A local Hilbert–Schmidt operator is an operator of the form \(\begin{aligned} (Tx)(t)=\int \limits _{-\infty }^{+\infty }k(t,s)x(s)ds \end{aligned}\) with a measurable kernel \(k:\mathbb {R}^2\rightarrow \mathbb {C}\) under the condition \(\begin{aligned} \int \limits _a^{b}\int \limits _a^{b}|k(t,s)|^2 ds dt<\infty \end{aligned}\) for all \(-\infty<a<b<+\infty \) . We prove that, under some additional conditions that provide the action of the operator T in \(L_2(\mathbb {R},\mathbb {C})\) , the invertibility of the operator \(\textbf{1}+T\) implies that the inverse operator has the form \(\textbf{1}+T_1\) , where \(T_1\) is also a local Hilbert–Schmidt operator whose kernel S satisfies the same conditions.
