Beurling's Theorem

Consider a sequence of complex numbers say a=(a_0,a_1,...,a_n,...) and a_0^2+a_1^2+a_2^2 + ... + a_n^2 +... is finite. We collect all such sequences of complex numbers and name the collection H. H is said to be the Hardy space on the unit disc and any element of it is represented by a(z) = a_0 + a_1 z + a_2 z^2 +... + a_n z^n + .... Let T: H --> H be an operator such that T(a_0,a_1,a_2,...,a_n,...)=(0,a_0,a_1,a_2,...,a_n,...). We call T unilateral shift on H.

A closed subspace M of H is invariant under T if (a_0,a_1,a_2,...,a_n,...) is in M, then (0,a_0,a_1,a_2,...,a_n,...) is also in M. We define an inner function to be a φ ∈ H_infinity with |φ| = 1 almost everywhere. 

Beurling's theorem: The non-zero invariant subspaces of the unilateral shift T on H are just φH , where φ is an inner function. 

The situation on the Hardy space of bidisc is pretty different. This is proved by Mandrekar.