Banach spaces with a unique greedy basis

Abstract

The purpose of this article is to undertake an in-depth study of the properties of existence and uniqueness of greedy bases in Banach spaces. We show that greedy bases fail to exist for a range of neo-classical spaces within the family of Nakano and Orlicz sequence spaces and find the first-known cases of non-trivial spaces (i.e., different from c 0 , ℓ 1 , and ℓ 2) with a unique greedy basis. The variety and nature of those examples evince that a complete classification of Banach spaces with a unique greedy basis cannot be expected.

On the permutative equivalence of squares of unconditional bases

“…Our results have applications also in connection with the topic of uniqueness of greedy basis (see [3]). Recall that a basis (x n ) n∈N is said to be democratic if…”

“…To the best of our knowledge, the list of known Banach spaces with a unique unconditional basis up to a permutation which in addition is democratic reduces to 1 , c 0 , 2 , T and T (2) . A warning is in order here: before the alert reader begins to do their math, we advance that the unit vector system of Bourgin-Nakano spaces is not democratic unless it is equivalent to the unit vector system of p for some p (see [3,Theorem 3.10]). Corollary 6.2 allows us to enlarge this scant list with one of the examples that we highlight from Corollary 6.2.…”

“…This problem was also settled in the aforementioned article by Gowers and Maurey. Indeed, the authors constructed in [18] a Banach space X with X X 3 but X X 2 . Then, if we put Y = X 2 , we have that X is isomorphic to a complemented subspace of Y , that Y is isomorphic to a subspace of X, that X 2 Y 2 , and that X Y .…”

Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces

“…Then, the unit vector system is a symmetric basis of d(w, 1). Proposition 3.1 (See [9, Theorems 2.4.14 and 2.5.10 and Corollary 2.4.26]; see also [2,Section 6]). Let w = (w j ) ∞ j=1 ∈ W, let s denote its primitive weight, and let w −1 = (1/w j ) ∞ j=1 be its inverse weight.…”