Note on the construction of the Bogolyubov scattering operator in the $({:}\varphi^4{:})_2$ theory

The Bogolyubov scattering operator $S(g)$ is constructed for the ${:}\varphi^4{:}_2$ theory in the special case when the space-time cutoff has a product structure that consists of two smooth functions: one corresponding to a time cutoff and the other to a space cutoff. Unitarity and causality are verified for this operator. The generalization of these results to a large class of cutoffs is discussed.