Essential pseudospectra and essential norms of band-dominated operators

TL;DR

This paper investigates the essential pseudospectra and norms of band-dominated operators on $l^p$-spaces, showing that their essential properties are preserved under limit operator identification within a generalized compact operator framework.

Contribution

It demonstrates that the essential pseudospectra and norms of band-dominated operators are preserved through limit operators using a generalized $ ext{P}$-compact framework, extending previous results.

Findings

Preservation of essential pseudospectra under limit operator identification.

Extension of the framework to $l^p$-spaces with $p eq 2$ and vector-valued spaces.

Generalization from compact to $ ext{P}$-compact operators.

Abstract

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.