Algebraic Multiplicity of Eigenvalues of Linear Operators (Operator Theory: Advances and Applications)

376438400X......Page 1
Contents......Page 7
Preface......Page 11
I. Finite-dimensional Classic Spectral Theory......Page 24
1.1 Basic concepts......Page 25
1.2 The Jordan theorem......Page 30
1.3 The Jordan canonical form......Page 41
1.4 The real canonical form......Page 45
1.5 An example......Page 48
1.6 Exercises......Page 53
1.7 Comments on Chapter 1......Page 57
2. Operator Calculus......Page 58
2.1 Normof a linear operator......Page 59
2.2 Introduction to operator calculus......Page 62
2.3 Resolvent operator. Dunford’s integral formula......Page 66
2.4 The spectralmapping theorem......Page 72
2.5 The exponentialmatrix......Page 74
2.6 An example......Page 77
2.7 Exercises......Page 79
2.8 Comments on Chapter 2......Page 83
3.1 Estimating the inverse of amatrix......Page 84
3.2 Vector-valued Laurent series......Page 86
3.3 The eigenvalues are poles of the resolvent......Page 87
3.4 Spectral projections......Page 89
3.5 Exercises......Page 91
3.6 Comments on Chapter 3......Page 93
II. Algebraic Multiplicities......Page 96
4. Algebraic Multiplicity Through Transversalization......Page 102
4.1 Motivating the concept of transversality......Page 103
4.2 The concept of transversal eigenvalue......Page 106
4.3 Algebraic eigenvalues and transversalization......Page 108
4.4 Perturbation fromsimple eigenvalues......Page 118
4.5 Exercises......Page 122
4.6 Comments on Chapter 4......Page 124
5. Algebraic Multiplicity Through Polynomial Factorization......Page 126
5.1 Derived families and factorization......Page 127
5.2 Connections between χ and μ......Page 133
5.3 Coincidence of the multiplicities χ and μ......Page 137
5.4 A formula for the partial μ-multiplicities......Page 140
5.5 Removable singularities......Page 141
5.6 The product formula......Page 142
5.7 Perturbation from simple eigenvalues revisited......Page 149
5.8 Exercises......Page 151
5.9 Comments on Chapter 5......Page 155
6. Uniqueness of the Algebraic Multiplicity......Page 157
6.1 Similarity of rank-one projections......Page 159
6.2 Proof of Theorem6.0.1......Page 160
6.3 Relaxing the regularity requirements......Page 162
6.4 A general uniqueness theorem......Page 165
6.5 Applications. Classical multiplicity formulae......Page 166
6.6 Exercises......Page 169
6.7 Comments on Chapter 6......Page 170
7. Algebraic Multiplicity Through Jordan Chains. Smith Form......Page 171
7.1 The concept of Jordan chain......Page 173
7.2 Canonical sets and κ-multiplicity......Page 175
7.3 Invariance by continuous families of isomorphisms......Page 179
7.4 Local Smith form for C[sup(∞)] matrix families......Page 185
7.5 Canonical sets at transversal eigenvalues......Page 189
7.6 Coincidence of the multiplicities κ and χ......Page 192
7.7 Labeling the vectors of the canonical sets......Page 198
7.8 Characterizing the existence of the Smith form......Page 199
7.9 Two illustrative examples......Page 207
7.10 Local equivalence of operator families......Page 211
7.11 Exercises......Page 217
7.12 Comments on Chapter 7......Page 222
8. Analytic and Classical Families. Stability......Page 226
8.1 Isolated eigenvalues......Page 227
8.2 The structure of the spectrum......Page 228
8.3 Classic algebraic multiplicity......Page 229
8.4 Stability of the complex algebraic multiplicity......Page 232
8.5 Exercises......Page 237
8.6 Comments on Chapter 8......Page 239
9. Algebraic Multiplicity Through Logarithmic Residues......Page 241
9.1 Finite Laurent developments of Σ[sup(–1)]......Page 242
9.2 The trace operator......Page 245
9.3 The multiplicity through a logarithmic residue......Page 249
9.4 Holomorphic and classical families......Page 253
9.5 Spectral projection......Page 254
9.6 Exercises......Page 257
9.7 Comments on Chapter 9......Page 262
10. The Spectral Theorem for Matrix Polynomials......Page 264
10.1 Linearization of a matrix polynomial......Page 265
10.2 Generalized Jordan theorem......Page 267
10.3 Remarks on scalar polynomials......Page 270
10.4 Constructing a basis in the phase space......Page 271
10.5 Exercises......Page 277
10.6 Comments on Chapter 10......Page 278
11.1 General Fredholm operator families......Page 280
11.2 Meromorphic families......Page 281
11.3 Unbounded operators......Page 282
11.4 Non-Fredholmoperators......Page 284
III. Nonlinear Spectral Theory......Page 287
12. Nonlinear Eigenvalues......Page 288
12.1 Bifurcation values. Nonlinear eigenvalues......Page 290
12.2 A short introduction to the topological degree......Page 293
12.3 The algebraic multiplicity as an indicator of the change of index......Page 299
12.4 Characterization of nonlinear eigenvalues......Page 301
12.5 Comments on Chapter 12......Page 306
Bibliography......Page 309
Notation......Page 317
E......Page 321
M......Page 322
S......Page 323
U......Page 324