  Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on
some separable infinite-dimensional Hilbert space and $V$ be a compact
non-selfadjoint perturbation. We relate the regularity properties of $V$ to
various spectral properties of the perturbed operator $H_0+V$. The structure of
the discrete spectrum and the embedded eigenvalues are analysed jointly with
the existence of limiting absorption principles in a unified framework. Our
results are based on a suitable combination of complex scaling techniques,
resonance theory and positive commutators methods. Various results scattered
throughout the literature are recovered and extended. For illustrative
purposes, the case of the one-dimensional discrete Laplacian is emphasized.
