On a Hilbert Space Reformulation of Riemann Hypothesis

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, B'{a}ez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace ${\mathcal{H}}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to ${\mathcal{H}}_0^{\mathrm{\perp }}$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_{\mathbb{N}}=\{T_k:k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypothesis is true if and only if ${\mathcal{H}}_0$ is $T_k^{\ast }$-invariant for all $k\in $\mathbb{N}$$.