M. A. Bastos

On a new algorithm for almost periodic factorization

An algorithm is proposed allowing to find necessary and sufficient conditions for existence of an almost periodic factorization for several new classes of triangular and block triangular matrix functions.

Convolution equations of the first kind on a finite interval in Sobolev spaces

The study of a class of operators associated with convolution equations of the first kind on a finite interval is reduced to the study of Wiener-Hopf operators with piecewise continuous symbol on R. Fredholm properties and invertibility conditions for this class of operators are investigated. An example from diffraction theory is considered.

Convolution Type Operators with Symbols Generated by Slowly Oscillating and Piecewise Continuous Matrix Functions

The paper deals with a local study of the Banach algebra A [SO, PC] generated by the convolution type operators W a b = a F −1bF with data a E ∈ [SO,PC] n× n and b∈ [SO p ,PC] p n× n which act on the Lebesgue space L p n (ℝ) (1 < p < ∞, n ≥ 1). Here [SO,PC] n× n means the C*-algebra generated by slowly oscillating (SO)and piecewise continuous(PC) n × n matrix functions, and SOBA11020200787 is a Fourier multiplier analogue onL p (ℝ)of [SO,PC] n× n The work is based on the study of Fourier multiplier analogue SO p ofSOon the characterization of the multiplicative linear functionals of slowly oscillating functions, and on the compactness of the commutators AW a, b — W a, b A, where A ∈ A PC] a ∈ SOandb ∈ SO p. Making use of the Allan-Douglas local principle we construct homomorphisms of A [SO PC]onto local Banach algebras and establish a Fredholm criterion for operators A ∈ A [SO PC] in terms of the invertibility of their images in the local algebras.

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.

Oscillatory Riemann–Hilbert problems and the corona theorem☆

The paper is devoted to the Riemann–Hilbert problem with matrix coefficient G∈[L∞(R)]2×2 having detG=1 in Hardy spaces [Hp±]2,1<p⩽∞, on half-planes C±. Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann–Hilbert problem in [H∞±]2 we study the solvability of the non-homogeneous Riemann–Hilbert problem in [Hp±]2,1<p<∞, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in C+ and C−). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H∞+ are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.

Operator algebras, operator theory and applications

Summer School Lecture Notes.- Subalgebras of Graph C*-algebras.- C*-algebras and Asymptotic Spectral Theory.- Toeplitz Operator Algebras and Complex Analysis.- Workshop Contributed Articles.- Rotation Algebras and Continued Fractions.- On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols.- Diffraction by a Strip and by a Half-plane with Variable Face Impedances.- Factorization Algorithm for Some Special Matrix Functions.- On a Radon Transform.- Extensions of ?-C* -algebras.- Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized Holder Spaces.- Nonlocal Singular Integral Operators with Slowly Oscillating Data.- Poly-Bergman Projections and Orthogonal Decompositions of L 2-spaces Over Bounded Domains.- Vekuas Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces.- On Homotopical Non-invertibility of C*-extensions.- Galois-fixed Points and K-theory for GL(n).- Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory.- Compact Linear Operators Between Probabilistic Normed Spaces.- Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrodinger Operators.- On Finite Sections of Band-dominated Operators.- Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent.- Orbit Representations and Circle Maps.- On Generalized Spherical Fractional Integration Operators in Weighted Generalized Holder Spaces on the Unit Sphere.

Generalized factorization for a class 2×2 matrix functions with rationally-independent entries ∗

Necessary and sufficient conditions for the existence of canonical generalized factorizations are given for a class of non-rational 2×2 matrix-valued functions. Explicit formulas for the factors are derived. The class considered is related to a class studied by Rawlins and Williams from the point of view of function-theoretic factorization. The method of analysis followed in the paper involves solving a certain non-linear singular integral equation. An example is included.

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton−1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.

On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onLp(ℝ+,ℂm) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL1(ℝ,ℂmxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.