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

Abstract

Let $N$ be an integral operator of the form $\left(Nu\right)(x)={\int }_{{\mathbb{R}}^c}n(x,x-y)u(y)dy$ 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 $1+N$ has an inverse, then $(1+N{)}^{-1}=1+M$, where $M$ is an integral operator possessing the same properties.

16 pages

References in corpus (2)

• Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group
• Symmetry and Inverse Closedness for Some Banach ${}^{\ast }$-Algebras Associated to Discrete Groups