Spectral Information and Structural Reconstruction for a Residual q-Difference Operator

This work investigates the structural information encoded in the spectral decomposition of residual q-difference operators. Rather than interpreting the spectrum solely as a collection of eigenvalues, the paper studies which structural properties of the underlying operator can be reconstructed exclusively from its spectral sequence. Fundamental reconstruction results are established for invertibility, kernel, image and iterative evolution, illustrating how spectral information acts as a mechanism for structural inference. This work constitutes the fourth contribution in the Residual q-Difference Operator research program.