Locally finite cycles of linear mappings in countable dimension

Abstract

Let n be a positive integer. An n-cycle of linear mappings is an n-tuple (u1,…,un) of linear maps u1∈Hom(U1,U2),u2∈Hom(U2,U3),…,un∈Hom(Un,U1), where U1,…,Un are vector spaces over a field. We classify such cycles, up to equivalence, when the spaces U1,…,Un have countable dimension and the composite un∘un−1∘⋯∘u1 is locally finite.When n=1, this problem amounts to classifying the reduced locally nilpotent endomorphisms of a countable-dimensional vector space up to similarity, and the known solution involves the so-called Kaplansky invariants of u. Here, we extend Kaplansky's results to cycles of arbitrary length. As an application, we prove that if un∘⋯∘u1 is locally nilpotent and the Ui spaces have countable dimension, then there are bases B1,…,Bn of U1,…,Un, respectively, such that, for every i∈〚1,n〛, ui maps every vector of Bi either to a vector of Bi+1 or to the zero vector of Ui+1 (where we convene that Un+1=U1 and Bn+1=B1).