Meaning
The restricted admissible Grassmannian keeps the admissible projections whose difference from a chosen reference projection lies in a declared operator ideal.
With reference polarization P⁰ and a transport-stable ideal J, the restricted fiber is {P∈Gr_adm(H):P−P⁰∈J}. The ideal specifies which deviations are small enough for the chosen Fredholm comparison framework.
Example (illustrative)
If J is the compact-operator ideal, a finite-rank change of P⁰ satisfies the ideal condition. It belongs to the restricted admissible Grassmannian only if it also meets the underlying admissibility requirements.
Scope and limits
The choice of reference and ideal is part of the definition. Ideal membership alone does not establish physical admissibility.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 3.9.