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Infinite Matrices and their Finite Sections

An Introduction to the Limit Operator Method

Marko Lindner (Autor)

Buch | Softcover

XV, 191 Seiten

2006 | 2006
Springer Basel (Verlag)
978-3-7643-7766-3 (ISBN)

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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.

Preliminaries.- Invertibility at Infinity.- Limit Operators.- Stability of the Finite Section Method.

From the reviews:

"The book under review introduces the reader to one of the central themes concerning infinite matrices: approximation by matrices of finite size. It is written for a broad audience starting with graduate students in mathematics and above. ... It is more introductory in nature and provides a very accessible summary of core themes, which are helpful in understanding properties of infinite matrices like Fredholmness, invertibility at infinity, stability and limit operators." (G. Feichtinger, Monatshefte für Mathematik, Vol. 159 (4), March, 2010)

Erscheint lt. Verlag	16.8.2006
Reihe/Serie	Frontiers in Mathematics
Zusatzinfo	XV, 191 p. 12 illus.
Verlagsort	Basel
Sprache	englisch
Maße	170 x 240 mm
Gewicht	410 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	band-dominated matrix • Finite Mathematik • Functional Analysis • limit operator • linear algebra • linear operator • Matrix • matrix theory • Matrizen • Numerical analysis • operator theory • Schrödinger operator
ISBN-10	3-7643-7766-6 / 3764377666
ISBN-13	978-3-7643-7766-3 / 9783764377663
Zustand	Neuware