A characterization of Schauder frames which are near-Schauder bases

TL;DR

This paper characterizes when Schauder frames in Banach spaces are essentially near-Schauder bases, linking this property to the absence of $c_0$ copies in associated sequence spaces and kernels.

Contribution

It provides a characterization of near-Schauder bases among Schauder frames using minimal-associated sequence spaces and reconstruction operators.

Findings

A Schauder frame is a near-Schauder basis iff the kernel of its minimal-associated reconstruction operator contains no $c_0$.

A Schauder frame in a space with no $c_0$ is a near-Schauder basis iff its associated sequence space contains no $c_0$.

The kernel of the reconstruction operator is finite dimensional, with dimension equal to the basis excess.

Abstract

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of $c_0$. In particular, a Schauder frame of a Banach space with no copy of $c_0$ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of $c_0$. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.