Critical ideals and covering widths of ideal Schauder bases

An ideal Schauder basis of a Banach space expands every vector as a series whose partial sums Sₙx converge along an ideal of subsets of ℕ, rather than along the cofinite sets. Its critical ideal is the least ideal with this property: it is generated by the finite sets and the error sets {n: ‖x − Sₙx‖ ≥ ε}. We determine the possible descriptive complexities of critical ideals and study when they are P-ideals. If the coordinate functionals are continuous, the critical ideal is complete for exactly one of the classes Fσ, Fσδ and Fσδσ, and all three cases occur; it is Fσ if and only if it is a P⁺-ideal, and it is always Fσ in a reflexive space. Since analyticity of a critical ideal forces continuity of the coordinates, an analytic critical ideal is Borel. A critical ideal need not be a P-ideal, even for a basis of a separable Hilbert space; the P-property is equivalent to a uniform bound on the number of vectors whose error sets are needed to cover a member of the ideal. Every proper analytic P-ideal is isomorphic to the critical ideal of a basis of ℓ₁ ⊕ c₀, but there is a pathological Fσ P-ideal on ℕ that is not the critical ideal of any basis of any Banach space. These results answer Questions 11.3 and 11.4 of Kwela and Swaczyna (2025) negatively, and their Question 11.2 negatively as stated and positively up to isomorphism. Elementary coordinatewise examples also illustrate a negative answer to their Question 11.1; the dense-span ℓ₁ existence phenomenon is already implicit in classical non-strong Markushevich basis constructions. We also show that every proper nonpathological Fσ P-ideal I is the critical ideal of a basis of a reflexive space and satisfies I = Fin(ψ) = Exh(ψ) for a nonpathological lower semicontinuous submeasure ψ; this answers a question of Lopez-Abad, Olmos-Prieto and Uzcátegui-Aylwin (2025).

GitHub repository: https://github.com/eulogica/critical-ideals-covering-widths