A version of Calderón–Mityagin theorem for the class of rearrangement invariant groups

Abstract

Let l0 be the group (with respect to the coordinate-wise addition) of all sequences of real numbers x=(xk)k=1∞ that are eventually zero, equipped with the quasi-norm ‖x‖0=card{suppx}. A description of orbits of elements in the pair (l0,l1) is given, which complements (in the sequence space setting) the classical Calderón–Mityagin theorem on a description of orbits of elements in the pair (l1,l∞). As a consequence, we obtain that the pair (l0,l1) is K-monotone.