Essential pseudospectra and essential norms of band-dominated operators

An operator A   on an lplp-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 PP-compact operators, allowing for a more flexible framework that naturally extends to lplp-spaces with p∈{1,∞}p∈{1,∞} and/or vector-valued lplp-spaces.