Victor Shulman, Lyudmila Turowska
We establish a version of the Beurling-Pollard theorem for operator synthesis and apply it to derive some results on linear operator equations and to prove a Beurling-Pollard type theorem for Varopoulos tensor algebras. Additionally we establish a Beurling-Pollard theorem for weighted Fourier algebras and use it to obtain ascent estimates for operators that are functions of generalized scalar operators.
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Suppose $α$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $β$ is a "nice" norm on $L^{\infty}\left(Ω,μ\right) $ where $μ$ is a $σ$-finite measure on $Ω$. We prove a version of Beurling's invariant subspace theorem for the space $L^β\left(μ,H^α\right) .$ Our proof uses the recent version of Beurling's theorem on $H^α\left(\mathbb{T}\right) $ proved by the first author and measurable cross-section techniques. Our result significantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.
Yanni Chen extended the classical Beurling-Helson-Lowdenslager Theorem for Hardy spaces on the unit circle $\mathbb{T}$ defined in terms of continuous gauge norms on $L^{\infty}$ that dominate $\Vert\cdot\Vert_{1}$. We extend Chen's result to a much larger class of continuous gauge norms. A key ingredient is our result that if $α$ is a continuous normalized gauge norm on $L^{\infty}$, then there is a probability measure $λ$, mutually absolutely continuous with respect to Lebesgue measure on $\mathbb{T}$, such that $α\geq c\Vert\cdot\Vert_{1,λ}$ for some $0<c\leq1.$
We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator $S_K$ on a vector-valued Hardy space $H^{2}(Ω,K)$ is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-inner divisors.
We extend the de Branges-Beurling theorem characterizing the shift-invariant spaces boundedly contained in the Hardy space of square-summable power series to the full Fock space over $\mathbb{C} ^d$. Here, the full Fock space is identified as the \emph{Non-commutative (NC) Hardy Space} of square-summable Taylor series in several non-commuting variables. We then proceed to study lattice operations on NC kernels and operator-valued multipliers between vector-valued Fock spaces. In particular, we demonstrate that the operator-valued Fock space multipliers with common coefficient range space form …
We extend Beurling's invariant subspace theorem, by characterizing subspaces $K$ of the noncommutative $L^p$ spaces which are invariant with respect to Arveson's maximal subdiagonal algebras, sometimes known as noncommutative $H^\infty$. It is significant that a certain subspace, and a certain quotient, of $K$ are $L^p({\mathcal D})$-modules in the recent sense of Junge and Sherman, and therefore have a nice decomposition into cyclic submodules. We also give general inner-outer factorization formulae for elements in the noncommutative $L^p$. These facts generalize the classical ones, and shoul…
We prove Beurling's theorem for the full group $SL(2,\R)$. This is the {\em master theorem} in the quantitative uncertainty principle as all the other theorems of this genre follow from it.
We prove Beurling's theorem and $L^p-L^q$ Morgan's theorem for step two nilpotent Lie groups
Some Versions of Beurling's Theorem on H-type Groups
本研究在建立H型群的Gutzmer公式基础上，证明了特定维数H型群上Beurling定理的类似结论，并借助修正Radon变换得到该定理的其他版本，拓展了相关调和分析理论。
We formulate and prove an analogue of Beurling's theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.
Kernel theorems for Beurling-Björck type spaces
本文针对由权函数ω和η定义的一般Beurling-Björck型空间，证明了新的核定理，特别覆盖了经典Beurling-Björck空间。
This paper has been withdrawn by the authors, due to the requirement of the Journal where a modified version will be published.
The classical Beurling-Helson-Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space $H^{2}$ and of the Lebesgue space $L^{2}$. In this paper, which is self-contained, we define a very general class of norms $α$ and define spaces $H^α$ and $L^α.$ We then extend the Beurling-Helson-Lowdenslager invariant subspace theorem. The idea of the proof is new and quite simple; most of the details involve extending basic well-known $\left \Vert {}\right \Vert _{p}$-results for our more general norms.
We give a simple proof of the Kernel theorem for the space of tempered ultradistributions of Beurling - Komatsu type, using the characterization of Fourier-Hermite coefficients of the elements of the space. We prove in details that the test space of tempered ultradistributions of Beurling - Komatsu type can be identified with the space of sequences of ultrapolynomal falloff and its dual space with the space of sequences of ultrapolynomial growth. As a consequence of the Kernel theorem we have that the Weyl transform can be extended on a space of tempered ultradistributions of Beurling - Komats…
We prove Beurling's theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space.
A generalization of the Beurling--Malliavin Majorant theorem
该研究推广了 Beurling–Malliavin 控制函数定理，提出函数成为控制函数的新的、更一般的充分条件，并证明其在多种意义下最优。
We define tensor product decompositions of $E_0$-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent.
In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_α(D)$ for any $-1<α< +\infty$. That is, every invariant subspace $H$ for the shift operator $S$ on $A^2_α(D)$ $(-1<α< +\infty)$ has the property $H=[H\ominus zH]_{S,A^2_α\left(D\right)}$.
In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $% α$ on a tracial von Neumann algebra $\left( \mathcal{M},τ\right) $ where $α$ is $\left\Vert \cdot \right\Vert _{1}$-dominating with respect to $τ$. In the paper, we first define a class of norms $% N_{Δ}\left( \mathcal{M},τ\right) $ on $\mathcal{M}$, called determinant, normalized, unitarily invariant continuous norms on $\mathcal{M}$. If $α\in N_{Δ}\left( \mathcal{M},τ\right) $, then there exists a faithful normal tracial state $ρ$ on $\mathcal{M}…
We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only $\mathcal{C}^{1}$ changes of variables that leave invariant the modulation spaces $\M{p,q}(\rd)$ are affine functions on $\rd$. A special case of our result involving the Sjöstrand algebra was considered earlier by A. Boulkhemair.
We give a generalization of the Beurling-Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.
A remark on the Beurling-Malliavin theorem in several variables
本文通过提升技巧证明多变量中Beurling-Malliavin定理可扩展至径向函数，并简化了Vasilyev的论证并回答了其关于Cartwright版本的问题。
We provide a short argument to establish a Beurling-Lax-Halmos theorem for reproducing kernel Hilbert spaces whose kernel has a complete Nevanlinna-Pick factor. We also record factorization results for pairs of nested invariant subspaces.
We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.
Completeness of Exponentials and Beurling's Theorem on $\mathbb{R}^n$ and $\mathbb{T}^n$
本文通过建立 Beurling 定理与指数系线性张成稠密性问题之间的等价性，证明了 Beurling 定理的多变量推广，并在 $\mathbb{R}^n$ 和 $\mathbb{T}^n$ 上得到新的相关结果。
The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in the complex plane. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the clos…
We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
Let $\mathfrak A$ be a maximal subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$. If every right invariant subspace of $\mathfrak A$ in the non-commutative Hardy space $H^2$ is of Beurling type, then we say $\mathfrak A$ to be type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the non-commutative $H^1$ space. We show that the right analytic Toeplitz algebra on the non-commutative Hardy space $H^p$ associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.
A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorem. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.
We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $ 0 < a < 1.$ We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.
Wiener type theorems for countable limits of quasi-Beurling algebras and maximizing results on weights
本文研究了可数极限的准Beurling代数的Wiener型定理，并通过权重分析得到逆闭代数的层次结构及非可接受加权Wiener定理的权重最大化结果。
The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces
本文构造反例证明加权Bergman空间中Beurling型定理成立的临界值为$\alpha=1$，并推广预定曲率构造，将常数改进为$1$。
In a recent paper, M. Raghupathi has extended the famous theorem of Beurling to the context of subspaces that are invariant under the class of subalgebras of $H^\infty$ of the form $IH^\infty$, where $I$ is an inner function. In this paper, we provide analouges of the above mentioned $IH^\infty$ related extension of Beurling's theorem to the context of uniform algebras, on compact abelian groups with ordered duals, the Lebesgue space on the real line and in the setting of the space $BMOA$. We also provide a significant simplification of the proof of the Beurling's theorem in the setting of uni…
Interpretations of the Beurling-Lax-Halmos Theorem on invariant subspaces of the unilateral shift are explored using the language of Hilbert modules. Extensions and consequences are considered in both the one and multivariate cases with an emphasis on the classical Hardy, Bergman and Drury-Arveson spaces.
A note on Bohr's theorem for Beurling integer systems
本文研究Beurling整数系统中Bohr定理的成立条件，证明对Beurling素数序列施加小扰动可使其满足Bohr条件，并结合概率方法构造出Bohr定理与Riemann假设同时成立的系统，从而给出Helson关于广义Dirichlet级数Hardy空间外函数猜想的反例。
We prove a version of Whitney's extension theorem in the ultradifferentiable Beurling setting with controlled loss of regularity. As a by-product we show the existence of continuous linear extension operators on certain spaces of Whitney ultrajets on arbitrary closed sets in $\mathbb R^n$.
Beurling Criteria for Reproducing Kernels
研究再生核的贝林准则，通过多种方式刻画贝林 - 拉克斯 - 哈尔莫斯（BLH）再生核对类别，统一扩展相关结果，举例说明理论并与完全皮卡对研究对比，指出完全皮卡核类别与BLH核类别反向包含关系不成立。
The Beurling-Lax-Halmos Theorem for Infinite Multiplicity
本文探讨了贝鲁林-拉克斯-哈洛姆定理的扩展问题，研究了移不变子空间的结构及与汉克尔算子核的关系，提出了贝鲁林度数的概念，并探讨了模型算子的谱乘积性。
Let $0<p\leq 1$, $ω$ be a weight on $\mathbb Z$, and let $\mathcal A$ be a unital Banach algebra. If $f$ is a continuous function from the unit circle $\mathbb T$ to $\mathcal A$ such that $\sum_{n\in \mathbb Z} \|\widehat f(n)\|^p ω(n)^p<\infty$ and $f(z)$ is left invertible for all $z \in \mathbb T$, then there is a weight $ν$ on $\mathbb Z$ and a continuous function $g:\mathbb T \to \mathcal A$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $g$ is a left inverse of $f$ and $\sum_{n\in \mathbb Z}\|\widehat g(n)\|^pν(n)^p<\infty$. We shall obtain a similar result w…
We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from $W$ onto the unit disk $D$ is almost 1-1 rwith respect to the Lebesgure on $\partial D$ and if the Riemann map belongs to the weak-star closure of the polynomials in $H^{\infty}(W)$. Our main theorem states: In order that for each $M\in Lat(M_{z})$, there exist $u\in H^{\infty}(G)$ such that $ M = \vee\{u H^{2}(G)\}$, it is necessary and sufficient that the …
We prove sectorial extension theorems for ultraholomorphic function classes of Beurling type defined by weight functions with a controlled loss of regularity. The proofs are based on a reduction lemma, due to the second author, which allows to extract the Beurling from the Roumieu case, which was treated recently by Jiménez-Garrido, Sanz, and the third author. In order to have control on the opening of the sectors, where the extensions exist, we use the (mixed) growth index and the order of quasianalyticity of weight functions. As a consequence we obtain corresponding extension results for cla…
We establish some multivariate generalizations of the Beurling-Lax-Halmos theorem.
The aim of this paper is to establish a canonical decomposition of operator-valued strong $L^2$-functions by the aid of the Beurling-Lax-Halmos Theorem which characterizes the shift-invariant subspaces of vector-valued Hardy space. This decomposition invites us to coin a new notion of the "Beurling degree" of the inner function. Eventually, we establish a deep connection between the spectral multiplicity of the model operator and the Beurling degree of the corresponding characteristic function.
Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications
本文在量子环上研究Hörmander型傅里叶乘子定理和Nikolskii不等式，并推导了相关空间的嵌入定理和插值性质。
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. Also we develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón.
Beurling density theorems for sampling and interpolation on the flat cylinder
本文研究了平圆柱上的Fock空间，利用适应其几何结构的上下Beurling密度，完整刻画了采样集与插值集，并将结果等价转化为带时间周期化高斯窗的Gabor框架与Riesz基序列的完整刻画。
In this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in $H^\infty(\mathbb{D}^n)$.
We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling-Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results, is the fact that the right spectrum of a quaternionic linear operator and the S-spectrum coincide. Finally, we study the Krein-Langer factorization for slice hyperholomorphic generalized Sc…
Liftings and invariant subspaces of Hankel operators
本文证明了Hankel变体的换位子提升定理，并利用核空间揭示了Hankel算子Beurling型约化与不变子空间的完整结构。
Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.
Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
本文在非交换Drury-Arveson空间中建立Sarason型交换子提升定理，解决非交换Nevanlinna-Pick插值问题，并给出移位不变子空间的Beurling-Lax-Halmos型刻画。
In this paper we extend vessel theory, or equivalently, the theory of overdetermined $2D$ systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beur…
We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(π, \mathcal{H}_π)$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_π$ of formal dimension $d_π$. If $g \in \mathcal{H}_π$ is an integrable vector and the set $\{ π(λ)g : λ\in Λ\}$ for a discrete subset $Λ\subseteq N / Z(N)$ forms a frame for $\mathcal{H}_π$, then its density satisfies the strict inequali…
Full Gabor frames, its existence problem, and a non-uniform Balian-Low type theorem
针对一类在数学和物理中重要的Delone集，证明了非均匀Balian-Low型定理并解决了Gabor框架存在的逆问题，引入完全Gabor框架并证明其存在等价于下Beurling密度严格大于1。
This is a survey paper concerned with strongly continuous semigroups in a Banach algebra (often itself simply the algebra of bounded linear operators on a Banach space). These are defined either on $(0,\infty)$ or on a sector in the complex plane, in which case they are supposed analytic. However, the semigroups are not assumed to be strongly continuous at 0. The results in this survey, beginning with classical theorems of Beurling, Kato and Neuberger and concluding with recent work by the authors, are of the nature of "0-1" laws, indicating that the quantitative behaviour of the semigroup at …
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