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.
Categories: math.FA math.OA