Bases equivalent to the unit vector basis of $c_0$ or $\ell_p$

TL;DR

This paper characterizes when an unconditional basis in a Banach space is equivalent to the standard basis of c0 or l_p, based on properties of finitely supported blocks and their uniform equivalence or complementability.

Contribution

It provides a new characterization of unconditional bases equivalent to classical bases using properties of finitely supported blocks and their uniform equivalence or complementability.

Findings

Unconditional basis in a Banach space is equivalent to the basis of c0 or l_p if and only if certain block conditions hold.

Finitely supported blocks generated by a unit vector and its dual are key to this equivalence.

The characterization involves uniform equivalence or complementability of these blocks.

Abstract

We will show that an unconditional basis in a Banach space is equivalent to the unit vector basis of $c_0$ or $\ell_p$ for $1\le p < \infty$ if and only if all finitely supported blocks of the basis generated by a unit vector and its dual basis are uniformly equivalent to the basis or all such blocks are uniformly complemented.