Besselian Schauder Frames and the Structure of Banach Spaces

Abstract

Schauder bases are fundamental tools for analyzing the structure of Banach spaces. In this work, we show that Besselian Schauder frames (BSF) play a similar role in certain contexts. BSF are a new class of Schauder frames, lying between unconditional and general frames. We first prove that every unconditional Schauder frame (USF) is BSF, but the reverse implication is false. Specifically, we extend several well-known results of Karlin and James to Banach spaces with BSF, particularly to those with USF. We prove that many classical Banach spaces do not admit BSF, and in particular, do not admit USF. Before establishing these results, for every Banach space $E$ with a finite dimensional decomposition, we provide an explicit method to construct a Schauder frame for $E$. In particular, Szarek's Banach space has a Schauder frame, which famously lacks a Schauder basis. This finding provides strong motivation for extending classical Schauder basis theory to the framework of Schauder frames.