Chandler-Wilde, Simon N. ; Lindner, Marko : Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices

Author(s):

Chandler-Wilde, Simon N.
Lindner, Marko

Title:

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

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 7, 2008

Mathematics Subject Classification:

47A53			[ Fredholm operators; index theories ]
46N20			[ Applications to differential and integral equations ]
46E40			[ Spaces of vector- and operator-valued functions ]
47B37			[ Operators on special spaces ]
47L80			[ Algebras of specific types of operators ]

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 and Silbermann in 2004 and Lindner in 2006) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang in 2002. We 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 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 $\ell^\infty$ and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the text. Results in this second half 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 Schrödinger 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 $\R^N$.

Keywords:

infinite matrices, limit operators, collective compactness, Fredholm operators, spectral theory

Language:

English

Publication time:

4 / 2008