  An operator $A$ on an $l^p$-space is called band-dominated if it can be
approximated, in the operator norm, by operators with a banded matrix
representation. The coset of $A$ in the Calkin algebra determines, for example,
the Fredholmness of $A$, the Fredholm index, the essential spectrum, the
essential norm and the so-called essential pseudospectrum of $A$. This coset
can be identified with the collection of all so-called limit operators of $A$.
It is known that this identification preserves invertibility (hence spectra).
We now show that it also preserves norms and in particular resolvent norms
(hence pseudospectra). In fact we work with a generalization of the ideal of
compact operators, so-called $\mathcal{P}$-compact operators, allowing for a
more flexible framework that naturally extends to $l^p$-spaces with
$p\in\{1,\infty\}$ and/or vector-valued $l^p$-spaces.
