Anatoly N. Kochubei
We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.
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We introduce and study non-Archimedean analogs of the operators of unilateral shift and backward shift playing crucial roles in the classical theory of nonselfadjoint operators. In particular, we find various functional models of these operators having both common and different features compared to their classical counterparts.
We describe some classes of linear operators on Banach spaces over non-Archimedean fields, which admit orthogonal spectral decompositions. Several examples are given.
We study some classes of algebras of operators on non-Archimedean Banach spaces. In particular, we propose a non-Archimedean version of the crossed product construction.
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…
Differentiations of operator algebras over non-archimedean spherically complete fields are investigated. Theorems about a differentiation being internal are demonstrated.
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。
Let $D^α, α>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^αD^{-α}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
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.
We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz Theorem of Dritschel-Rovnyak and positivstellensätze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for *-algebras that is interesting in its own right.
Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields
本文研究了非阿基米德域上的非均匀周期小波框架，通过傅里叶变换技术和正交扩展原理，提出了一种构造方法。
In this work the problem about an existence of non-measurable automorphisms of Lie groups finite and as well infinite dimensional over the field of real numbers and also over the non-archimedean local fields is investigated. Non-measurability of automorphisms is considered relative to real-valued measures and also measures with values in non-archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application for a construction of non-measurable irreducible unitary representations is demonstrated.
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。
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…
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。
Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields
本文提出非阿基米德局部域上非均匀小波框架的构造方法，发展了斜扩展和正交扩展原理，并展示了应用示例。
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。
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。
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。
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.
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 …
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…
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。
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。
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.
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…
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。
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 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。
We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological groups, and unitary operators given by the evolution group of a self-adjoint operator or by powers of a unitary operator. Our results are illustrated with a wide range of examples in quantum mechanics and dynamical systems, as for instance Schrödinger operators, Dirac operators, quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walk…
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。
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generator…
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) …
We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.
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。
We prove that there exists a rank one perturbation of a unitary operator on a complex separable infinite dimensional Hilbert space which is hypercyclic.
The objective of this paper to give a formula for unitary operators that corresponds to the Heler-Sj{ö}strand formula for self-adjoint operators.
Let $A$ and $B$ be compact operators over a topological space $X$ and suppose that these operators are normal and have same distinct eigenvalues at each point. By obstruction theory, we establish a necessary and sufficient condition for $A$ and $B$ to be strongly unitarily equivalent. When $X=S^1$, we also give a sufficient condition for $A$ and $B$ to be approximately unitarily equivalent with some assumption on their eigenvalues.
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.
In this paper, we investigate operator-valued frames with the structure of group-like unitary system. We show the commutant of the group-like unitary system can be characterized in terms of analysis operators associated with all the operator-valued Bessel generators and give some sufficient and necessary conditions to describe when an operator-valued frame generator admits a Parseval dual.
Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any Carleson set on the circle can be the spectrum of a perturbed (hypercyclic) operator.
A unitary operator $V$ and a rank $2$ operator $R$ acting on a Hilbert space $\H$ are constructed such that $V+R$ is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.
Geometry of spectral bounds of curves of unitary operators
研究利用酉算子群及其作用球面上的度量几何方法，重新证明酉算子乘积的谱界定理，并将其推广至可微曲线。
A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.
Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary operators.
This paper is concerned with a certain aspect of the spectral theory of unitary operators in a Hilbert space and its aim is to give an explicit construction of continuous functions of unitary operators. Starting from a given unitary operator we give a family of sequences of trigonometric polynomials converging weakly to the complex measures which allow us to define functions of the operator.
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…
As applications of Kadison's Pythageorean and carpenter's theorems, the Schur-Horn theorem, and Thompson's theorem, we obtain an extension of Thompsons theorem to compact operators and use these ideas to give a characterization of diagonals of unitary operators. Thompson's mysterious inequality concerning the last terms of the diagonal and singular value sequences plays a central role.
We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable Hilbert space. The samples are defined by means of suitable discrete convolution systems which generalize some usual sampling settings; here regular sampling means that the samples are taken at a countable discrete subgroup.
In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.
We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form $(1-<z,w>)^{-γ}$ for $γ>0$. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.
The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz functions on the unit circle.
The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions coincides with the class of operator Lipschitz functions.
We consider a weighted form of the Poisson summation formula. We prove that under certain decay rate conditions on the weights, there exists a unique unitary Fourier-Poisson operator which satisfies this formula. We next find the diagonal form of this operator, and prove that under weaker conditions on the weights, a unique unitary operator still exists which satisfies a Poisson summation formula in operator form. We also generalize the interplay between the Fourier transform and derivative to those Fourier-Poisson operators.
In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen.
We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.
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