  It is well known that, given an equivariant and continuous (in a suitable
sense) family of selfadjoint operators in a Hilbert space over a minimal
dynamical system, the spectrum of all operators from that family coincides. As
shown recently similar results also hold for suitable families of
non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a
large class of bounded linear operator families on Banach-space valued
$\ell^p$-spaces over countable discrete groups. We also provide equality of the
pseudospectra for operators in such a family. A main tool for our analysis are
techniques from limit operator theory.
