Yu. I. Karlovich

Banach Algebras Generated by N Idempotents and Applications

It is well known that for Banach algebras generated by two idempotents and the identity all irreducible representations are of order not greater than two. These representations have been described completely and have found important applications to symbol theory. It is also well known that without additional restrictions on the idempotents these results do not admit a natural generalization to algebras generated by more than two idempotents and the identity. In this paper we describe all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations. These representations are of order not greater than N and allow us to construct a symbol theory with applications to singular integral operators.

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.

Toeplitz and Singular Integral Operators on General Carleson Jordan Curves

This paper is concerned with the spectra of Toeplitz operators with piecewise continuous symbols and with the symbol calculus for singular integral operators with piecewise continuous coefficients on L P (Γ) where 1 < p < ∞ and Γ is a Carleson Jordan curve. It is well known that piecewise smooth curves lead to the appearance of circular arcs in the essential spectra of Toeplitz operators, and only recently the authors discovered that certain Carleson curves metamorphose these circular arcs into logarithmic double-spirals. In the present paper we dispose of the matter by determining the local spectra produced by a general Carleson curve. These spectra are of a qualitatively new type and may, in particular, be heavy sets — until now such a phenomenon has only be observed for spaces with general Muckenhoupt weights.

A Local-trajectory Method and Isomorphism Theorems for Nonlocal C*-algebras

A nonlocal version of the Allan-Douglas local principle applicable to nonlocal C*-algebras \( \mathcal{B}\) associated with C*-dynamical systems is elaborated. This local-trajectory method allows one to study the invertibility of elements b e \( \mathcal{B}\) in terms of invertibility of their local representatives. Isomorphism theorems for nonlocal C*-algebras are established.

Almost Periodic Factorization of Some Triangular Matrix Functions

The paper is devoted to matrices of the form \( G(x) = \left[ {\begin{array}{*{20}c} {e^{i\lambda x} } & 0 \\ {f(x)} & {e^{ - i\lambda x} } \\ \end{array} } \right], \), with almost periodic off-diagonal entry f. Some new cases are found, in terms of the Bohr-Fourier spectrum of f, in which G is factorable. Formulas for the partial indices are derived and, under additional constraints, the factorization itself is constructed explicitly. Some a priori conditions on the Bohr-Fourier spectra of the factorization factors (provided that a canonical factorization exists) are also given.

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.

Singular integral operators with complex conjugation from the viewpoint of pseudodifferential operators

We study the algebra A generated by singular integral operators a I+b S and the operator of complex conjugation V on the weighted Lebesgue space L p(Γ,w). Our approach is based on transforming the operators in A locally into Mellin pseudodifferential operators. By having recourse to the Fredholm and index theory of the latter class of operators, we can establish Fredholm criteria and index formulas for operators in A with slowly oscillating coefficients in the case of slowly oscillating composed curves F and slowly oscillating Muckenhoupt weights w. We are in particular able to consider curves with whirl points, in which case massive local spectra may emerge even for constant coefficients and weights.

Factorizations, Riemann–Hilbert problems and the corona theorem

The solvability of the Riemann–Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener–Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x ): =e iλx , λ ∈ R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2

Invertibility of Functional Operators with Slowly Oscillating Non-Carleman Shifts

We prove criteria for the invertibility of the binomial functional operator