  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 $z$. In this paper, 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.
