  We study the properties of the entanglement spectrum in gapped
non-interacting non-Hermitian systems, and its relation to the topological
properties of the system Hamiltonian. Two different families of entanglement
Hamiltonians can be defined in non-Hermitian systems, depending on whether we
consider only right (or equivalently only left) eigenstates or a combination of
both left and right eigenstates. We show that their entanglement spectra can
still be computed efficiently, as in the Hermitian limit. We discuss how
symmetries of the Hamiltonian map into symmetries of the entanglement spectrum
depending on the choice of the many-body state. Through several examples in one
and two dimensions, we show that the biorthogonal entanglement Hamiltonian
directly inherits the topological properties of the Hamiltonian for line gapped
phases, with characteristic singular and energy zero modes. The right (left)
density matrix carries distinct information on the topological properties of
the many-body right (left) eigenstates themselves. In purely point gapped
phases, when the energy bands are not separable, the relation between the
entanglement Hamiltonian and the system Hamiltonian breaks down.
