Infinite Matrices and their Finite Sections An Introduction to the Limit Operator Method /

• Main Author: Lindner, Marko (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Basel : Birkhäuser Basel, 2006.
• Series: Frontiers in Mathematics,
• Subjects: Mathematics.; Matrix theory.; Algebra.; Functional analysis.; Numerical analysis.; Functional Analysis.; Linear and Multilinear Algebras, Matrix Theory.; Numerical Analysis.
• Physical Description: XV, 191 p. 12 illus. online resource.
• ISBN: 9783764377670
• ISSN: 1660-8046

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.