Spectral Properties of a Noēsis Hilbert–Schmidt Operator and Their Implications for the Riemann Hypothesis

This paper introduces a compact Hilbert–Schmidt operator, the Noēsis operatorTNT_{\mathcal N}TN, constructed to encode the critical-line behaviour of theRiemann zeta function through its Mellin symbolm(t)=w(t)ζ(12+it)m(t)=w(t)\zeta(\tfrac12+it)m(t)=w(t)ζ(21+it). Using the unitary Mellin transform, theoperator is shown to be unitarily equivalent to a multiplication operator,yielding a complete spectral decomposition and explicit oscillatory integralformulae for its eigenvalues. A Mellin-conjugated family TN(s)T_{\mathcal N}^{(s)}TN(s) is analysed, and self-adjointness is proved to occur precisely onthe critical line ℜ(s)=12\Re(s)=\tfrac12ℜ(s)=21. Within this framework, the reality ofthe multiplier is equivalent to the critical-line symmetry classicallyassociated with the Hilbert–Pólya approach, placing the Riemann Hypothesisinto a compact, explicitly diagonalisable operator-theoretic setting. The Noēsis operator provides a structurally transparent and analyticallytractable framework for exploring spectral symmetry in zeta-function theory.