Locally finite cycles of linear mappings in countable dimension

Let $n$ be a positive integer. An $n$-cycle of linear mappings is an $n$-tuple $(u_1,\dots ,u_n)$ of linear maps $u_1\in \mathrm{H}\mathrm{o}\mathrm{m}(U_1,U_2),u_2\in \mathrm{H}\mathrm{o}\mathrm{m}(U_2,U_3),\dots ,u_n\in \mathrm{H}\mathrm{o}\mathrm{m}(U_n,U_1)$, where $U_1,\dots ,U_n$ are vector spaces over a field. We classify such cycles, up to equivalence, when the spaces $U_1,\dots ,U_n$ have countable dimension and the composite $u_n\circ u_{n-1}\circ \cdots \circ u_1$ 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 $u_n\circ \cdots \circ u_1$ is locally nilpotent and the $U_i$ spaces have countable dimension, then there are bases ${\mathbf{B}}_1,\dots ,{\mathbf{B}}_n$ of $U_1,\dots ,U_n$, respectively, such that, for every $i\in \{1,\dots ,n\}$, $u_i$ maps every vector of ${\mathbf{B}}_i$ either to a vector of ${\mathbf{B}}_{i+1}$ or to the zero vector of $U_{i+1}$ (where we convene that $U_{n+1}=U_1$ and ${\mathbf{B}}_{n+1}={\mathbf{B}}_1$).