The orthogonality problem and shift-invariance in the $H^2$ approach to the RH

A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced by Noor [9]. It states that the RH holds if and only if the linear manifold $\mathcal{N}$ generated by a sequence of functions $(h_k)_{k\geq 2}$ is dense in the Hardy space $H^2$. In this article we consider the following three questions: $\mathbf{A.}$ How small is the orthogonal complement of $\mathcal{N}$? $\mathbf{B.}$ Is the closure of $\mathcal{N}$ a shift-invariant subspace of $H^2$? $\mathbf{C.}$ Is $\mathcal{N}$ dense in $H^2$ under a weaker topology?. We use tools from local Dirichlet spaces, the de Branges-Rovnyak spaces and the Smirnov class to address these questions.Comment: 7 page