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 $\mathcal{ H}_0^\bot$ 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 $k\in \mathbb{ N}$.
Categories: math.NT