Weak Greedy Algorithms and the Equivalence Between Semi-greedy and Almost Greedy Markushevich Bases

Abstract

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