A Functional Characterization of Almost Greedy and Partially Greedy Bases in Banach Spaces

Pablo Manuel Berná,Diego Mondéjar

doi:10.3390/math9151827

Abstract

In 2003, S. J. Dilworth, N. J. Kalton, D. Kutzarova and V. N. Temlyakov introduced the notion of almost greedy (respectively partially greedy) bases. These bases were characterized in terms of quasi-greediness and democracy (respectively conservativeness). In this paper, we show a new functional characterization of these type of bases in general Banach spaces following the spirit of the characterization of greediness proved in 2017 by P. M. Berná and Ó. Blasco.

Highlights

Introduction and BackgroundCitation: Berná, P.M.; Mondéjar, D
All the characterizations were given under the assumption of unconditionality and one of the democracy-like properties, but in [8], we found a new and interesting property that is very useful to give a new characterization of greediness
We focused our attention in a closed inequality to characterize the so-called almost greedy and partially greedy bases

Summary

Introduction and Background

The basis is democratic (or symmetric for the largest coefficients) and quasi-greedy if and only if the basis has Property (F). F ≤ Cq (1 + (1 + Cq )∆d ); If B is ∆-symmetric for the largest coefficients and Cq -quasi-greedy, the basis has Property (F) with constant:. Since Property (F) implies quasi-greediness with constant Cq ≤ F by (1), if we take the element h := f + 1ηB with k fk∞ ≤ 1, we have:. (3)Assume that B is Cq -quasi-greedy and ∆d -democratic, and take f , g ∈ X f in with f · g = 0, infn∈supp( g) |en∗ ( g)| ≥ k fk∞ , A ∩ B = ∅, | A| ≤ | B| < ∞, and supp( f + g) ∩.

Proof of Theorem 5

Proof of Theorem 6