Singular integral operators. The case of an unlimited contour

Abstract

Let $\mathrm{\Gamma }$be a closed or unclosed unlimited contour, a shift $\alpha (t)$ maps homeomorphicly the contour $\mathrm{\Gamma }$ onto itself with preserving or reversing the direction on $\mathrm{\Gamma }$ and also satisfies the conditions: for some natural $n\ge 2$, ${\alpha }_n(t)\equiv t$, and ${\alpha }_j(t)\not\equiv t$ for $1\le j<n$. In this work we study subalgebra $\mathrm{\Sigma }$ of algebra$L(L_p(\mathrm{\Gamma },\rho ))$, which contains all operators of the form$$\left(M\phi \right)(t)=\sum _{k=0}^{n-1}\{a_k(t)\phi ({\alpha }_k(t))+\frac{b_k(t)}{\pi \mathrm{i}}{\int }_{\mathrm{\Gamma }}\frac{\phi (\tau )}{\tau -{\alpha }_k(t)}d\tau \}$$with piecewise-continuous coefficients. The existence of such an isomorphism between $\mathrm{\Sigma }$ and some algebra $\mathfrak{A}$ of singular operators with Cauchy kernel that an arbitrary operator from $\mathrm{\Sigma }$ and its image are Noetherian or not Noetherian simultaneously is proved. It allows to introduce the concept of a symbol for all operators from $\mathrm{\Sigma }$ and, using the known results for algebra $\mathfrak{A}$, in terms of a symbol to receive conditions of Noetherian property.