On the inverse closedness of the subalgebra of local Hilbert–Schmidt operators

A local Hilbert–Schmidt operator is an operator of the form
\begin{equation*} (Tx)(t)=\int\limits_{-\infty}^{+\infty}k(t,s)x(s)ds \end{equation*}
with a measurable kernel $k:\mathbb{R}^2\to\mathbb{C}$ under the condition that
\begin{equation*} \int\limits_a^{b}\int\limits_a^{b}|k(t,s)|^2 ds dt<\infty \end{equation*}
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 $\mathbf{1}+T$ implies that the inverse operator has the form $\mathbf{1}+T_1$, where $T_1$ is also a local Hilbert–Schmidt operator whose kernel $S$ satisfies the same conditions.