On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Holder coefficients in a Holder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Holder coefficients in a Holder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.