Criteria for One-Sided Invertibility of a Functional Operator in Rearrangement-Invariant Spaces of Fundamental Type

Let γ be a simple open smooth curve and be an orientation-preserving diffeomorphism of α onto itself which has only two fixed points.Criteria for one-sided invertibility of thefunctional operator A = aI — bW, where a and b are continuous functions, I is the identity operator, W is the shift operator: (Wf ) (t) =f[α (t)], in a reflexive rearrangement-invariant space of fundamental type X(γ) with nontrivial Boyd indices, are obtained.     

An example of a reflexive Lorentz Gamma space with trivial Boyd and Zippin indices

Czechoslovak Mathematical Journal - We show that for every p ∈ (1, ∞) there exists a weight w such that the Lorentz Gamma space Γp,w is reflexive, its lower Boyd and Zippin indices...     

Algebras of Singular Integral Operators with Piecewise Continuous Coefficients on Reflexive Orlicz Spaces

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces L_m(σ) which are generalizations of the Lebesgue spaces L_P(σ), 1 < p < ∞. We suppose that σ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient G, we establish a Fredholm criterion and an index formula in terms of the essential range of G complemented by spiralic horns depending on the Boyd indices of L_M(σ) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix - valued coefficients on L_Mⁿ(σ).     

Higher order asymptotics of Toeplitz determinants with symbols in weighted Wiener algebras

We extend a result of Böttcher and Silbermann on higher order asymptotics of determinants of block Toeplitz matrices with symbols in Wiener algebras with power weights to the case of Wiener algebras with general weights satisfying natural submultiplicativity, monotonicity, and regularity conditions.     

Fluorescence spectrum of a one-atom laser in the strong-coupling regime

The multicomponent spectrum of the resonance fluorescence is studied based on the expansion of the Green function with respect to the coherent states for a one-atom laser. An analytical solution is derived for the spectrum in the frequency range corresponding to the vacuum Rabi oscillations in the approximation of the strong atom-field coupling and a low incoherent pump intensity.     

Correction to: One-sided invertibility of discrete operators with bounded coefficients

Aequationes mathematicae - A Correction to this paper has been published: https://doi.org/10.1007/s00010-020-00773-8     

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.Work partially supported by F. C. T. (Portugal) Project Praxis XXI/2/2.1/MAT/441/94 and by CONACYT (México) grant, Cátedra Patrimonial, No. 990017-EX.     

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

We prove Szegös strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary N-functions both satisfying the ⁰ ₂-condition and by weight sequences satisfying some regularity and compatibility conditions.