A version of Calder\'{o}n-Mityagin theorem for the class of rearrangement invariant groups

Abstract

Let $l_0$ be the group (with respect to the coordinate-wise addition) of all sequences of real numbers $x=(x_k)_{k=1}^\infty$ that are eventually zero, equipped with the quasi-norm $\|x\|_0={\rm card}\{supp\,x\}$. A description of orbits of elements in the pair $(l_0,l_1)$ is given, which complements (in the sequence space setting) the classical Calder\'{o}n-Mityagin theorem on a description of orbits of elements in the pair $(l_1,l_\infty)$. As a consequence, we obtain that the pair $(l_0,l_1)$ is ${\mathcal K}$-monotone.