On the index of a bisingular operator with an involutive shift

In the theory of singular operators with an involutive shift, the issues of Noether (Fredholm) property and the index of an operator of the form $A+VB$ are fully studied, where $A$ and $B$ are singular operators, and $V$ is an operator of an involutive shift in the space of $p$-summable functions on a simple closed contour of the Lyapunov type. Together with the operator $A+VB$, the corresponding matrix singular operator without shift $M=\left(
\begin{array}{cc}A&{VBV}\\ B&{VAV}\end{array}
\right)$ is considered. It is well known that the operators $A+VB$ and $M$ are Noetherian operators or not simultaneously, and their indices are related as $1:2$. Similar questions about simultaneous Noetherian property and proportionality of indices arise for bisingular operators with an involutive shift $A+WB$ and their corresponding matrix operators $M=\left(
\begin{array}{cc}A&{WBW}\\ B &{WAW}\end{array}
\right)$, where $A$ and $B$ are bisingular operators, and $W$ is an operator of an involutive shift in the space of $p$-summable functions on the direct product of simple closed contours of the Lyapunov type. In this paper, we study bisingular operators with an involutive shift that decomposes into one-dimensional components. Two types of such shifts are considered — coordinate-wise and cross. In these cases, the corresponding matrix operators are matrix bisingular operators without shift. The simultaneous Noetherian property of the bisingular operator with a shift and the corresponding matrix bisingular operator without shift is obtained. The proportionality of the indices of bisingular operators with a coordinate-wise shift and the corresponding matrix operators is established. Namely: it is proved that the indices of these operators are related as $1:2$. In a special case, the same result about the indices is obtained for the cross shift.