Chapter 1. Introduction: Band-dominated operators ............... 1
Chapter 2. Stability ............................................ 5
2.1. Predholm band-dominated operators and their indices ...... 6
2.2. Algebras of matrix sequences ............................. 7
2.3. Stability vs. Fredholmness ............................... 9
2.4. Stability of the finite sections method .................. 9
2.5. Band-dominated operators with slowly oscillating
coefficients ............................................ 11
2.6. The role of the index ................................... 13
2.7. The C*-algebra of the finite sections method ............ 14
2.8. Another perspective on localization ..................... 19
Chapter 3. Stable regularizability ............................. 23
3.1. Moore-Penrose invertibility in C*-algebras .............. 23
3.2. Stable regularizability vs. Moore-Penrose
invertibility ........................................... 25
3.3. Stable regularizability of the finite sections method ... 26
Chapter 4. Compactness ......................................... 30
4.1. Compact sequences ....................................... 30
4.2. Characterization via singular values .................... 32
4.3. Central rank characterizations .......................... 35
4.4. Minimal and maximal characterizations ................... 38
4.5. Compact sequences in S(N) and S(Z) ...................... 41
Chapter 5. Fredholmness ........................................ 47
5.1. Fredholm sequences ...................................... 47
5.2. Fredholmness and stability .............................. 50
5.3. Fredholmness and stable regularizability ................ 51
5.4. Fredholmness of the finite sections method .............. 52
5.5. Slowly oscillating coefficients ......................... 54
Chapter 6. Essential fractality ................................ 57
6.1. Fractality of quotient maps ............................. 57
6.2. J-fractal algebras ...................................... 59
6.3. Essential fractality and singular values ................ 61
6.4. Essential fractality of the finite sections algebras .... 62
Chapter 7. Applications ........................................ 64
7.1. Approximation numbers ................................... 64
7.2. Rank-preserving discretizations ......................... 65
7.3. Arveson dichotomy: band-dominated operators ............. 67
7.4. Arveson dichotomy: the general setting .................. 70
7.5. Essential spectra ....................................... 71
7.6. Essential pseudospectra ................................. 73
7.7. Pseudomodes ............................................. 76
7.8. Determinants ............................................ 80
Bibliography ................................................... 83
Index .......................................................... 86
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