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

Suppose $α$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right)$ and $β$ is a "nice" norm on $L^{\infty}\left(Ω,μ\right)$ where $μ$ is a $σ$-finite measure on $Ω$. We prove a version of Beurling's invariant subspace theorem for the space $L^β\left(μ,H^α\right) .$ Our proof uses the recent version of Beurling's theorem on $H^α\left(\mathbb{T}\right)$ 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.

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