2020/11/19 by Richard Lechner, Lechner, Richard · 1 citation
Mathematics · #15A09 #46B07 #46B25 #46B26 #47A68 #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2011.09915
openalex publication_date 2020/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that every subsymmetric Schauder basis (ej) of a Banach space X has the factorization property, i.e. IX factors through every bounded operator T\colon X→ X with a δ-large diagonal (that is infj |⟨ Tej, ej^*⟩| ≥ δ> 0, where the (ej^*) are the biorthogonal functionals to (ej)). Even if X is a non-separable dual space with a subsymmetric weak^* Schauder basis (ej), we prove that if (ej) is non-ℓ1-splicing (there is no disjointly supported ℓ1-sequence in X), then (ej) has the factorization property. The same is true for ℓp-direct sums of such Banach spaces for all 1≤ p≤ ∞. Moreover, we find a condition for an unconditional basis (ej)j=1n of a Banach space Xn in terms of the quantities ‖e1+…+en‖ and ‖e1^*+…+en^*‖ under which an operator T\colon Xn→ Xn with δ-large diagonal can be inverted when restricted to Xσ= [ej : j∈σ] for a "large" set σ⊂ \1,…,n\ (restricted invertibility of T; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators T with a δ-large diagonal defined on any space Xn with a subsymmetric basis (ej) can be inverted on Xσ for some σ with |σ|≥ c n1/4.