  We prove necessary and sufficient conditions for the asymptotic normality of
multiple integrals with respect to a Poisson measure on a general measure
space, expressed both in terms of norms of contraction kernels and of variances
of carr\'e-du-champ operators. Our results substantially complete the fourth
moment theorems recently obtained by D\"obler and Peccati (2018) and D\"obler,
Vidotto and Zheng (2018). An important tool for achieving our goals is a novel
product formula for multiple integrals under minimal conditions.
