Paired integral operators with homogeneous kernels perturbated by operators of multiplicative shift

In the space $L_p(\mathbb{R}^n)$, where $1\leqslant p\leqslant \infty$, we consider an operator $B$, which is the sum of two terms. The first term is a paired multidimensional integral operator, whose kernels are homogeneous of degree $(-n)$ and invariant with respect to the rotation group of $\mathbb{R}^n$-space, and the second term is a series, convergent in the operator norm, composed of multidimensional multiplicative shift operators with complex coefficients. We impose some additional conditions on the kernels and coefficients of the operator $B$, and these conditions ensure the boundedness of this operator in the space of summable functions. The main aim of the paper is to study the invertibility of the operator $B$. To solve this problem we use a special method that allows the reduction of the multidimensional paired operator to an infinite sequence of one-dimensional paired operators $B_m$, where $m\in\mathbb{Z}_+$. It is shown that the operator $B$ is invertible if and only if all the operators $B_m$ are invertible, where $m$ runs through all values from zero to some finite number $m_0$. In turn, the operators $B_m$ reduce to integral-difference convolution operators whose theory is well known. All this allowed us to determine the symbol of the operator $B$. This symbol represents the pair of functions $(\beta_1(m,\xi),\beta_2(m,\xi))$, defined on the set $\mathbb{Z}_+\times\mathbb{R}$. If the symbol is non-degenerate, then we define in a natural way a real number $\nu$ and integers $\varkappa_m$, where $m\in\mathbb{Z}_+$. Numbers $\nu$ and $\varkappa_m$ are called indices. The main result of the work is the invertibility criterion of the multidimensional paired operator $B$ in the space $L_p(\mathbb{R}^n)$. According to this criterion, the operator $B$ is invertible if and only if its symbol is non-degenerate, and all its indices are zero.