  Spectral functions of non-Hermitian Hamiltonians can reveal the existence of
topologically non-trivial line gaps and the associated topological edge modes.
However, the computation of spectral functions in a non-Hermitian many-body
system remains an open challenge. Here, we put forward a numerical approach to
compute spectral functions of a non-Hermitian many-body Hamiltonian based on
the kernel polynomial method and the matrix-product state formalism. We show
that the local spectral functions computed with our algorithm reveal
topological spin excitations in a non-Hermitian spin model, faithfully
reflecting the non-trivial line gap topology in a many-body model. We further
show that the algorithm works in the presence of the non-Hermitian skin effect.
Our method offers an efficient way to compute local spectral functions in
non-Hermitian many-body systems with tensor-networks, allowing to characterize
line gap topology in non-Hermitian quantum many-body models.
