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

Abstract

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.