Renormings and symmetry properties of 1-greedy bases

Abstract

We continue the study of 1 -greedy bases initiated by Albiac and Wojtaszczyk (2006) [1]. We answer several open problems that they raised concerning symmetry properties of 1 -greedy bases and the improving of the greedy constant by renorming. We show that 1 -greedy bases need not be symmetric or subsymmetric. We also prove that one cannot in general make a greedy basis 1 -greedy as demonstrated for the Haar basis of the dyadic Hardy space H 1 ( R ) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of L p ( 1 < p < ∞ ) that makes the Haar basis 1 -unconditional and 1 -democratic. Other results in this paper clarify the relationship between various basis constants that arise in the context of greedy bases.