  In 2016 and 2017, Haihui Fan, Don Hadwin and Wenjing Liu proved a commutative
and noncommutative version of Beurling's theorems for a continuous unitarily
invariant norm $\alpha $ on $L^{\infty}(\mathbb{T},\mu)$ and tracial finite von
Neumann algebras $\left( \mathcal{M},\tau \right) $, respectively. In the
paper, we study unitarily $\|\|_{1}$-dominating invariant norms $\alpha $ on
finite von Neumann algebras. First we get a Burling theorem in commutative von
Neumann algebras by defining $H^{\alpha}(\mathbb{T},\mu)=\overline
{H^{\infty}(\mathbb{T},\mu)}^{\sigma(L^{\alpha}\left(
  \mathbb{T} \right),\mathcal{L}^{\alpha^{'}}\left(
  \mathbb{T} \right))}\cap L^{\alpha}(\mathbb{T},\mu)$, then prove that the
generalized Beurling theorem holds. Moreover, we get similar result in
noncommutative case. The key ingredients in the proof of our result include a
factorization theorem and a density theorem for $L^{\alpha
}\left(\mathcal{M},\tau \right) $.
