In 1993, Alcantara-Bode showed ([2]) that Riemann Hypothesis
holds if and only if the integral operator on the Hilbert space L2(0; 1)
having the kernel function defined by the fractional part of (y/x), is
injective. Since then, the injectivity of the integral operator used in
equivalent formulation of RH has not been addressed nor has been
dissociated from RH.
We provided in this paper methods for investigating the injectivity
of linear bounded operators on separable Hilbert spaces using their
approximations on dense families of subspaces.
On the separable Hilbert space L2(0,1), an linear bounded operator
(or its associated Hermitian), strict positive definite on a dense family
of including approximation subspaces in built on simple functions, is
injective if the rate of convergence of its sequence of injectivity pa-
rameters on approximation subspaces is inferior bounded by a not null
constant, that is the case with the Beurling - Alcantara-Bode integral
operator.
We applied these methods to the integral operator used in RH
equivalence proving its injectivity.