2017/09/07 by Richard Lechner, Lechner, Richard
Mathematics · #46B07 #46B25 #46B26 #60G46 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B07 #msc:46B25 #msc:46B26 #msc:60G46
paper · pdf · doi:10.48550/arxiv.1709.02297
29 pages, 2 figures
arxiv created 2017/09/07 · openalex publication_date 2017/09/07 · arxiv updated 2017/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
SL^∞ denotes the space of functions whose square function is in L^∞, and the subspaces SL^∞n, n∈ℕ, are the finite dimensional building blocks of SL^∞. We show that the identity operator ISL^∞n on SL^∞n well factors through operators T : SL^∞N→ SL^∞N having large diagonal with respect to the standard Haar system. Moreover, we prove that ISL^∞n well factors either through any given operator T : SL^∞N→ SL^∞N, or through ISL^∞N-T. Let X(r) denote the direct sum (∑n∈ℕ0 SL^∞n)r, where 1≤ r ≤ ∞. Using Bourgain's localization method, we obtain from the finite dimensional factorization result that for each 1≤ r≤ ∞, the identity operator IX(r) on X(r) factors either through any given operator T : X(r)→ X(r), or through IX(r) - T. Consequently, the spaces (∑n∈ℕ0 SL^∞n)r, 1≤ r≤ ∞, are all primary.