Lebesgue and Hardy Spaces for Symmetric Norms II: A Vector-Valued Beurling Theorem

Suppose $\alpha$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right)$ and $\beta$ is a "nice" norm on $L^{\infty}\left(\Omega,\mu \right)$ where $\mu$ is a $\sigma$-finite measure on $\Omega$. We prove a version of Beurling's invariant subspace theorem for the space $L^{\beta}\left(\mu,H^{\alpha}\right) .$ Our proof uses the recent version of Beurling's theorem on $H^{\alpha}\left(\mathbb{T}\right)$ proved by the first author and measurable cross-section techniques. Our result significantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.