Using the equivalent formulation of RH given by Beurling ([4],
1955), Alcantara-Bode showed ([2], 1993) that Riemann Hypothesis
holds if and only if the integral operator on the Hilbert space L2(0; 1)
having the kernel defined by fractional part function of the expression
between brackets {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 and, a pure mathematics solution for RH is not ready yet.
Here is a numerical analysis approach of the injectivity of the linear
bounded operators on separable Hilbert spaces addressing the problems
like the one presented in [2]. Apart of proving the injectivity of the
Beurling - Alcantara-Bode integral operator, we obtained the following
result: every linear bounded operator (or its associated Hermitian),
strict positive definite on a dense family of including approximation
subspaces in L2(0,1) built on simple functions, is injective if the rate
of convergence to zero of its unbounded sequence of inverse condition
numbers on approximation subspaces is o(n-s) for some s ≥ 0. When
s = 0, the sequence is inferior bounded by a not null constant, that is
the case in the Beurling - Alcantara-Bode integral operator.
In the Theorem 4.1 we addressed with numerical analysis tools
the injectivity of the integral operator in [2] claiming that - even if a
solution in pure mathematics is desired, together with the Theorem 1,
pg. 153 in [2], the RH holds.