Complete latticeability in vector-valued sequence spaces

TL;DR

This paper demonstrates the complete latticeability of certain disjoint null sequences and tensor product differences in Banach lattices, using advanced techniques to extend known results in vector-valued sequence spaces.

Contribution

It adapts a technique to prove latticeability of specific null sequence sets and applies the mother vector method to tensor product differences in Banach lattices.

Findings

Proves latticeability of disjoint null sequences in Banach lattices.

Establishes latticeability of differences of sequence spaces involving tensor products.

Extends techniques to new classes of vector-valued sequence spaces.

Abstract

First we adjust a technique due to Jim\'enez-Rodr\'iguez to prove the complete latticeability of the set of disjoint non-norm null weakly null sequences and of the set of disjoint non-norm null regular-polynomially null sequences in Banach lattices. Then we apply the mother vector technique to prove the complete latticeability of $\lambda_\pi(E) \setminus \lambda_s(E)$, which implies the complete latticeability of $(\ell_p\widehat{\otimes}_{|\pi|}E)\setminus(\ell_p\widehat{\otimes}_{\pi}E)$, where $E$ is a Banach lattice and $1 < p < \infty$.