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

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

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Contents

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

XIV
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XV
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XVI
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XVII
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XVIII
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XIX
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XX
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XXII
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XXIV
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XXV
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XXVI
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XXVII
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XXVIII
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XXIX
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XXX
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XXXI
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XXXIII
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XXXIV
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XXXV
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XXXVI
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XXXVII
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XXXVIII
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XXXIX
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XLI
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XLII
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XLIII
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XLIV
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XLVI
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XLVII
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LVII
116
LX
118
LXI
125
LXII
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LXIV
128
LXV
130
LXVI
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LXVII
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LXVIII
134
LXIX
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LXX
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LXXI
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LXXII
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LXXIII
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LXXIV
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LXXV
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LXXVI
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LXXVII
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LXXVIII
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LXXIX
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LXXXI
159
LXXXII
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LXXXIII
166
LXXXV
172
LXXXVI
176
LXXXVIII
179
LXXXIX
181
XC
185
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