  This work explores and develops elements of Stein's method of approximation,
in the infinitely divisible setting, and its connections to functional
analysis. It is mainly concerned with multivariate self-decomposable laws
without finite first moment and, in particular, with $\alpha$-stable ones,
$\alpha \in (0,1]$. At first, several characterizations of these laws via
covariance identities are presented. In turn, these characterizations lead to
integro-differential equations which are solved with the help of both semigroup
and Fourier methodologies. Then, Poincar\'e-type inequalities for
self-decomposable laws having finite first moment are revisited. In this
non-local setting, several algebraic quantities (such as the carr\'e du champs
and its iterates) originating in the theory of Markov diffusion operators are
computed. Finally, rigidity and stability results for the Poincar\'e-ratio
functional of the rotationally invariant $\alpha$-stable laws, $\alpha\in
(1,2)$, are obtained; and as such they recover the classical Gaussian setting
as $\alpha \to 2$.
