This might be obvious, but I don't see why...
Let $\lambda \in \mathbb{C}\backslash\mathbb{R}_+$ and $m(x) = \lambda/(\lambda - e^x)$. If $f \in L^2$, then
$$ mf = \int^\infty_{-\infty} \frac{\lambda e^x}{(\lambda - e^x)^2} \, f \chi_{(-\infty, x)} \, dx $$
as a Bochner integral in $L^2$.
How could one derive the above equality?