This is an announcement for the paper "Renorming spaces with greedy bases" by S. J. Dilworth, D. Kutzarova, E. Odell, Th. Schlumprecht and A. Zsak.
Abstract: We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given $\vare>0$, so that the basis becomes $(1+\vare)$-democratic, and hence $(2+\vare)$-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is $(1+\vare)$-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1<p<\infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.
Archive classification: math.FA
Mathematics Subject Classification: 41A65, 41A44, 41A50, 46B03
Submitted from: schlump@math.tamu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1403.3777
or