Upper and lower estimates for Schauder frames and atomic decompositions

Kevin Beanland, Daniel Freeman, and Rui Liu. "Upper and lower estimates for Schauder frames and atomic decompositions." Fundamenta Mathematicae 231.2 (2015): 161-188. <http://eudml.org/doc/282861>.

@article{KevinBeanland2015,
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.},
author = {Kevin Beanland, Daniel Freeman, Rui Liu},
journal = {Fundamenta Mathematicae},
keywords = {Banach spaces; frames; Schauder frames; atomic decompositions},
language = {eng},
number = {2},
pages = {161-188},
title = {Upper and lower estimates for Schauder frames and atomic decompositions},
url = {http://eudml.org/doc/282861},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Kevin Beanland
AU - Daniel Freeman
AU - Rui Liu
TI - Upper and lower estimates for Schauder frames and atomic decompositions
JO - Fundamenta Mathematicae
PY - 2015
VL - 231
IS - 2
SP - 161
EP - 188
AB - 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.
LA - eng
KW - Banach spaces; frames; Schauder frames; atomic decompositions
UR - http://eudml.org/doc/282861
ER -