Nontrivial Riemann Zeros as Spectrum

Let $\mathrm{\Lambda }(s):=\mathrm{\Gamma }(s+1)(1-2^{1-s})\zeta (s)$, and denote its set of zeros by $Z_{\mathrm{\Lambda }}:=Z_{\zeta }\cup Z_{\mathrm{p}}$, where $Z_{\zeta }$ consists of the nontrivial zeros of $\zeta (s)$ and $Z_{\mathrm{p}}$ those of the prefactor $(1-2^{1-s})$, with $s\mathrm{\ne }1$. We introduce a non-symmetric operator $R$ on $L^2([0,\mathrm{\infty }))$ with spectrum $$\sigma (R)=\{i(1\mathrm{/}2-\lambda )\mid \lambda \in Z_{\mathrm{\Lambda }}\}\mathrm{.}$$ Assuming the simplicity of all nontrivial Riemann zeros, we construct the compression $R_{Z_{\zeta }}$ of $R$ to the spectral subspace associated with $Z_{\zeta }$, and show that $R_{Z_{\zeta }}$ is intertwined with its adjoint by a positive semidefinite operator $W$; i.e., $W{R}_{Z_{\zeta }}=R_{Z_{\zeta }}^{†}W$ with $W\ge 0$. The positivity of $W$, viewed as an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces $\mathrm{\Re }(\rho )=1\mathrm{/}2$ for all $\rho \in Z_{\zeta }$, 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 $\{\mathrm{\Im }(\rho )\mid \rho \in Z_{\zeta }\}$. We further extend the framework to accommodate higher-order nontrivial Riemann zeros, should they exist, and to cover any Mellin-transformable $L$-function satisfying a functional equation.