Discrete convolution operators with almost-stabilized coefficients

For a convolution operator of the form $K=\sum_j=0^{N-1}P_jA_j$, where $A_j$ are discrete Wiener—Hopf operators and $P_j$ are coordinate projections whose indices are congruent to $j$ modulo $N$, necessary and sufficient conditions for it to be Nötherian in $L_{p+}$ ($1\le p\le\infty$), $c_+^0$ and formulas for the index are obtained.