Inverse-closedness of subalgebras of integral operators with almost periodic kernels

Abstract

The integral operator of the form $$\left(Nu\right)(x)=\sum _{k=1}^{\infty }{e}^{i⟨{\omega }_k,x⟩}{\int }_{{\mathbb{R}}^c}{n}_k(x-y)u(y)dy$$ acting in $L_p({\mathbb{R}}^c)$, $1\le p\le \infty$, is considered. It is assumed that ${\omega }_k\in {\mathbb{R}}^c$, $n_k\in L_1({\mathbb{R}}^c)$, and $$\sum _{k=1}^{\infty }\Vert n_k{\Vert }_{L_1}<\infty .$$ We prove that if the operator $1+N$ is invertible, then $(1+N{)}^{-1}=1+M$, where $M$ is an integral operator possessing the analogous representation.

21 pages