Algebraic Multiplicity of Eigenvalues of Linear Operators (eBook)

Contents 8
Preface 12
Part I: Finite-dimensional Classic Spectral Theory 24
The Jordan Theorem 26
1.1 Basic concepts 26
1.2 The Jordan theorem 31
1.3 The Jordan canonical form 42
1.4 The real canonical form 46
1.5 An example 49
1.6 Exercises 54
1.7 Comments on Chapter 1 58
Operator Calculus 59
2.1 Norm of a linear operator 60
2.2 Introduction to operator calculus 63
2.3 Resolvent operator. Dunford’s integral formula 67
2.4 The spectral mapping theorem 73
2.5 The exponential matrix 75
2.6 An example 78
2.7 Exercises 80
2.8 Comments on Chapter 2 84
Spectral Projections 85
3.1 Estimating the inverse of a matrix 85
3.2 Vector-valued Laurent series 87
3.3 The eigenvalues are poles of the resolvent 88
3.4 Spectral projections 90
3.5 Exercises 92
3.6 Comments on Chapter 3 94
Part II: Algebraic Multiplicities 96
Algebraic Multiplicity Through Transversalization 103
4.1 Motivating the concept of transversality 104
4.2 The concept of transversal eigenvalue 107
4.3 Algebraic eigenvalues and transversalization 109
4.4 Perturbation from simple eigenvalues 119
4.5 Exercises 123
4.6 Comments on Chapter 4 125
Algebraic Multiplicity Through Polynomial Factorization 127
5.1 Derived families and factorization 128
5.2 Connections between . and µ 134
5.3 Coincidence of the multiplicities . and µ 138
5.4 A formula for the partial µ-multiplicities 141
5.5 Removable singularities 142
5.6 The product formula 143
5.7 Perturbation from simple eigenvalues revisited 150
5.8 Exercises 152
5.9 Comments on Chapter 5 156
Uniqueness of the Algebraic Multiplicity 158
6.1 Similarity of rank-one projections 160
6.2 Proof of Theorem 6.0.1 161
6.3 Relaxing the regularity requirements 163
6.4 A general uniqueness theorem 166
6.5 Applications. Classical multiplicity formulae 167
6.6 Exercises 170
6.7 Comments on Chapter 6 171
Algebraic Multiplicity Through Jordan Chains. Smith Form 172
7.1 The concept of Jordan chain 174
7.2 Canonical sets and .-multiplicity 176
7.3 Invariance by continuous families of isomorphisms 180
7.4 Local Smith form for C8 matrix families 186
7.5 Canonical sets at transversal eigenvalues 190
7.6 Coincidence of the multiplicities . and . 193
7.7 Labeling the vectors of the canonical sets 199
7.8 Characterizing the existence of the Smith form 200
7.9 Two illustrative examples 208
7.10 Local equivalence of operator families 212
7.11 Exercises 218
7.12 Comments on Chapter 7 223
Analytic and Classical Families. Stability 227
8.1 Isolated eigenvalues 228
8.2 The structure of the spectrum 229
8.3 Classic algebraic multiplicity 230
8.4 Stability of the complex algebraic multiplicity 233
8.5 Exercises 238
8.6 Comments on Chapter 8 240
Algebraic Multiplicity Through Logarithmic Residues 242
9.1 Finite Laurent developments of 243
9.2 The trace operator 246
9.3 The multiplicity through a logarithmic residue 250
9.4 Holomorphic and classical families 254
9.5 Spectral projection 255
9.6 Exercises 258
9.7 Comments on Chapter 9 263
The Spectral Theorem for Matrix Polynomials 265
10.1 Linearization of a matrix polynomial 266
10.2 Generalized Jordan theorem 268
10.3 Remarks on scalar polynomials 271
10.4 Constructing a basis in the phase space 272
10.5 Exercises 278
10.6 Comments on Chapter 10 279
Further Developments of the Algebraic Multiplicity 281
11.1 General Fredholm operator families 281
11.2 Meromorphic families 282
11.3 Unbounded operators 283
11.4 Non-Fredholm operators 285
Part III: Nonlinear Spectral Theory 287
Nonlinear Eigenvalues 289
12.1 Bifurcation values. Nonlinear eigenvalues 291
12.2 A short introduction to the topological degree 294
12.3 The algebraic multiplicity as an indicator of the change of index 300
12.4 Characterization of nonlinear eigenvalues 302
12.5 Comments on Chapter 12 307
Bibliography 310
Notation 318
Index 322