Inverse-closedness of the set of integral operators with $L_1$-continuously varying kernels

TL;DR

This paper proves that integral operators with kernels varying continuously in $L_1$ are inverse-closed, meaning their inverses also have integral kernels with similar properties, under certain conditions.

Contribution

It establishes inverse-closedness for a class of integral operators with $L_1$-continuously varying kernels, extending the understanding of their algebraic structure.

Findings

Inverse of $oldsymbol{1+N}$ is also an integral operator with similar properties.

Continuity in $L_1$-norm of the kernel function ensures inverse-closedness.

The result applies to operators with kernels satisfying an $L_1$ bound.

Abstract

Let $N$ be an integral operator of the form $\bigl(Nu\bigr)(x)=\int_{\mathbb R^c}n(x,x-y)\,u(y)\,dy$ acting in $L_p(\mathbb R^c)$ with a measurable kernel $n$ satisfying the estimate $|n(x,y)|\le\beta(y)$, where $\beta\in L_1$. It is proved that if the function $t\mapsto n(t,\cdot)$ is continuous in the norm of $L_1$ and the operator $\mathbf1+N$ has an inverse, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the same properties.