We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L2(T ℂn). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial isometries. The other three major advantages provided by our characterization are: (i) we provide precise necessary and sufficient conditions for the presence of reducing subspaces inside simply invariant subspaces; (ii) we give a complete description of the wandering vectors; (iii) we prove the theorem in the setting of all the Lebesgue spaces Lp (0 < p ≤ ∞). Our second theorem generalizes the first theorem along the lines of de Branges' generalization of Beurling's theorem by characterizing those Hilbert spaces that are simply invariant under multiplication by zn and which are contractively contained in Lp (1 ≤ p ≤ ∞). This also generalizes a theorem of Paulsen and Singh [Proc. Amer. Math. Soc. 129 (2000)] as well as the main theorem of Redett [Bull. London Math. Soc. 37 (2005)]. © Instytut Matematyczny PAN, 2010.