2012/02/12 by Kevin Beanland, Daniel Freeman, Beanland, Kevin +3 · 2 citations
Mathematics · #46B20 (Primary) 41A65 (Secondary) #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1202.2492
openalex publication_date 2012/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite dimensional Banach space has a Schauder frame, then it also has a Schauder frame which is not shrinking.