We study interpolation of the measure of noncompactness of bilinear operators. We prove a result of a general nature which states that for a large class of interpolation functors preserving bilinear interpolation estimates of measures of noncompactness of interpolated linear operators between Banach couples can be lifted to bilinear operators. As an application, we show that the measure of noncompactness of bilinear operators behave well under the real method of interpolation.

Given 0 < s ≤ 1 and ψ an s-convex function, s – ψ -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s – ψ -spaces. For these spaces, new measures of noncompactness are also defined, related to the Hausdorff measure of noncompactness. As an application, compact sets in s – ψ -interpolation spaces of a quasi-Banach couple are studied.

This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of Banach spaces generated by abstract methods of interpolation, in the sense of Aronszajn and Gagliardo. To get these results, we prove a key one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an extra approximation property. We give applications to general cases, including Peetre’s method and the general real interpolation methods.

In this work, we show that an n-dimensional sublattice ${\Lambda }^{\prime }=m\Lambda$ of an n-dimensional lattice $\Lambda$ induces a $G={\mathbb{Z}}_m^n$ tessellation in the flat torus $T_{{\beta }^{\prime }}={\mathbb{R}}^n/{\Lambda }^{\prime }$, where the group G is isomorphic to the lattice partition $\Lambda /{\Lambda }^{\prime }$. As a consequence, we obtain, via this technique, toric codes of parameters $\left[[2m^2,2,m]\right]$, $\left[[3m^3,3,m]\right]$ and $\left[[6m^4,6,m^2]\right]$ from the lattices ${\mathbb{Z}}^2$, ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^4$, respectively. In particular, for $n=2$, if ${\Lambda }_1$ is either the lattice ${\mathbb{Z}}^2$ or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell ${\mathcal{P}}_0^{\prime }$ of each hexagonal sublattice ${\Lambda }^{\prime }$ of hexagonal lattices $\Lambda$, using either the fundamental cell ${\mathcal{P}}_0$ or the Voronoi cell ${\mathcal{V}}_0$. These partitions allow us to present new classes of toric codes with parameters $\left[[3m^2,2,m]\right]$ and color codes with parameters $\left[[18m^2,4,4m]\right]$ in the flat torus from families of hexagonal lattices in ${\mathbb{R}}^2$.