certain Banach commutative algebras with unit. In order to do that, we study the spectrum of this algebras and we show that, under special conditions, these algebras are isometrically isomorphic to certain spaces of continuous functions defined over compacts. Such isometries preserve projections and allow to define associated measures which are known as spectral measures. We also show that each element of the algebras can be represented as an integral defined by these measures.

Representation theorems for operators on free Banach spaces of type countable

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on a Free Banach spaces of countable type. The main goal of this work wil be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact developing the Gelfand space theory in the non-archimedean setting…

On normal, formally normal and quasinormal composition operators in $\ell^2$-spaces

Unbounded composition operators in $L^2$-space over discrete measure spaces are investigated. Normal, formally normal and quasinormal composition operators acting in $L^2$-spaces of this kind are characterized.

$C$--normal operators

A new class of operators, larger than $C$-symmetric operators and different than normal one, named $C$--normal operators is introduced. Basic properties are given. Characterizations of this operators in finite dimensional spaces using a relation with conjugate normal matrices are presented.

算子代数范畴中遗忘函子的局部可表现性与单子性

Local presentability and monadicity of forgetful functors for operator algebraic categories

本研究将完备算子空间范畴的局部可数可表现性结论推广到非完备情形，证明算子系统等范畴的该性质，厘清多类算子代数相关范畴间遗忘函子的伴随性与单子性，并借助Kadison对偶定理研究函数论子范畴的相关性质。

具有算子值正规符号的次正规Toeplitz算子

Hyponormal Toeplitz operators with operator valued normal symbols

本文刻画了具有算子值正规符号的次正规Toeplitz算子，并探讨了该类算子何时为解析或正规。主要贡献在于给出了此类算子的结构特征。研究问题涉及算子理论中的次正规性，方法基于符号的算子值性质分析。结果对理解Toeplitz算子的分类有重要意义。

Decomposition of Normal Operators and Its Application to Spectral Theorem

A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators. This extension gives a new proof of the spectral theorem for unbounded normal operators.

Normaloid 算子与根问题

Normaloid Operators and the Root Problem

本文将n次根问题的先前结果推广到一大类Hilbert空间算子，即具有normaloid部分的normaloid算子（包括亚正规算子和k-亚正规算子），证明若此类算子的n次幂正规，则其本身正规。

与正规算子平方根相关算子的模型理论

Model Theory for Operators Related to Square Roots of Normal Operators

研究与正规算子平方根相关算子，证明循环正规算子平方根及相关循环算子与特定向量值空间上块乘法算子酉等价，考虑向量值哈代空间界点赋值，得出相关算子有非平凡不变子空间及最小n-正规扩张唯一性等结论。

Normal Truncated Toeplitz Operators

The characterization of normal truncated Toepltiz operators is first given by Chalendar and Timotin. We give an elementary proof of their result without using the algebraic properties of truncated Toeplitz operators.

Bilateral weighted shift operators similar to normal operators

We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$.

$(α,β)$-A-Normal Operators in Semi-Hilbertian Spaces

In this paper we introduce and prove some properties of $(α;β)$-normal operators according to semi-Hilbertian space structures. Furthermore we s,ate various inequalities between the A-operator norm and A-numerical radius of $(α,β)$-normal operators in semi Hilbertian spaces.

Products of Unbounded Normal Operators

The present paper partly constitutes an "unbounded" follow-up of a paper by I. Kaplansky dealing with bounded products of normal operators. Results on the normality of unbounded products are also included.

On the similarity of complex symmetric operators to perturbations of restrictions of normal operators

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in $\mathbb{C}$, and some objects similar to objects from the theory of spectral problems for Jacobi matrices.

Algebraic Sum of Unbounded Normal Operators and the Square Root Problem of Kato

We prove that the algebraic sum of unbounded normal operators satisfies the square root problem of Kato under appropriate hypotheses. As application, we consider perturbed Schrodinger operators.

一些最小拓扑空间，以及向量格

Some minimum topological spaces, and vector lattices

本文研究了最小拓扑空间和向量格的存在性，通过 Yosida 定理探讨了覆盖算子与 hull 算子的关系，并给出了几种特定算子下的结果。

More on the Normality of the Unbounded Product of Two Normal Operators

Let $A$ and $B$ be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.

Unbounded normal operators in octonion Hilbert spaces and their spectra

Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.

$C$-normality of rank-one perturbations of normal operators

For a separable complex Hilbert space $H$, we say that a bounded linear operator $T$ acting on $H$ is $C$-normal, where $C$ is a conjugation on $H$, if it satisfies $CT^*TC=TT^*$. For a normal operator, we give geometric conditions which guarantee that its rank-one perturbation is a $C$-normal for some conjugation $C$.