Nontrivial Riemann Zeros as Spectrum

Let $\Lambda (s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s)$, and denote its set of zeros by $Z_\Lambda := Z_\zeta \cup Z_\mathrm{p}$, where $Z_\zeta$ consists of the nontrivial zeros of $\zeta(s)$ and $Z_\mathrm{p}$ those of the prefactor $( 1-2^{1-s} )$, with $s \neq 1$. We introduce a non-symmetric operator $R$ on $L^2([0,\infty))$ with spectrum $$\sigma(R) = \left\{ i\left(1/2- \lambda \right) \mid \lambda \in Z_\Lambda \right\} \, .$$ 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., $W \, R_{Z_\zeta} = R_{Z_\zeta}^\dagger \, W$ with $W \ge 0$. The positivity of $W$, viewed as an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces $\Re(\rho)=1/2$ 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 $\{ \Im(\rho) \mid \rho \in Z_\zeta\}$. 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.