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.
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.
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.
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.
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 $L^p(\mathbb R)$.
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].
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.
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.
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…
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…
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…
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.
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…
In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction an…
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…
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-…
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.
We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $Ξ= (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2Ξ$, consider the Hankel operator $$Γ_f (g)(x)=\int_Ξ f(x+y) g(y) \, dy, \quad x \inΞ.$$ Then $Γ_f$ extends to a bounded operator on $L^2(Ξ)$ if and only if there is a bounded function $b$ on $\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2…
For bounded Lebesgue measurable functions $f,g,ϕ$ and $ψ$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}ϕP_{-}+P_{-}ψP_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or qu…
We give alternative proofs to (block case versions of) some formulas for Toeplitz and Fredholm determinants established recently by the authors of the title. Our proof of the Borodin-Okounkov formula is very short and direct. The proof of the Baik-Deift-Rains formulas is based on standard manipulations with Wiener-Hopf factorizations.
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.
