The \(C^{*}\)-algebra of singular integral operators with semi-almost periodic coefficients.
Let \((Sf)(x)=\frac{1}{\pi i}\int_R \frac{f(t)}{t-x}\,dt\) be the Cauchy singular integral operator on the real line \(\mathbb{R}\) and \(P=\frac 12 (I+S), Q=\frac 12 (I-S)\) be the projections generated by \(S\). In the space \(L^2_N\), the authors consider singular integral operators of the form \(A=PaP+Q+cH_++H_-dI\), where \(a,b,d \in SAP_{N\times N}\) (algebra of semi-almost periodic matrix functions) and \(H_\pm\in H_\infty.\) Here \(H_\infty\) is the closed two-sided ideal of the \(C^\ast\)-algebra \(\text{alg}(S, [C(\overline {\mathbb{R}})_{N\times N}])\) that is generated by all commutators \(uS-SuI\) with \(u\in C(\overline {\mathbb{R}})\), where \(C(\overline {\mathbb{R}})\) consists of complex valued continuous functions on \(\mathbb{R}\) which have finite limits at \(-\infty\) and \(\infty\). The authors give a Fredholm criterion for the operator \(A\) and a formula for the calculation of \(\operatorname{ind} A.\) The case of more general operators of the form \(aP+bQ+\sum_{k=1}^m c_kH_{k,+}+\sum_{j=1}^n H_{j,-}d_jI\) is also discussed. These results are applied to Toeplitz-like operators.
scientific article; zbMATH DE number 2037120
scientific article