  Black holes in anti-de Sitter spacetime provide an important testing ground
for both gravitational and field-theoretic phenomena. In particular, the study
of perturbations can be useful to further our understanding regarding certain
physical processes, such as superradiance, or the dynamics of strongly coupled
conformal field theories through the holographic principle. In this work we
continue our systematic study of the ultraviolet instabilities of black-hole
quasinormal modes, built on the characterization of the latter as eigenvalues
of a (spectrally unstable) non-selfadjoint operator and using the
pseudospectrum as a main analysis tool, extending our previous studies in the
asymptotically flat setting to Anti-de Sitter asymptotics. Very importantly,
this step provides a singularly well-suited probe into some of the key
structural aspects of the pseudospectrum. This is a consequence of the specific
features of the Schwarzschild-anti-de Sitter geometry, together with the
existence of a sound characterization by Warnick of quasinormal modes as
eigenvalues, that is still absent in asymptotic flatness. This work focuses on
such structural aspects, with an emphasis on the convergence issues of the
pseudospectrum and, in particular, the comparison between the hyperboloidal and
null slicing cases. As a physical by-product of this structural analysis we
assess, in particular, the spectral stability of purely imaginary
``hydrodynamic" modes, which appear for axial gravitational perturbations, that
become dominant when the black-hole horizon is larger than the anti-de Sitter
radius. We find that their spectral stability, under perturbations, depends on
how close they are to the real axis, or conversely how distant they are from
the first oscillatory overtone.
