The study of invariant subspaces of Hilbert space operators is a classical problem in analysis. It is an open question whether every operator on an Hilbert space has an invariant subspace other than the trivial ones, the zero subspace and the whole space. A. Beurling completely characterized the invariant subspaces of the unilateral shift $(a_0,a_1,\ldots)\mapsto(0,a_0,a_1,\ldots)$ on $\ell^2(\mathbb{N})$ modeling the shift operator on $\ell^2(\mathbb{N})$ with the multiplication by $z$ on the Hardy space of the unit disc. Few years later P. Lax proved an analogous result, that is, he characterized the translation invariant subspace of $L^2(0,\infty)$. In this seminar I will illustrate Beurling and Lax's result. Time permitting, I will also present an analogous of Beurling's result in the setting of the quaternionic Hardy space of the unit ball. This result was recently obtained in a joint work with G. Sarfatti.