# Infinite Matrices and their Finite Sections

## An Introduction to the Limit Operator Method

• Marko Lindner
Book

Part of the Frontiers in Mathematics book series (FM)

1. Front Matter
Pages i-xv
2. Pages 1-49
3. Pages 51-74
4. Pages 75-148
5. Pages 149-179
6. Back Matter
Pages 181-191

### Introduction

In this book we are concerned with the study of a certain class of in?nite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their ?nite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We think of our in?nite matrices as bounded linear operators on a Banach space E of two-sided in?nite sequences. Probably the simplest case to start with 2 +? is the space E = of all complex-valued sequences u=(u ) for which m m=?? 2 |u | is summable over m? Z. m Theclassofoperatorsweareinterestedinconsistsofthoseboundedandlinear operatorsonE whichcanbeapproximatedintheoperatornormbybandmatrices. We refer to them as band-dominated operators. Of course, these considerations 2 are not limited to the space E = . We will widen the selection of the underlying space E in three directions: p • We pass to the classical sequence spaces with 1? p??. n • Our elements u=(u )? E have indices m? Z rather than just m? Z. m • We allow values u in an arbitrary ?xed Banach spaceX rather than C.

### Keywords

Matrix Operator theory Schrödinger operator band-dominated matrix functional analysis limit operator linear algebra linear operator numerical analysis

#### Authors and affiliations

• Marko Lindner
• 1
• 2
2. 2.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-7643-7767-0
• Copyright Information Birkhäuser Verlag 2006
• Publisher Name Birkhäuser Basel
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-7643-7766-3
• Online ISBN 978-3-7643-7767-0
• Series Print ISSN 1660-8046