Two-Sided Response Reduction in Non-Selfadjoint Systems

Reduction of a self-adjoint system can often be organized around a single energetic lifting or extremal principle. In a non-selfadjoint system this symmetry is absent: the original problem and its adjoint generate different hidden-sector conditions, and a faithful effective description must preserve both. This work develops a two-sided operator framework for response reduction beyond the self-adjoint setting. Left and right elimination are treated simultaneously for block operators and for sectorial forms. The construction separates algebraic reduction from any assumption of an underlying energy minimum and provides a common language for nonnormal and dissipative systems. A Hermitian enlargement is used to compare both sides within a single geometric carrier. The evolution of left and right response data is then studied together with the correlations between them and with the stability of the reduction near loss of hidden-sector solvability. The main structural message is that non-selfadjoint reduction is intrinsically two-sided. Keeping only one lifting, one norm, or one positive diagnostic can erase information that remains present in the coupled left–right response. A complete effective description must retain their relative geometry as well as their separate magnitudes.