Let \(\Gamma \) be a closed or non closed unlimited contour in \(\mathbb{C}\), and let \(\alpha (t)\) a shift applying \(\Gamma \) homeomorphicaly onto itself with the properties: for some natural \(n\geq 2,\) \(\alpha_{n}(t)=t\) and \( \alpha _{j}(t)=\alpha [ \alpha _{j-1}(t)]\neq t,\) \((1\leq j<n),\) \(\alpha _{0}(t)=t.\)
In this paper one considers the set of singular integral operators of the form \[ (M\varphi )(t)=\sum_{k=0}^{n-1}\left\{ a_{k}(t)\cdot\varphi (\alpha _{k}(t))+ \frac{b_{k}(t)}{\pi i}\int_{\Gamma}\frac {\varphi (\tau)}{\tau -\alpha_{k}(t)}d\tau \right\} ,\tag{1} \] where the coefficients \(a_{k}(t),\) \(b_{k}(t)\) are piecewise continuous functions on \(\Gamma ,\) and one studies the algebra \(\Sigma\) containing all the operators of the form (1)
The author proves the existence of an isomorphism \(\triangle \) between \(\Sigma\) and some algebra \(\mathcal{U}\) of singular integral operators with Cauchy kernel, such that every \(A\in\Sigma\) and its image \(\triangle (A)\in \mathcal{U}\) are simultaneously Noetherian or not Noetherian. Using the known symbol concept of an operator in \(\mathcal{U},\) one introduces the symbol for \(A\in \Sigma \) and the index of \(A.\)
Characterizations of the operators in algebra \(\Sigma \) ( as a subalgebra of \(L(L_{p}(\Gamma ,\rho)))\) which are Noetherian in the space \(L_{p}(\Gamma ,\rho )\) are given, and illustrative examples are considered.