Besselian Schauder Frames and the Structure of Banach Spaces
Rafik Karkri
TL;DR
The paper investigates Besselian Schauder frames ($BSF$) as a flexible extension of Schauder frame theory to Banach spaces, establishing that every $USF$ is a $BSF$ but not conversely. It generalizes key results from Karlin and James to spaces endowed with $BSF$, revealing precise characterizations of shrinkage, bounded completeness, and reflexivity in terms of the absence of subspaces isomorphic to $ extit{$oldsymbol{\oldmathell}_1$}$ or $c_0$, and showing how these properties interplay with the dual and bidual. A constructive framework is developed to build frames from any finite dimensional decomposition ($FDD$), enabling explicit frames even when a Schauder basis fails, as demonstrated by Szarek's space. The work also delineates which classical Banach spaces admit $BSF$ or $USF$ and discusses a strict hierarchy among frames, bases, and $BSF$, with implications for weak sequential completeness, property (u), and reflexivity in 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. 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.