Nontrivial Riemann Zeros as Spectrum

Let $\Lambda(s) := \Gamma(s+1) \, (1-2^{1-s}) \, \zeta(s)$ , and denote its zero set by $\mathcal{Z}_\Lambda := \mathcal{Z}_\zeta \cup \mathcal{Z}_\mathrm{p}$ , where $\mathcal{Z}_\zeta$ consists of the nontrivial zeros of $\zeta(s)$ and $\mathcal{Z}_\mathrm{p}$ of the zeros of the prefactor $( 1-2^{1-s} )$ , excluding $s = 1$ . We introduce a non-symmetric operator $\hat{\mathcal{R}}$ on a dense domain $\mathcal{D}(\hat{\mathcal{R}}) \subset L^2([0,\infty))$ with point spectrum $\displaystyle \sigma_{\mathrm{p}}(\hat{\mathcal{R}}) = \left\{ i\left(1/2- \lambda \right) \mid \lambda \in \mathcal{Z}_\Lambda \right\} \, .$ Assuming the simplicity of all nontrivial Riemann zeros, we construct the compression $\hat{\mathcal{R}}_\zeta$ of $\hat{\mathcal{R}}$ onto the spectral subspace associated with $\mathcal{Z}_\zeta$ , and show that $\hat{\mathcal{R}}_\zeta$ is intertwined with its adjoint by a positive semidefinite operator $\hat{W}$ ; i.e., $\hat{W} \, \hat{\mathcal{R}}_\zeta = \hat{\mathcal{R}}_\zeta^\dagger \, \hat{W}$ with $\hat{W} \ge 0$ . The positivity of $\hat{W}$ , viewed as an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces $\Re(\rho)=1/2$ for all $\rho \in \mathcal{Z}_\zeta$ , in accordance with the Riemann Hypothesis. Under the same positivity condition, the intertwining relation yields a self-adjoint operator whose spectrum coincides with the set $\{ \Im(\rho) \mid \rho \in \mathcal{Z}_\zeta\}$ . We further extend the framework to potential higher-order Riemann zeros and outline its generalization to any Mellin-transformable $L$ -function satisfying a reflection-type functional equation.