A Helson-Lowdenslager-deBranges theorem in \(L^2\) (Q2701586)

The paper presents a generalization of the well-known Helson-Lowdenslager generalization of the Beurling theorem. The result is analogous to \textit{L. de Branges} and \textit{J. Rovnyak}'s generalization of the Beurling theorem [Square summable power series, New York etc.: Holt, Rinehart and Winston (1966; Zbl 0153.39602)]. Let \(M\neq\{0\}\) be a simply invariant Hilbert space contractively contained in \(L^2\) (i.e., \(M\subset L^2\), \(\|h\|_{L^2}\leq\|h\|_{M}\)) on which the unilateral shift \(Sf=e^{it}f\) acts isometrically. The authors give a condition under which there exists a unique \(b\) in the unit ball of \(L^{\infty}\) such that \(M=bH^2\).