RH Toolkit 1: Deferral Operators on ℓ² and Their Continuous Realization

We develop a functional–analytic framework for a class of linear operators acting on discrete Hilbert sequence spaces, referred to as deferral operators. These operators are defined via admissible kernels on  and are shown to be bounded and compact, and self-adjoint under a natural symmetry condition on the kernel. As a consequence, compact self-adjoint deferral operators admit a standard spectral decomposition with discrete real eigenvalues accumulating only at zero.

A systematic correspondence between discrete deferral operators and integral operators on continuous  spaces is established through explicit kernel constructions and unitary equivalence. This discrete-to-continuous realization preserves compactness, self-adjointness, and nonzero spectral data, allowing discrete kernel-based operators to be studied within continuous functional-analytic settings without loss of spectral information.

In addition, several intrinsic structural invariances of deferral operators are identified and classified, including translation invariance and Fourier diagonalization, a complete gauge classification of Hermitian phase structure via cycle flux, normalization via constant eigenvectors, and stability under composition within the Hilbert–Schmidt ideal. Taken together, these results provide an operator–theoretic framework for analyzing discrete operators with localized kernels and their continuous realizations.