  In this paper, we analyze a virtual element method (VEM) for solving a
non-selfadjoint fourth-order eigenvalue problem derived from the transmission
eigenvalue problem. We write a variational formulation and propose a
$C^1$-conforming discretization by means of the VEM. We use the classical
approximation theory for compact non-selfadjoint operators to obtain optimal
order error estimates for the eigenfunctions and a double order for the
eigenvalues. Finally, we present some numerical experiments illustrating the
behavior of the virtual scheme on different families of meshes.
