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

Abstract: In the first half of this text we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch & Silbermann and Lindner) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang. We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator $A$. In the second half of this text we study bounded linear operators on the generalised sequence space $\ell^p(\Z^N,U)$, where $p\in [1,\infty]$ and $U$ is some complex Banach space. We 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, we obtain stronger results than previously known for the subtle limiting cases of $p=1$ and $\infty$. Our tools in this study are the results from the first half of the text and an exploitation of the partial duality between $\ell^1$ and $\ell^\infty$. Results in this second half of the text include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schr\"odinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on $\R^N$.