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

Let a be the Banach algebra of all bounded linear operators on the weighted Lebesgue space L-p(T,w) with an arbitrary Muckenhoupt weight w on the unit circle T, and U the Banach subalgebra of B generated by the operators of m ultiplication by piecewise continuous coefficients and the operators e(h,la mbda)S(T)e(h,lambda)(-1) I (h is an element of R, lambda is an element of T ) where S-T is the Cauchy singular integral operator and e(h,lambda)(t) = e xp(h(t+lambda)/(t-lambda)), t is an element of T. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra U and its matrix analogue U-NxN These shift-invariant algebr as arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms of T onto itself with second Taylor derivatives.