A General 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 version of Beurling’s theorem on \(H^{\alpha }\left( \mathbb {T}\right) \) in Chen (Adv Appl Math, 2016) and measurable cross-section techniques. Our result significantly extends a result of Rezaei, Talebzadeh, and Shin (Int J Math Anal 6:701–707, 2012).