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.
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A G L(E aop(A approximation method Banach algebra Banach space Banach subalgebra band matrix band operator band-dominated operators BDOp bounded bounded linear operator bounded set clearly compact compact operators Consequently continuous functions convolution operators Corollary deﬁned Deﬁnition diagonal diﬀerent dimX discrete equation equivalent Example exists ﬁnite section method ﬁrst Fredholm operator function f G BUC hence holds implies index set integral operator invertible at inﬁnity isometry Lemma limit operator limit operators Ah lp(Zn,X mapping maximal ideal Moreover multiplication operator norm operator norm operator of Mb operator spectrum P-convergence periodic functions Proof Proposition pseudo-ergodic respectively sequence h shift shows stable stacked operator strong convergence Subsection suﬃcient suppose take an arbitrary tending to inﬁnity Toeplitz operators uniformly invertible