Upper and lower estimates for Schauder frames and atomic decompositions
Volume 231 / 2015
Fundamenta Mathematicae 231 (2015), 161-188
MSC: Primary 46B20; Secondary 41A65.
DOI: 10.4064/fm231-2-4
Abstract
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.
