# Beurling–Lax theorem

In mathematics, the Beurling–Lax theorem is a theorem due to Beurling (1949) and Lax (1959) which characterizes the shift-invariant subspaces of the Hardy space ${\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} )}$. It states that each such space is of the form

${\displaystyle \theta H^{2}(\mathbb {D} ,\mathbb {C} ),}$

for some inner function ${\displaystyle \theta }$.