Multiplication by finite Blaschke factors on a general class of Hardy spaces

TL;DR

This paper extends Beurling's theorem to Hardy spaces with generalized norms, analyzing multiplication by finite Blaschke factors and characterizing invariant subspaces, leading to a new factorization framework.

Contribution

It generalizes Beurling's theorem to broader Hardy spaces with continuous gauge norms and characterizes invariant subspaces for specific Blaschke products.

Findings

Generalized Beurling's theorem for finite Blaschke factors

Characterization of invariant subspaces of B^2(z) and B^3(z)

Factorization results for functions in these Hardy spaces

Abstract

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous $\|.\|_1$-dominating normalized gauge norms instead of the classical norms on measurable functions and a Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor $B(z)$ and also derive the common invariant subspaces of $B^2(z)$ and $B^3(z)$. These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm.