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

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.

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)