Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices

In the first half of this memoir the authors explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). They build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator $A$ (its operator spectrum). In the second half of this memoir the authors study bounded linear operators on the generalised sequence space $\ell^p(\mathbb{Z}^N,U)$, where $p\in [1,\infty]$ and $U$ is some complex Banach space. They make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator $A$ is a locally compact perturbation of the identity. Especially, they obtain stronger results than previously known for the subtle limiting cases of $p=1$ and $\infty$.