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

Springer Science & Business Media, Nov 10, 2006 - Mathematics - 191 pages
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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### Contents

 II 1 III 2 IV 3 V 4 VIII 5 XI 6 XII 8 XIII 9
 XLVIII 95 XLIX 97 L 101 LII 102 LIII 106 LIV 108 LV 109 LVI 114

 XIV 16 XV 20 XVI 27 XVII 36 XVIII 38 XIX 39 XX 40 XXII 41 XXIV 42 XXV 44 XXVI 46 XXVII 47 XXVIII 49 XXIX 50 XXX 53 XXXI 56 XXXIII 59 XXXIV 60 XXXV 63 XXXVI 74 XXXVII 75 XXXVIII 77 XXXIX 81 XLI 82 XLII 86 XLIII 89 XLIV 90 XLVI 93 XLVII 94
 LVII 116 LX 118 LXI 125 LXII 126 LXIV 128 LXV 130 LXVI 132 LXVII 133 LXVIII 134 LXIX 136 LXX 138 LXXI 139 LXXII 140 LXXIII 144 LXXIV 145 LXXV 147 LXXVI 149 LXXVII 152 LXXVIII 154 LXXIX 156 LXXXI 159 LXXXII 162 LXXXIII 166 LXXXV 172 LXXXVI 176 LXXXVIII 179 LXXXIX 181 XC 185 Copyright