Bibasic sequences in Banach lattices

TL;DR

This paper introduces and studies bibasic sequences in Banach lattices, exploring their properties, stability, and various special types, extending previous foundational work and providing new characterizations.

Contribution

It extends the theory of bibasic sequences in Banach lattices, characterizing them via the bibasis inequality and analyzing their stability and special subclasses.

Findings

Bibasic sequences can be characterized by the bibasis inequality.

They are stable under small perturbations and embeddings.

Various special types, like permutable and absolute sequences, are analyzed.

Abstract

Given a Schauder basic sequence $(x_k)$ in a Banach lattice, we say that $(x_k)$ is bibasic if the expansion of every vector in $[x_k]$ converges not only in norm, but also in order. We prove that, in this definition, order convergence may be replaced with uniform convergence, with order boundedness of the partial sums, or with norm boundedness of finite suprema of the partial sums. The results in this paper extend and unify those from the pioneering paper "Order Schauder bases in Banach lattices" by A.Gumenchuk et al. In particular, we are able to characterize bibasic sequences in terms of the bibasis inequality, a result they obtained under certain additional assumptions. We then embark on a deeper study of their properties. We show that they are independent of ambient space, stable under small perturbations, and preserved under sequentially uniformly continuous norm isomorphic embeddings. We consider several special kinds of bibasic sequences, including permutable sequences, i.e., sequences for which every permutation is bibasic, and absolute sequences, i.e., sequences where expansions remain convergent after we replace every term with its modulus. We provide several equivalent characterizations of absolute sequences, showing how they relate to bibases and to further modifications of the basis inequality. We further consider bibasic sequences with unique order expansions. We show that this property does generally depend on ambient space, but not for the inclusion of $c_0$ into $\ell_\infty$. We also show that small perturbations of bibases with unique order expansions have unique order expansions, but this is not true if "bibases" is replaced with "bibasic sequences". Finally, we consider uo-bibasic sequences, which are obtained by replacing order convergence with uo-convergence in the definition of a bibasic sequence. We show that such sequences are very common.