On the refined properties of uniformly quasi-greedy bases

Abstract: In this paper, we study some refined properties of uniformly quasi-greedy bases. In particular, we characterize the stability of uniformly quasi-greediness under scalings with bounded quotient and show that unconditionality is sufficient to guarantee this property. For the "isometric" case, i.e., when the uniformly quasi-greedy constant is $1$, we prove that 1-uniformly quasi-greediness implies disjointness if the underlying norm is strictly monotone. We also introduce a stronger notion of quasi-greediness by requiring uniform order boundedness of the greedy sums over greedy orderings. The characterization and properties of such bases are derived along the lines of uniformly quasi-greedy bases.