Shrinking Schauder Frames and their Associated Bases

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.