Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases

BERASATEGUI, MIGUEL; LASSALLE, SILVIA

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS

BIRKHAUSER BOSTON INC

Año: 2023 vol. 29

1069-5869

We introduce and study the notion of weak semi-greedy systems—which is inspired inthe concepts of semi-greedy and branch semi-greedy systems and weak thresholdingsets-, and prove that in infinite dimensional Banach spaces, the notions of semi-greedy,branch semi-greedy, weak semi-greedy, and almost greedy Markushevich bases areall equivalent. This completes and extends some results from (Berná in J Math AnalAppl 470:218–225, 2019; Dilworth et al. in Studia Math 159:67–101, 2003; J FunctAnal 263:3900–3921, 2012).We also exhibit an example of a semi-greedy system thatis neither almost greedy nor a Markushevich basis, showing that the Markushevichcondition cannot be dropped from the equivalence result. In some cases, we obtainimproved upper bounds for the corresponding constants of the systems.