Spectral approximation of Wiener-Hopf operators with almost periodic generating function

The paper deals with spectral approximation of Wiener-Hopf operators acting on L^p -spaces by their

finite sections. The generating functions of the Wiener-Hopf operators are supposed to be continuous plus almost

periodic.While the usual spectra of the finite sections drastically fail to converge to the spectrum of the Wiener-Hopf

operator,it turns out that other spectral approximants, viz. the pseudospectra and the numerical ranges, do converge

perfectly.The proof requires a modified approach to the finite section method for Wiener-Hopf operators. This note

generalizes results obtained by Böttcher, Grudsky and Silbermann for the case of continuous generating

functions.     

Spectral asymptotics of differential and pseudodifferential operators. II

In the article we obtain asymptotic formulae with remainder estimates for the distribution function of the eigenvalues of degenerate elliptic differential operators and Schrödinger operators with singular potential.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 78–106, 1984.     

Wiener-Hopf operators and generalized analytic functions

An almost periodic generalization of H+C is defined and analyzed. Applications of this analysis are made to the type II index theory of Wiener-Hopf operators with almost periodic symbols.Supported in part by the National Science Foundation.     

Wiener-Hopf operators on U2

In this paper, we describe the symbol calculus and index theorem for Wiener-Hopf operators on the group of complex 2×2 unitary matrices U₂.Research supported by grants of the National Science Foundation.     

Multivariable wiener-hopf operators I. Representations

Integral Equations and Operator Theory -     

On non-square sections of Wiener-Hopf operators

In this paper we study the left, right, and two-side invertibility of non-square finite sections of one-dimensional convolution operators.     

Asymptotic formulas for Toeplitz and Wiener-Hopf operators

For certain analytic functions f, the expression trace {f(T_n[]) – f(T_n[]T_n[])} is computed asymptotically. Here T_n[] is the finite Toeplitz matrix generated by the function . The analogous expression for Wiener-Hopf operators is also computed asymptotically. These results in turn yield information concerning the asymptotic behavior of determinants of finite Toeplitz and Wiener-Hopf operators with discontinuous generating function.     

Tangible convex bodies. Appendix to “Multivariable Wiener-Hopf Operators I”

Study in a local geometry of non-smooth convex bodies via their supporting cones. The supporting cones are differential objects if the convex bodies are tangible. Examples of completely tangible and non-tangible convex bodies are presented.     

An index theorem for Wiener-Hopf operators

We study the multivariate generalisation of the classical Wiener-Hopf algebra, which is the C^∗-algebra generated by the Wiener-Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C^∗-algebra is known to be isomorphic to the reduced C^∗-algebra of a certain restricted action groupoid. It admits a composition series, and therefore, a ‘symbol’ calculus. Using groupoid methods, we obtain, in the framework of Kasparov's bivariant KK-theory, a topological expression of the index maps associated to these symbol maps in terms of geometric-topological data of the underlying convex cone. This generalises an index theorem by Upmeier concerning Wiener-Hopf operators on symmetric cones. Our result covers a wide class of cones containing polyhedral and homogeneous cones.     

Factorization of Wiener-Hopf conservative integral operators

Translated from Matematicheskie Zametki, Vol. 46, No. 1, pp. 3–10, July, 1989.     

Global solvability for first order real linear partial differential operators II

Let L be a real $C^{\infty }$ vector field on a smooth manifold X, vanishing at exactly one point $x_0$. From the pioneering work of B. Malgrange (1955–1956) [6], we know that solvability of $P=L+c$ on $C^{\infty }(X)$, for $c\in C^{\infty }(X,\mathbb{C})$, implies that: (a) X is L-convex. Also, it follows: (b) a non-resonance condition for the jet-solvability at $x_0$.In a previous paper, in addition to (a) and (b), the authors showed that P is globally solvable on $C^{\infty }$ if we assume: (c) a non-resonance condition in order to linearize L near $x_0$; that (d) the only relatively compact orbit of L is $\{x_0\}$; and that (e) c is real.Here we obtain the same conclusion without (c) and (e).     

Multipliers and Wiener-Hopf operators on weighted L p spaces

We study multipliers M (bounded operators commuting with translations) on weighted spaces L _ω ^p (?), and establish the existence of a symbol µ _M for M, and some spectral results for translations S _t and multipliers. We also study operators T on the weighted space L _ω ^p (?⁺) commuting either with the right translations S _t , t ∈ ?⁺, or left translations P ⁺ S _?t , t ∈ ?⁺, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S _t ) of the operator S _t proving that $\sigma (S_t ) = \{ z \in \mathbb{C}:|z| \leqslant e^{t\alpha _0 } \} ,$ where α ₀ is the growth bound of (S _t ) _t≥0. A similar result is obtained for the spectrum of (P ⁺ S _?t ), t ≥ 0. Moreover, for an operator T commuting with S _t , t ≥ 0, we establish the inclusion , where $\mathcal{O}$ = {z ∈ ?: Im z < α ₀}.     

Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals

The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

Normal and compact solvability of linear operators

Dedicated to Yu. G. Reshetnyak on his sixtieth birthday.     

Circulant integral operators as preconditioners for Wiener-Hopf equations

In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {C }. Letk(t) denote the kernel function of the Wiener-Hopf equation and be its Fourier transform. We prove that for sufficiently large if {C } is uniformly bounded on the real lineR and the convolution product of the Fourier transform ofC with uniformly onR, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.Research supported in part by HKRGC grant no. 221600070.