  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 $k\in
\mathbb{N}$.
