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

TL;DR

This paper proves that for a class of integral operators with almost periodic kernels, invertibility of the operator implies the inverse also has a similar almost periodic structure, establishing inverse-closedness.

Contribution

It demonstrates that the algebra of such integral operators is inverse-closed, meaning the inverse of an invertible operator within this algebra also belongs to the same algebra.

Findings

Invertibility implies the inverse has a similar integral representation.

The algebra of these integral operators is inverse-closed.

The result applies to operators with almost periodic kernels in $L_p$ spaces.

Abstract

The integral operator of the form $$\bigl(Nu\bigr)(x)=\sum_{k=1}^\infty e^{i\langle\omega_k,x\rangle} \int_{\mathbb R^c}n_k(x-y)\,u(y)\,dy$$ 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 $$\sum_{k=1}^\infty\lVert n_k\rVert_{L_1}<\infty.$$ We prove that if the operator $\mathbf1+N$ is invertible, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the analogous representation.