Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(mathbb{R},w)}$ , where ${pin(1,infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${mathfrak{U}_{p,w}}$ of ${mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${ain PSO^diamond}$ ) and all convolution operators W ⁰(b) ( ${bin PSO_{p,w}^diamond}$ ), where ${PSO^diamondsubset L^infty(mathbb{R})}$ and ${PSO_{p,w}^diamondsubset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${mathbb{R}cup{infty}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${Ainmathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${mathfrak{U}_{p,w}}$ more effective results.