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

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous ‖·‖1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _1$$\end{document}-dominating normalized gauge norms instead of the classical norms on measurable functions. 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 B2(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^2(z)$$\end{document} and B3(z).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^3(z).$$\end{document} These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm.