A shift-invariant algebra of singular integral operators with oscillating coefficients

Abstract

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ω) with an arbitrary Muckenhoupt weight ω on the unit circleT, and$\mathfrak{A}$ the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,λS _T e ⁻¹ _h,λ I (h∈R, λ∈T) whereS _T is the Cauchy singular integral operator ande _h,λ(t)=exp(h(t+λ)/(t−λ)),t∈T. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra$\mathfrak{A}$ and its matrix analogue$\mathfrak{A}_{NxN}$. These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.