Near Self-Adjointness and Phase Structure in Hilbert–Schmidt Operators Arising from the Riemann Zeta Function

This paper investigates the spectral structure of a compact Hilbert–Schmidt operator arising from the Mellin diagonalisation of the Riemann zeta function on the critical line. Building on earlier work, the operator is shown to be compact and normal, admitting an explicit decomposition into commuting self-adjoint and skew-adjoint components determined by the real and imaginary parts of its Mellin multiplier. The analysis isolates the precise operator-theoretic obstruction to self-adjointness, demonstrating that all deviations from Hermitian symmetry are governed by a single skew-adjoint Hilbert–Schmidt component associated with the phase of ζ(12+it)\zeta(\tfrac12+it)ζ(21+it). This yields a natural and quantitative notion of near self-adjointness and leads to explicit bounds on the resulting spectral distortion. Rather than attempting to enforce self-adjointness by construction, the compact Mellin-diagonal framework provides a transparent setting in which structural symmetry and analytic obstruction are cleanly separated. The results clarify the role of phase effects in Hilbert–Pólya-type operator models . This work contributes to a broader compact operator programme aimed at understanding structural and spectral aspects of operators derived from arithmetic functions within a fully controlled functional-analytic setting.