Algebra of singular integral operators with a Carleman backward slowly oscillating shift

In this paper we study the Banach algebra of singular integral operators wi th piecewise continuous coefficients and a Carleman orientation-reversing s lowly oscillating shift on the Lebesgue space with a power weight on the un it circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operato rs in this algebra in terms of their symbols.