Characterization of 1-quasi-greedy bases

In this note we continue the study initiated in Albiac and Wojtaszczyk (2006) of greedy-like bases in the "isometric case", i.e., in the case that the constants that arise in the context of greedy bases in their different forms are 1. Here we settle the problem to find a satisfactory characterization of 1-quasi-greedy bases in Banach spaces. We show that a semi-normalized basis in a Banach space is quasi-greedy with quasi-greedy constant 1 if and only if it is unconditional with suppression-unconditional constant 1. (C) 2015 Elsevier Inc. All rights reserved.