Let $T>0$, $X := L^2([0, T])$ and $\Gamma: X \rightarrow X$ be defined as $$ (\Gamma u)(t) := \int^t_0 k(t, s)u(s)~\mathrm{d}s, $$ where $k \in L^2([0, T]^2)$.
I am rather certain that $\mathrm{Id} - \Gamma$ is invertible (I know that it is if $k \in L^\infty([0, T]^2)$). Does anyone have a literature reference for this result?