Upper and lower estimates for schauder frames and atomic decompositions

TL;DR

This paper characterizes when Schauder frames in separable Banach spaces are shrinking or reflexive, linking these properties to associated spaces with specific basis properties, and extends finite dimensional decomposition estimates to Schauder frames.

Contribution

It establishes new equivalences between Schauder frame properties and associated space bases, extending finite dimensional decomposition theorems to Schauder frames.

Findings

A Schauder frame is shrinking iff it has an associated space with a shrinking basis.

A Schauder frame is shrinking and boundedly complete iff it has a reflexive associated space.

Separable Banach spaces with Schauder frames can have non-shrinking frames.

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.