Weak forms of unconditionality of bases in greedy approximation

We study a new class of bases, weaker than quasi-greedy bases, which retain their unconditionality properties and can provide the same optimality for the thresholding greedy algorithm. We measure how far these bases are from being unconditional and use this concept to give a new characterization of nearly unconditional bases.F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal y Aproximacion. M. Berasategui and S. Lassalle were supported by ANPCyT PICT-2018-04104 and CONICET PIP 1609. P. Berna by Grant PID2019-105599GB-I00 (Agencia Estatal de Investigacion, Spain) and 20906/PI/18 from Fundacion Seneca (Region de Murcia, Spain). S. Lassalle was also supported in part by CONICET PIP 0483 and PAI UdeSA 2020-2021