Abstract:
In the space $L_p(\mathscr L)$, $p>1$, we consider the operator $A\varphi=a\varphi+bS\varphi+cP\varphi+T\varphi$, where $a(t)$, $b(t)$, and $c(t)$ are piecewise-continuous functions on the contour $\mathscr L$, $T$ is a completely continuous operator,
$$
P_\varphi=\frac1{2\pi i}\int_\mathscr L\frac{\varphi(\tau)\,d\tau}{\tau-t-1},\quad S_\varphi=\frac1{\pi i}\int_{\mathscr L}\frac{\varphi(\tau)\,d\tau}{\tau-t},
$$ $\mathscr L$ is a closed convex Lyapunov contour having no rectilinear portions. We study the properties of the operator $P$ and we show that the Noether property conditions and the index of the operator $A$ do not depend on the term $c_P$.
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