On the index of a bisingular operator with an involutive shift

S. V. Efimov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 66-77

Résumé

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.