A finite multiplicity Helson–Lowdenslager–de Branges theorem
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 $L^2(\mathbb T;\mathbb C^n)$. 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 $L^p$ $(0< p \le \infty)$. 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 $z^n$ and which are contractively contained in $L^p$ $( 1\le p \le \infty)$. 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)].