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

TL;DR

This paper extends Beurling's invariant subspace theorem to vector-valued Lebesgue and Hardy spaces with symmetric norms, using recent advances and measurable cross-section techniques, broadening previous results in the field.

Contribution

It introduces a generalized Beurling theorem for vector-valued spaces with symmetric norms, expanding the scope of invariant subspace characterizations.

Findings

Proves a version of Beurling's theorem for $L^{eta}(\mu,H^{\alpha})$ spaces.

Utilizes recent Beurling theorem on $H^{\alpha}(\mathbb{T})$ and measurable cross-section methods.

Significantly extends prior results by Rezaei, Talebzadeh, and Shin.

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.