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A Helson-Lowdenslager type result has been proved by Chen in the context ofLebesgue spaces of the unit circle equipped with a continuous rotationallysymmetric norm by studying the simply invariant subspaces of the operator ofmultiplication by the coordinate function $z$. In this paper, we generalizeChen's result by obtaining a description of simply invariant subspaces formultiplication by $z^n$. A de Branges type result is also proved for Hardyspaces equipped with continuous rotationally symmetric norms.

Helson-Lowdenslager and de Branges type theorems in the setting of continuous rotationally symmetric norms