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Fredholm Theory and Stable Approximation of Band Operators and Their Generalisations

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Fredholm Theory and Stable Approximation of Band Operators and Their Generalisations Fredholm Theory and Stable Approximation of Band Operators and Their Generalisations Habilitationsschrift zur Erlangung des akademischen Grades Doctor rerum naturalium habilitatus (Dr. rer. nat. habil.) vorgelegt der Fakult¨at f¨urMathematik der Technischen Universit¨at Chemnitz von Dr. Marko Lindner, geboren am 20. Oktober 1973 in Zschopau. Chemnitz, den 23. Februar 2009 Contents 1 Introduction 7 2 Preliminaries 15 2.1 Numbers and Vectors ......................... 15 2.2 Banach Spaces and Banach Algebras ................ 16 2.3 Linear Operators ........................... 17 2.4 Spaces of Sequences .......................... 18 2.5 An Approximate Identity ....................... 19 2.6 Different Topologies on E ...................... 21 2.6.1 The Norm Topology ..................... 21 2.6.2 The Local Topology ..................... 22 2.6.3 The Strict Topology ..................... 22 2.7 Comments and References ...................... 25 3 Classes of Operators 27 3.1 Continuous Operators on (E, s) ................... 27 3.2 Compact Operators and Generalisations .............. 29 3.2.1 Compact Operators on (E, k · k) and Generalisations ... 29 3.2.2 Restrict and Extend Operators to and from E0 ...... 35 3.2.3 Compact Operators on (E, s) and Generalisations ..... 36 3.2.4 Algebraic Properties ..................... 39 3.3 Duality: Adjoint and Preadjoint Operators ............. 41 3.3.1 Definitions ........................... 41 3 4 CONTENTS 3.3.2 Duality in Action: Fredholm Operators on E∞ ....... 43 3.4 Operator Chemistry ......................... 48 3.5 Notions of Operator Convergence .................. 50 3.6 Infinite Matrices ............................ 56 3.6.1 Inducing Matrix vs. Representation Matrix ......... 56 3.6.2 Our Operator Classes from the Matrix Point of View ... 57 3.7 Band- and Band-Dominated Operators ............... 60 3.7.1 Measures of Off-Diagonal Decay ............... 60 3.7.2 Characterisations of BO(E) and BDO(E) ......... 62 3.7.3 The Wiener Algebra ..................... 65 3.8 Comments and References ...................... 71 4 Key Concepts 73 4.1 Fredholmness and Invertibility at Infinity .............. 73 4.1.1 Fredholmness Revisited .................... 73 4.1.2 P-Fredholmness and Invertibility at Infinity ........ 74 4.1.3 Invertibility at Infinity in BDO(E) ............. 77 4.1.4 Invertibility at Infinity vs. Fredholmness .......... 78 4.2 Collectively Compact Operator Theory ............... 82 4.2.1 Collective Compactness: A Short Intro ........... 82 4.2.2 The Chandler-Wilde/Zhang Approach in our Case ..... 83 4.3 Limit Operators ............................ 86 4.4 Collective Compactness Meets Limit Operators .......... 91 4.5 Comments and References ...................... 95 5 Band-Dominated Fredholm Operators 99 5.1 More Preliminaries .......................... 99 5.1.1 Rich Band-Dominated Operators .............. 99 5.1.2 When is T (K) Uniformly Montel? .............. 100 ∗ 5.1.3 The Operator Spectra of A and A|E0 ............ 104 5.2 Main Theorems ............................ 106 CONTENTS 5 5.2.1 The General Case, E = Ep(X) ................ 106 5.2.2 The Case E = E∞(X) .................... 109 5.2.3 The One-Dimensional Case, E = `∞(Z,X) ......... 110 5.3 Fredholmness in the Wiener Algebra ................ 116 5.4 Limit Operators and the Fredholm Index .............. 121 5.5 Different Types of Diagonal Behaviour ............... 123 5.5.1 Periodic and Almost Periodic Operators .......... 123 5.5.2 Slowly Oscillating Operators ................. 134 5.5.3 Pseudoergodic Operators ................... 136 5.6 Comments and References ...................... 140 6 Stable Approximation of Infinite Matrices 141 6.1 Approximation Methods ....................... 141 6.1.1 Definitions ........................... 141 6.1.2 Applicability vs. Stability .................. 144 6.1.3 Stability vs. Invertibility at Infinity ............. 145 6.2 The Finite Section Method ...................... 149 6.3 Strategy 1: Passing to Subsequences ................ 151 6.3.1 The Philosophy ........................ 152 6.3.2 The Stability Theorem for Subsequences of the FSM ... 153 6.3.3 Starlike Sets Instead of Convex Polytopes ......... 156 6.3.4 Examples ........................... 157 6.3.5 Some Specialities in the Case N = 1 ............ 161 6.3.6 On the Uniform Boundedness Condition in (iii) ...... 164 6.4 Strategy 2: Rectangular Finite Sections ............... 165 6.5 Comments and References ...................... 167 7 Applications 169 7.1 Fredholm and Spectral Studies .................... 169 7.1.1 Discrete Schr¨odinger Operators ............... 169 7.1.2 A Bidigonal Random Matrix ................. 177 6 CONTENTS 7.1.3 A Tridigonal Random Matrix ................ 184 7.1.4 A Class of Integral Operators ................ 199 7.2 Approximation Methods ....................... 204 7.2.1 The FSM for Slowly Oscillating Operators ......... 204 7.2.2 A Special Finite Section Method for BC .......... 205 7.2.3 Boundary Integral Equations on Unbounded Rough Surfaces209 7.2.4 Rough Surface Scattering in 3D ............... 227 Bibliography 240 Index 255 Theses 259 Chapter 1 Introduction This text is written to summarise my research activities over the last years. Parts of this research have been summarised before in the monographs “Infinite Matri- ces and Their Finite Sections: An Introduction to the Limit Operator Method” [106] in 2006 and “Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices” [39] in 2008, the first single-authored and the lat- ter co-authored by Simon Chandler-Wilde from the University of Reading, UK. The current text is therefore naturally a hybrid between these two monographs enriched with more recent results, both published and so far unpublished ones. Classes of infinite matrices. The main theme of the body of work to be presented here is the Fredholm theory of bounded linear operators generated by a class of infinite matrices (aij) that are either banded or have certain decay properties as one goes away from the main diagonal. In the simplest case to be considered, the indices i and j run through the integers Z and the matrix entries aij are complex numbers. Under certain conditions on the entries aij, the matrix (aij) then induces, via matrix-vector multiplication, a linear operator A on the space E = `2(Z, C) of two-sided infinite complex sequences with absolutely summable squares. We call A a band operator if (aij) is a band matrix with uniformly bounded entries, and we call it a band-dominated operator if it is the limit, in the operator norm on E, of a sequence of band operators. The set of all operators A whose matrix (aij) has a summable off-diagonal decay, that means X δk < ∞ with δk = sup |aj+k,k|, j∈Z k∈Z is called the Wiener algebra. This is a particularly nice class of bounded linear operators containing all band operators. A matrix with this property generates a bounded, and in fact band-dominated, linear operator on all spaces `p(Z, C) with p ∈ [1, ∞]. 7 8 CHAPTER 1. INTRODUCTION Fredholmness and limit operators. For the Fredholm theory of a band- dominated operator A, the values of any finite collection of matrix entries aij (say {aij : −100 ≤ i, j ≤ 100}) is completely irrelevant as changing these values only perturbs A by a finite rank operator. It is therefore clear that the key to the Fredholm properties of A is to understand the behaviour of the entries aij as (i, j) → ∞. Since we generally do not assume convergence of our matrix entries at infinity, this asymptotic behaviour1 cannot be reflected by a single number; it has much more complexity and needs a more involved storage device: the so- called limit operators of A, each of which is an operator on E itself. Precisely, with every sequence h = (h1, h2, ...) ⊂ Z going to infinity for which the sequence of matrices (ai+hk, j+hk )i,j, k = 1, 2, ... (1.1) converges entrywise as k → ∞, we associate the operator that is induced by the limit of this matrix sequence (1.1) and call it the limit operator of A with respect to h, denoted by Ah. The collection of all limit operators of A is denoted by σop(A); it carries all the information about the Fredholm properties of A. In fact, one can show [96, 140] that a band-dominated operator A is a Fredholm operator, in which case its Fredholm index can be calculated [139] by looking at two members of σop(A), if and only if all members of σop(A) are invertible and their inverses are uniformly bounded. If A is even in the Wiener algebra then this uniform boundedness condition can be dropped, yielding the formula [ specessA = spec Ah (1.2) op Ah∈σ (A) for the essential spectrum of A in terms of the spectra of its limit operators. More general spaces. Many of these ideas generalise to the case when A acts on E = `p(ZN ,X), where p ∈ [1, ∞], N is a natural number and X is a complex Banach space. The elements of E are functions ZN → X, thought of as p N generalised sequences (uk)k∈Z with values in X, such that kukkX is summable over ZN . In this setting we are interested in band-dominated operators A that are N induced by a matrix (aij)i,j∈Z with operator entries aij : X → X. If dim X = ∞ then the Fredholm theory of A changes; now a single matrix entry aij, being an infinite-dimensional operator itself, can change the Fredholm properties of A. 1 What we call “asymptotic behaviour” here is actually the coset of the matrix (aij) modulo the ideal K, where K is the closure, in the operator norm, of the set of all such matrices with only finitely many non-zero entries. In the setting of E = `2(Z, C), this ideal K exactly corresponds with the compact operators on E, and the Fredholm property of A is equivalent to the invertibility of the coset (aij) + K in a suitable factor algebra. 9 One can however prove an analogous theorem as before: Under an additional condition on the band-dominated operator A, the coset (aij)+K is invertible, in which case we now call A invertible (1.3) at infinity, if and only if all limit operators of A are invertible with their inverses uniformly bounded.
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