Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to $c_0$

TL;DR

This paper characterizes when Hilbert space frames contain Riesz bases and identifies Banach spaces lacking subspaces isomorphic to c_0, leading to improvements in frame theory.

Contribution

It provides new conditions for Hilbert space frames to contain Riesz bases and characterizes Banach spaces without c_0 subspaces, advancing frame and Banach space theory.

Findings

A Hilbert space frame contains a Riesz basis if every subfamily is a frame for its span.

New characterization of Banach spaces with no c_0 subspace.

Improvement of Holub's theorem on frames with Riesz bases plus finitely many elements.

Abstract

We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements.