Shrinking Schauder Frames and their Associated Bases

TL;DR

This paper develops methods to construct shrinking bases from Schauder frames in Banach spaces and characterizes spaces with such frames via the $w^*$-bounded approximation property.

Contribution

It introduces an explicit construction for shrinking associated bases and characterizes Banach spaces with shrinking Schauder frames.

Findings

Constructed explicit shrinking associated bases from Schauder frames.

Proved domination of uncountably many incomparable shrinking bases.

Characterized spaces with shrinking Schauder frames via the $w^*$-bounded approximation property.

Abstract

For a Banach space $X$ with a shrinking Schauder frame $(x_i,f_i)$ we provide an explicit method for constructing a shrinking associated basis. In the case that the minimal associated basis is not shrinking, we prove that every shrinking associated basis of $(x_i,f_i)$ dominates an uncountable family of incomparable shrinking associated bases of $(x_i,f_i)$. By adapting a construction of Pe{\l}czy{\'n}ski, we characterize spaces with shrinking Schauder frames as spaces having the $w^*$-bounded approximation property.