Wiener-Hopf Operators Research Articles

Index calculation for Wiener-Hopf operators with slowly oscillating coefficients

Index calculation for Wiener-Hopf operators with slowly oscillating coefficients

The Widom–Sobolev formula for discontinuous matrix-valued symbols

We prove the Widom–Sobolev formula for the asymptotic behaviour of truncated Wiener–Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.

The Brown–Halmos theorem for discrete Wiener–Hopf operators

We prove an analogue of the Brown–Halmos theorem for discrete Wiener–Hopf operators acting on separable rearrangement-invariant Banach sequence spaces.

Conjugations on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{L}}^\ extbf{2}$$\\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{H}}^\ extbf{2}$$\\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.

Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^3$$\\end{document} subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2\\ln (L)$$\\end{document}, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L\\Lambda \\subset {{\\mathbb {R}}}^3$$\\end{document} as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln (L)$$\\end{document} factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}, is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the relevant trace for piecewise C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{C}^{1,\\alpha }$$\\end{document} smooth surfaces ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}.

$${\mathcal{L}}$$-Wiener–Hopf Operators in Weighted Spaces in Case of Reflectionless Potential

$${\mathcal{L}}$$-Wiener–Hopf Operators in Weighted Spaces in Case of Reflectionless Potential

Unbounded Toeplitz Operators with Rational Symbols

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed 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, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

Unbounded Wiener\u2013Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

On complementable operators in the sense of T. Ando

Consider an operator A:H→K between Hilbert spaces and closed subspaces S⊂H and T⊂K. If there exist projections E on H and F on K such that R(E)=S, R(F)=T and AE=F⁎A then A is called (S,T)-complementable. The origin of this notion comes from the idea of T. Ando of defining Schur complements in terms of operators. In this paper we present some characterizations of these triples (A,S,T) and applications to bilateral Schur complements and generalized Wiener-Hopf operators.

On Determinant Expansions for Hankel Operators

Abstract Let w be a semiclassical weight that is generic in Magnus’s sense, and ( p n ) n = 0 ∞ ({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ 1 Γϕ 2) where Γϕ 1 and Γϕ 2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2m F 2m-1. These results extend those of Basor and Chen [2], who obtained 4 F 3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.

A Wiener-Hopf Integral Equation with a Nonsymmetric Kernel in the Supercritical Case

The paper is devoted to the solvability questions of the Wiener-Hopf integral equation in the case where the kernel K satisfies the conditions 0 ≤ K ∈ L1(ℝ), \(\int_{-\infty}^{\infty} K(t)dt>1\), K(±x) ∈ C(3)(ℝ+), (−1)nK(±x)(n)(x) ≥ 0, x ∈ ℝ+, n =1, 2, 3. Based on Volterra factorization of the Wiener-Hopf operator, and invoking the technique of nonlinear functional equations, we construct real-valued solutions both for homogeneous and non-homogeneous Wiener-Hopf equations, assuming that the function g is real-valued and summable, and the corresponding conditions are satisfied. The behavior at infinity of the corresponding solutions is also studied.

On using convolutions with exponential distributions for solving a Kolmogorov backward equation

We propose a numerical method to solve 3-dimensional partial differential equations with variable coefficients, which appear in applications focused on studying random walks in the presence of an absorbing barrier. We restrict ourselves to the case of the Kolmogorov backward equation, which most commonly arises in mathematical finance. A probabilistic interpretation of the problem we use allows to apply Carr’s randomization for dimensionality reduction and employ the Wiener-Hopf factorization to solve the arising 1-dimensional problems. The key idea of the approach proposed is to realize the expected present value operators, which appear in the factorization identity, as convolutions with certain exponential distributions. We generalize our method for the case of exponentially distributed jumps presence using the Russian splitting method. The Wiener-Hopf factors related to the jump component of the operator again involve a convolution with exponential kernels. The numerical iterative scheme suggested to evaluate Wiener-Hopf operators is much less computationally expensive than a standard Fast Fourier transform-based approach.

On the factorization of matrix and operator Wiener–Hopf integral equations

Let be a Wiener–Hopf operator, , , and let be the adjoint operator, , , where belongs to the Banach space of Bochner strongly integrable functions with values in a Banach algebra . We consider the canonical factorization problem , where is the identity operator and (resp. ) is a left (resp. right) triangular convolution operator such that the operators are invertible in the spaces , . We put forward a semi-inverse factorization method and prove that the canonical factorization exists if and only if the operators and are invertible in .

Schmidt Decomposable Products of Projections

Fil: Corach, Gustavo. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

Trace formulas for Wiener–Hopf operators with applications to entropies of free fermionic equilibrium states

We consider non-smooth functions of (truncated) Wiener–Hopf type operators on the Hilbert space L2(Rd). Our main results are uniform estimates for trace norms (d≥1) and quasiclassical asymptotic formulas for traces of the resulting operators (d=1). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space Rd (d≥1) at positive temperature, T>0. In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small T and large scaling parameter α>0 provided that the product Tα remains bounded from below. Here α is the reciprocal quasiclassical parameter. For d=1 we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at T=0), previously established by the authors for d≥1.

Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization

Let $\mu$ be an even measure on the real line $\mathbb{R}$ such that $$c_1 \int_{\mathbb{R}}|f|^2\,dx \le \int_{\mathbb{R}}|f|^2\,d\mu \le c_2\int_{\mathbb{R}}|f|^2\,dx$$ for all functions $f$ in the Paley-Wiener space $\mathrm{PW}_{a}$. We prove that $\mu$ is the spectral measure for the unique Hamiltonian $\mathcal{H}=\left(w&00&\frac{1}{w}\right)$ on $[0,a]$ generated by a weight $w$ from the Muckenhoupt class $A_2[0,a]$. As a consequence of this result, we construct Krein's orthogonal entire functions with respect to $\mu$ and prove that every positive, bounded, invertible Wiener-Hopf operator on $[0,a]$ with real symbol admits triangular factorization.

On semibounded Wiener-Hopf operators

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.

Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators $A, B$, a bounded operator $J$, and a function $f:\mathbb R\to\mathbb C$ we obtain bounds in quasi-normed ideals of compact operators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$. The focus is on functions $f$ that are smooth everywhere except for finitely many points. A typical example is the function $f(t) = |t|^\gamma$ with $\gamma \in (0, 1)$. The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace $tr f(S)$ with $S$ being a Wiener-Hopf operator with a discontinuous symbol.

On a coefficient in trace formulas for Wiener–Hopf operators

Let a = a(\xi), \xi \in \mathbb R, be a smooth function quickly decreasing at infinity. For the Wiener–Hopf operator W(a) with the symbol a , and a smooth function g\colon \mathbb C \to \mathbb C , H. Widom in 1982 established the following trace formula: \mathrm {tr}(g(W(a)) - W(g \circ a)) = \mathcal B(a; g), where \mathcal B(a; g) is given explicitly in terms of the functions a and g . The paper analyses the coefficient \mathcal B (a; g) for a class of non-smooth functions g assuming that a is real-valued. A representative example of one such function is g(t) = |t|^{\gamma} with some \gamma \in (0, 1] .

On the symmetrization of general Wiener-Hopf operators

On the symmetrization of general Wiener-Hopf operators