On Certain Spectral Operators Associated with the Riemann Zeta Function - with consequences for the Riemann Hypothesis

We develop a spectral–operator framework for the Riemann zeta function based on renormalized trace identities and distributional analysis. For a self-adjoint operator with discrete spectrum, we show that the validity of the Riemann–Weil explicit formula at the level of the renormalized trace rigidly determines the associated spectral measure. A heat-trace injectivity theorem and a prime-power decomposition via Möbius inversion are used to establish uniqueness of the arithmetic reconstruction. As a consequence, any self-adjoint realization satisfying the explicit formula must have spectrum equal to the imaginary parts of the nontrivial zeros of $\zeta(s)$, forcing all such zeros to lie on the critical line.