Compositions of projections in Banach spaces and relations between approximation properties

TL;DR

This paper establishes a precise condition involving norms of projection compositions that characterizes when a Banach space with a finite dimensional decomposition lacks the cpie-property, advancing understanding of approximation properties.

Contribution

It provides a necessary and sufficient condition based on projection composition norms for the absence of the cpie-property in Banach spaces with finite dimensional decompositions.

Findings

Characterization of Banach spaces without the cpie-property

Condition involving norms of compositions of projections

Insight into approximation properties in Banach spaces

Abstract

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\pi$-property in terms of norms of compositions of projections is found.