Preface ...................................................... vii
Preface to the Second Edition ................................. ix
I. Banach Algebras
1. Basic Concepts ......................................... 1
2. Commutative Banach algebras ........................... 18
3. Approximate point spectrum in commutative
Banach algebras ....................................... 26
4. Permanently singular elements and removability
of spectrum ........................................... 34
5. Non-removable ideals .................................. 42
6. Axiomatic theory of spectrum .......................... 51
7. Spectral systems ...................................... 58
8. Basic spectral systems in Banach algebras ............. 68
Comments on Chapter I ..................................... 72
II. Operators
9. Spectrum of operators ................................. 85
10. Operators with closed range ........................... 97
11. Factorization of vector-valued functions ............. 106
12. Kato operators ....................................... 117
13. General inverses and Saphar operators ................ 130
14. Local spectrum ....................................... 135
Comments on Chapter II ................................... 143
III. Essential Spectrum
15. Compact operators .................................... 149
16. Fredholm and semi-Fredholm operators ................. 155
17. Construction of Sadovskii/Buoni, Harte, Wickstead .... 164
18. Perturbation properties of Fredholm and
semi-Fredholm operators .............................. 169
19. Essential spectra .................................... 172
20. Ascent, descent and Browder operators ................ 178
21. Essentially Kato operators ........................... 187
22. Classes of operators defined by means of kernels
and ranges ........................................... 197
23. Semiregularities and miscellaneous spectra ........... 211
24. Measures of non-compactness and other operator
quantities ........................................... 220
Comments on Chapter III .................................. 228
IV. Taylor Spectrum
25. Basic properties ..................................... 237
26. Split spectrum .................................... 245
27. Some non-linear results .............................. 252
28. Taylor functional calculus for the split spectrum .... 260
29. Local spectrum for n-tuples of operators ............. 266
30. Taylor functional calculus ........................... 271
31. Taylor functional calculus in Banach algebras ........ 283
32. k-regular functions .................................. 285
33. Stability of index of complexes ...................... 293
34. Essential Taylor spectrum ............................ 299
Comments on Chapter IV ................................... 305
V. Orbits and Capacity
35. Joint spectral radius ................................ 311
36. Capacity ............................................. 321
37. Invariant subset problem and large orbits ............ 327
38. Hypercyclic vectors .................................. 341
39. Weak orbits .......................................... 348
40. Scott Brown technique ................................ 359
41. Kaplansky's type theorems ............................ 375
42. Polynomial orbits and local capacity ................. 378
Comments on Chapter V .................................... 387
Appendix ...................................................... 393
A.l. Banach spaces ............................................ 393
A.2. Analytic vector-valued functions ......................... 398
A.3. C∞-functions ............................................. 401
A.4. Semicontinuous set-valued functions ...................... 403
A.5. Some geometric properties of Banach spaces ............... 404
A.6. Basic properties of H∞ ................................... 405
Bibliography .................................................. 407
Index ......................................................... 429
List of Symbols ............................................... 437
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