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 and Silbermann [85] and Lindner [63]) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang [26]. 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 p (Z N , U ), where p ∈ [1, ∞] 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 ∞. Our tools in this study are the results from the first half of the text and an exploitation of the partial duality between 1 and ∞ 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 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 Schrödinger operators (both selfadjoint 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 .

Extreme Cases of Limit Operator Theory on Metric Spaces

“…We collect several background notions in operator algebra theory and establish our settings in this section. See [3,10] for more information.…”

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…The third generation works in cases where σ ess (A) can have gaps. This approach appeared (more or less independently) in Chandler-Wilde-Lindner [86,87], Georgescu-Iftimovici [186], Last-Simon [411,412], Mǎntoiu [440] and Rabinovich [488]. Perhaps the cleanest result from [412] With the HVZ theorem in hand, one can easily carry Kato's argument to its logical conclusion Theorem 11.9 (Simon [569]) Let H be an N -body Hamiltonian with center of mass removed.…”

Spectral Properties of Non-Unitary Band Matrices

“…For example, consider U (C) on K given by (7). For any C ∈ U (4), x ⊗ τ |U (C) 2n+1 x ⊗ τ = 0 for any n ∈ Z and x ⊗ τ ∈ K, because U (C) is offdiagonal.…”

“…However, the spectral theory of non-self-adjoint operators has been the object of many works, in various setups of regimes, as can be seen from the papers [6][7][8][10][11][12]16,31,32,34] and references therein. In particular, several analyses of non-self-adjoint operators focus on tri-diagonal operators, when expressed in a certain basis; see [7,8,10,11]. Since Jacobi matrices provide generic models of self-adjoint operators, it is quite natural to deal with non-self-adjoint tri-diagonal matrices which are deformations of Jacobi matrices.…”