Estelle L. Basor, Torsten Ehrhardt
In this paper we determine the asymptotics of the determinants of truncated Wiener-Hopf plus Hankel operators $\det(W_R(a)\pm H_R(a))$ as $R$ tends to infinity for symbols $a(x)=(x^2/(1+x^2))^β$ with the parameter $β$ being of small size.
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A factorization theory is proposed for Wiener-Hopf plus Hankel operators with almost periodic Fourier symbols. We introduce a factorization concept for the almost periodic Fourier symbols such that the properties of the factors will allow corresponding operator factorizations. Conditions for left, right, or both-sided invertibility of the Wiener-Hopf plus Hankel operators are therefore obtained upon certain indices of the factorizations. Under such conditions, the one-sided and two-sided inverses of the operators in study are also obtained.
We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.
Wiener-Hopf plus Hankel operators acting between Lebesgue spaces on the real line are studied in view of their invertibility, one sided-invertibility, Fredholm, and semi-Fredholm properties. This is done in two different cases: (i) when the Fourier symbols of the operators are unitary functions, and (ii) when the Fourier symbols are related with sectorial elements appearing in factorizations of functions originated by the Fourier symbols of the operators. The obtained result for the case (i) may be viewed as a Douglas-Sarason type theorem for Wiener-Hopf plus Hankel operators.
The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, \end{equation*} we establish necessary and also sufficient conditions for the operators $W(a)+H(b)$ to be one-sided invertible, invertible or generalized invertible. Besides, efficient representations for the corresponding inverses are given.
Considered are Wiener--Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions. If the generating functions $a$ and $b$ satisfy the matching condition \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, \end{equation*} an explicit description for the kernels and cokernels of the operators mentioned is given.
Wiener-Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions are studied. For $a$ and $b$ satisfying the so-called matching condition \begin{equation*} a(t) a(-t)=b(t) b(-t), \quad t\in \mathbb{R}, \end{equation*} we single out some classes of operators $W(a)+H(b)$ which are subject to Coburn-Simonenko theorem.
In this paper we are going to prove two asymptotic formulas for determinants det(I-K_s), as s goes to infinity, where K_s are the Wiener-Hopf-Hankel operators acting on L^2[0,s] with the kernels K(x-y)+K(x+y) and K(x-y)-K(x+y), respectively, and K(t):=sin(t)/(π*t). These formulas were conjectured by Dyson. The identification of the constant term in the asymptotics was an open problem for a long time.
A Wiener-Hopf operator on a Banach space of functions on R+ is a bounded operator T such that P^+S_{-a}TS_a=T, for every positive a, where S_a is the operator of translation by a. We obtain a representation theorem for the Wiener-Hopf operators on a large class of functions on R+ with values in a separable Hilbert space.
The solution of a problem arising in integrable systems requires sharp asymptotics for the inverses and determinants of truncated Wiener-Hopf operators, both in the regular case (where the non-truncated Wiener-Hopf operator is invertible) and in singular cases. This paper treats the singular cases where the symbol of the Wiener-Hopf operator has one double zero or two simple zeros. We find formulas for the inverses that hold uniformly throughout the underlying interval with very small error, and formulas for the determinants with very small error.
Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains
本文针对不含仿射直线的凸集证明Helson不等式，由此导出Paley-Wiener空间的弱分解，并将多面体上Schatten类Hankel算子的刻画从$1\leq p\leq 2$推广到所有$1\leq p<\infty$。
Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$.
Wiener-Hopf indices of unimodular functions on the unit circle, revisited
本文受Groenewald等人工作启发，用算子理论方法重新研究单位圆上幺模有理矩阵值函数的Wiener-Hopf指标，给出了更深入的理解和更简单的计算公式。
The spectral representation of the Wiener-Hopf operator K with kernel $1/π$ sinc is given determining explicitly the Hilbert space isomorphism, which transforms K into the multiplication operator by the identity on $L^2(0,1)$. Several related integral operators are studied. A close relationship of K to the finite Hilbert transformation is revealed yielding the spectral representation of the latter. This is of particular interest as it concerns a general feature of self-adjoint Wiener-Hopf operators [15].
Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions
该研究针对高维凸区域上的Paley–Wiener空间，建立Paley–Wiener型Besov空间理论，明确Hankel算子属于Schatten类Sᵖ的充要条件，并拓展了相关结果的适用范围。
We derive the asymptotic behavior of determinants of truncated Wiener-Hopf operators generated by symbols having Fisher-Hartwig singularities. This task is achieved thanks to an asymptotic resolution of the Riemann-Hilbert problem associated to some generalized sine kernel. As a byproduct, we give yet another derivation of the asymptotic behavior of Toeplitz determinants having Fisher-Hartwig singularities. The Riemann-Hilbert problem approach to these asymptotics yields a systematic although quickly cumbersome way to compute their sub-leading asymptotics.
Wiener-Hopf indices of unimodular functions on the imaginary axis
本文研究虚轴上单模有理矩阵函数的 Wiener-Hopf 指标，利用 Douglas-Shapiro-Shields 分解中因子实现的矩阵给出指标公式，并通过算子论直接方法与 Cayley 变换两种途径建立结果。
Canonical Wiener-Hopf factorization on the unit circle: matching subspaces versus Riccati equations
本文比较单位圆上典范右 Wiener-Hopf 分解的匹配不变子空间方法与非对称 Riccati 方程方法，将后者推广到非有理算子值函数，并证明两类方程虽不同却共享同一稳定解。
There is a bounded Hankel operator on the Paley--Wiener space of a disc in $\mathbb{R}^2$ which does not arise from a bounded symbol.
The present paper deals with an application of the image normalization technique for certain classes of Wiener-Hopf operators (WHOs) associated to ill-posed boundary-transmission value problems. We briefly describe the method of normalization and then apply it to boundary-transmission value problems issued from diffraction problems for a junction of two half-planes, which are relevant in mathematical physics applications. We consider different boundary-transmission conditions on the junction of the two semi-infinite half-planes and analyze the not normally solvability of the corresponding oper…
We study the bounded operators on weighted spaces Lw^2 on R^+ commuting either with the right translations St or left translations and we establish the existence of a symbol for these operators. We characterize completely the spectrum of St. We give also a spectral results for the left translations and for the bounded operators commuting with the semi-group (St) or with the semi-group of the left translations.
We study the operators T on the weighted space L^p commuting either with the right translations St or left translations P^+S_{-t} and we establish the existence of a symbol of T. We characterize completely the spectrum of St. We obtain a similar result for the spectrum of P^+S_{-t} and some spectral results for the bounded operators commuting with (St), t>0 or with (P^+St), t<0.
Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces
本文研究了Paley-Wiener空间中Hankel算子的Schatten类条件对Nehari定理的影响，证明在特定条件下Nehari定理不成立，并展示了多面体相关的Hardy不等式及其应用。
Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains and ranges are explicitly determined. A further topic concerns semibounded WH. Expressing a semibounded WH by a product of a closable operator and its adjoint this representation allows for a natural self-adjoint extension. It is shown that it coincides with the Friedrichs extension. Polar decomposition gives r…
Maximal Noncompactness of Wiener-Hopf Operators
本文研究了维纳-霍普夫算子在特定函数空间上的最大非紧性性质，证明其非紧性测度、本质范数和范数相等，扩展了该类算子在L^p空间中的理论成果。
On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators
本文研究了线性有序离散阿贝尔群上的维纳-霍普夫算子的Fredholm指数和谱性质，通过符号的逆傅里叶变换进行分析。
The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and $l^p$-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. Special attention is paid to a newly developed method, which allows to establish necessary, sufficient and also necessary and sufficient conditions of invertibility, one-sided and generalized invertibility for wide classes of operators and derive efficient formulas for the corresponding inverses. The work also contains a number of problems whose solution would be of inte…
The uniqueness of the Wiener-Hopf factorisation of Lévy processes and random walks
本文证明了无杀死的Lévy过程或随机游走的空间Wiener-Hopf分解的唯一性。
As in the paper [G. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators. \emph{Indag. Math.} 28 (2017) 694--710] our aim is to obtain explicitly the Wiener-Hopf indices of a rational $m\times m$ matrix function $R(z)$ that has no poles and no zeros on the unit circle $\mathbb{T}$ but, in contrast with that paper, the function $R(z)$ is not required to be unitary on the unit circle. On the other hand, using a Douglas-Shapiro-Shields type of factorization, we show that $R(z)$ factors…
Fredholm criteria for Wiener-Hopf operators with continuous symbols acting on some Banach function spaces
本文将Duduchava在1970年代关于Lebesgue空间上具有连续符号的Wiener-Hopf算子的Fredholm准则，推广至Lorentz、自反Orlicz和变Lebesgue空间三种Banach函数空间。
We prove that every positive bounded invertible Wiener-Hopf operator admits triangular factorization. This answers the question posed by L.Sakhnovich in 1994.
We show that a semibounded Wiener-Hopf quadratic form is closable in the space $L^2({\Bbb R}_{+})$ if and only if its integral kernel is the Fourier transform of an absolutely continuous measure. This allows us to define semibounded Wiener-Hopf operators and their symbols under minimal assumptions on their integral kernels. Our proof relies on a continuous analogue of the Riesz Brothers theorem obtained in the paper.
A new systems theory perspective on canonical Wiener-Hopf factorization on the unit circle
本文从系统理论视角研究单位圆上Hilbert空间算子值函数的典范Wiener-Hopf分解，利用无限维二分系统的严格有界实引理和Krein空间理论，给出了左、右典范分解的显式公式。
This paper reviews the modern state of the Wiener--Hopf factorization method and its generalizations. The main constructive results for matrix Wiener--Hopf are presented, approximation methods are outlined and the main areas of applications are mentioned. The aim of the paper is to sketched some perspective of the development of this method, importance of bringing together pure and applied analysis to most effectively use of the Wiener--Hopf technique.
Considered is the equation $$ (T(a)+H(b))ϕ=f, $$ where $T(a)$ and $H(b)$, $a,b\in L^\infty(\mathbb{T})$ are, respectively, Toeplitz and Hankel operators acting on the classical Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the generating functions $a$ and $b$ satisfy the so-called matching condition [1,2], $$ a(t) a(1/t)=b(t)b(1/t), \, t\in \mathbb{T}, $$ an efficient method for solving equations with Toeplitz plus Hankel operators is proposed. The method is based on the Wiener--Hopf factorization of the scalar functions $c(t)=a(t)b^{-1}(t)$ and $d(t)=a(t)b^{-1}(1/t)$ and allows one to find…
It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More general, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which are Fredholm can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem. In this paper we establish corresponding results for Toeplitz + Hankel operators and singular integral operators with flip under the assumption th…
Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators
本文研究了Wiener-Hopf型算子在Banach函数空间中的范数及非紧性测度的下界估计，引入了弱倍增和分离倍增性质，得出了算子范数与非紧性测度的明确下界关系。
With localization techniques one can obtain general limit theorems for Toeplitz determinants with Fisher-Hartwig singularities from the asymptotics for any symbol with one singularity of general type. There exists a family of these for which the determinants can be evaluated explicitly and their asymptotics determined. But for the Wiener-Hopf analogue, although there are likely analogous localization techniques, there is not a single example known of a symbol with Fisher-Hartwig singularity for which the determinant can be evaluated explicitly. In this paper we determine the asymptotics of Wie…
Gohberg-Krupnik Localisation for Discrete Wiener-Hopf Operators on Orlicz Sequence Spaces
本文利用 Matuszewska-Orlicz 指标引入 Fourier 乘子的“插值友好”类，并通过在 Calkin 代数中应用 Gohberg-Krupnik 局部化，将离散 Wiener-Hopf 算子的 Fredholm 性质研究归结为对局部代表算子的研究。
Some Wiener--Hopf determinants on [0,s] are calculated explicitly for all s>0. Their symbols are zero on an interval and they are related to the determinant with the sine-kernel appearing in the random matrix theory. The determinants are calculated by taking limits of Toeplitz determinants, which in turn are found from the related systems of polynomials orthogonal on an arc of the unit circle. As is known, the latter polynomials are connected to those orthogonal on an interval of the real axis. This connection is somewhat extended here. The determinants we compute originate from the Bernstein-…
Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond
本文通过将爱因斯坦场方程视为可积系统，利用维纳-霍普夫因子分解获得场方程及其关联Lax对的精确解，强调了多学科方法在引力场方程研究中的重要性。
On determinant expansions for Hankel operators
本文将Christoffel-Darboux核表示为Hankel积分算子乘积之和，给出了Wiener-Hopf算子行列式可化为Hankel算子乘积行列式的充分条件，并利用Barnes积分导出其按广义超几何函数的展开，推广了Basor和Chen的相关结果。
In this paper we determine the asymptotics of the determinant of Bessel operators for sufficiently smooth generating functions. These operators are similar to Wiener-Hopf operators with the Fourier transform replaced by the Hankel transform and thus the asymptotics of the determinanst are similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants. In order to compute the above, we also show that the Bessel operators differ from the Wiener-Hopf by a Hilbert-Schmidt operator.
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics
本文研究单位圆上含极点的有理矩阵符号所生成无界类 Toeplitz 算子的核维数，在 Wiener–Hopf 型分解的附加条件下给出公式，并推广相关核刻画。
Riemann-Hilbert problems, Toeplitz operators and ergosurfaces
本文利用Wiener-Hopf分解与Toeplitz算子的联系，证明有理monodromy矩阵缺失标准分解时产生的发散曲线对应旋转黑洞的能层表面。
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations
针对单位圆上含极点的有理矩阵符号所诱导的无界类 Toeplitz 算子，本文用状态空间方法将其可逆性归结为相关非对称离散代数 Riccati 方程的稳定解，并给出伪典范分解与逆算子公式。
Generating new gravitational solutions by matrix multiplication
本文提出一种基于不变性问题和矩阵乘法的引力解生成新方法，通过乘法形变和叠加构造出传统 Wiener-Hopf 分解无法获得的新解。
The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.
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