具有$L_1$ -连续变化核的积分算子集的逆闭性
Inverse-closedness of the set of integral operators with $L_1$-continuously varying kernels
摘要：
令$N$是作用于$L_p(\mathbb R^c)$中的形式为$\bigl(Nu\bigr)(x)=\int_{\mathbb R^c}n(x,x-y)\,u(y)\,dy$的积分算子，其可测核$n$满足估计$|n(x,y)|\leβ(y)$，其中$β\in L_1$。证明了如果函数$t\mapsto n(t,\cdot)$在$L_1$的范数下连续，且算子$\mathbf1+N$有逆算子，则有$(\mathbf1+N)^{-1}=\mathbf1+M$，其中$M$是具有相同性质的积分算子.
英文摘要：
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 $|n(x,y)|\leβ(y)$, where $β\in L_1$. 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.
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