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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-4,
author = {Kevin Beanland and Daniel Freeman and Rui Liu},
title = {Upper and lower estimates for Schauder frames and atomic decompositions},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {161-188},
zbl = {06451638},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-4}
}
Kevin Beanland; Daniel Freeman; Rui Liu. Upper and lower estimates for Schauder frames and atomic decompositions. Fundamenta Mathematicae, Tome 228 (2015) pp. 161-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-4/