On weighted Lebesgue spaces over \(\mathbb {R}_+\) with power weights, the Fredholmness of operators from the Banach algebra \({\mathfrak D}\) generated by multiplication operators, Wiener-Hopf operators and Mellin convolution operators with piecewise slowly oscillating data is studied. The present paper is a continuation of [3], where we described the maximal ideal space \({\mathfrak M}\) of a central subalgebra \({\mathfrak C}^\pi \) of the quotient Banach algebra \({\mathfrak D}^\pi \) with respect to the ideal of compact operators and identified the local Banach algebras \({\mathfrak D}^\pi _{\xi ,\eta ,\mu }\) associated with the points \((\xi ,\eta ,\mu )\in {\mathfrak M}\) by the Allan-Douglas local principle. Establishing isomorphisms of algebras \({\mathfrak D}^\pi _{\xi ,\eta ,\mu }\) and some operator algebras, studying the invertibility of the cosets in the algebras \({\mathfrak D}^\pi _{\xi ,\eta ,\mu }\) for each of the four subsets of \({\mathfrak M}\) with the help of the two idempotent theorem and the Gelfand transform, we construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak D}\) and establish Fredholm criteria for the operators \(A\in {\mathfrak D}\) .
