2. Latest development

It contains the generic injectivity theorem and separates the analysis of the restrictions of the linear operator from its finite rank approximations on a family of finite dimension subspaces whose union is dense.

The Injectivity Criteria used in [3] is used for the operator restrictions involving the operator adjoint and the finite rank approximations are considered when the operator is Hilbert-Schmidt.

Taking the integral operator utilized by Alcantara-Bode in his equivalent formulation of the Riemann Hypothesis (RH), both methods show that its null space contains only the null element, meaning that RH is true. The result is published for now as a preprint in

https://doi.org/10.20944/preprints202411.1062.v8

Numerical computations.

Given an integral operator on $\latex L^2(0,1)$, its positivity as well the positivity of the operator approximations on the finite dimension subspaces whose union is dense, are subject of the values of diagonal entries in the matrix representations of the operator restrictions defined by:

d_{k,k}^h = \int_{(k-1)h}^{kh}\int_{(k-1)h}^{kh} \rho(y,x)dxdy, k=1,n

corresponding to a partition of mesh h (nh=1) of the domain.

Here is the hint of the 1-diagonal sparsity of the matrices involved. Defining

I_{h,k}(t)= 1 \text{ for } t \in \Delta_{h,k} \text{ and } 0 \text{ otherwise}, k=1,n, nh=1,

from:

\cdot  \text{for any}  f\in L^2(0,1),  I_{h,k}(t)f(t) = 0  and f(t)I_{h,k}(t) = 0 \text{ if  } t\notin \Delta_{h,k}; 
\cdot \text{for any} \rho\in L^2(0,1)^2, I_{h,j}(y) \rho = 0 \text{ if } y \notin \Delta_{h,j} and, \rho I_{h,k}(x) = 0 \text{ if } x\notin \Delta_{h,k}  
we obtain:
\cdot for any \rho \in L^2(0,1)^2, I_{h,j}(y)\rho(y,x)I_{h,k}(x) = 0 \text{ if } k\neq j,   k,j=1,n.

Now, for \rho satisfying the Fubini Theorem,

d_{k,j}^h := \langle T_\rho I_{h,k}, I_{h,j}\rangle = \int_0^1\int_0^1 I_{h,j}(y)\rho(y,x)I_{h,k}(x) dxdy = \int_{(j-1)h}^{jh}\int_{(k-1)h}^{kh}I_{h,j}(y) \rho(y,x)I_{h,k}(x)dxdy = ^{Fubini \text{Theorem}} \int_{(j-1)h}^{jh}I_{h,j}(y)\big[\int_{(k-1)h}^{kh} \rho(y,x)I_{h,k}(x)dx\big]dy = 0 for j\neq k.

Thus, on the finite dimension subspaces spanned by collection of indicator interval functions associated to a partition of the domain, the matrix representations of integral operators are 1-dimensional sparse matrices.

In our case of the Alcantara-Bode integral operator whose injectivity attracts the Riemann Hypothesis to be true, has the diagonal of the form mentioned before.