In 1993, using the Beurling equivalent formulation of the RH ([3]),
Alcantara-Bode proved ([2]): the RH 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 this integral operator has not been
addressed even if the investigation of the injectivity of this operator
could be made out of context, by other means than of pure
mathematics. In line with this observation is our approach using the
operator approximations on a dense family of subspaces on separable
Hilbert spaces in order to address its injectivity.
One of the results obtained in this paper, Theorem 2.1 states that given an linear, bounded operator strict positive definite on a dense family of subspaces, having its sequence of injectivity parameters bounded inferior by a strict positive constant, is injective. (The parameters are the inverse condition numbers of the operator restrictions on the family of the subspaces.)
Using a version of this theorem on L2(0; 1) we proved (Theorem
4.1) the injectivity of the integral operator used by Alcantara-Bode in
[2] for his equivalent formulation of RH.