An algebra of singular integral operators with kernels of bounded oscillation, and application to periodic differential operators (Q909983)

This work is concerned with the Fredholm properties of the operators of the \(C^*\)-subalgebra \({\mathcal A}\) of \({\mathcal L}({\mathcal H})\), \({\mathcal H}=L^ 2({\mathbb{R}})\), generated by multiplication and convolution operators as follows: i) multiplication by function in CS(\({\mathbb{R}});\) ii) multiplication by \(e^{ikx}\), \(k\in {\mathbb{Z}};\) iii) the Fourier multipliers \(b(D)=F^{-1}b(x)F\) where \(b\in CS({\mathbb{R}})\) and F is the Fourier transform. A characterization of Fredholm operators \(A\in {\mathcal A}\) is given in terms of their symbols \(\sigma_ A\) and \(\nu_ A\) along with a general index formula.