Invariance in general Hardy and Lebesgue Spaces

Abstract

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous and#8214; and#12539; and#8214;_1-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 dissertation, 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. A Helson-Lowdenslager type result has been proved by Chen in the context of Lebesgue spaces of the unit circle equipped with a continuous rotationally symmetric norm by studying the simply invariant subspaces of the operator of multiplication by the coordinate function z. We generalize Chen s result by obtaining a description of simply invariant subspaces for multiplication by z^n. A de Branges type result is also proved for Hardy spaces equipped with continuous rotationally symmetric norms.