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Preface | p. xi |
Finite-dimensional Classic Spectral Theory | |
Summary | p. 2 |
The Jordan Theorem | |
Basic concepts | p. 3 |
The Jordan theorem | p. 8 |
The Jordan canonical form | p. 19 |
The real canonical form | p. 23 |
An example | p. 26 |
The complex Jordan form of A | p. 29 |
The real Jordan form of A | p. 30 |
Exercises | p. 31 |
Comments on Chapter 1 | p. 35 |
Operator Calculus | |
Norm of a linear operator | p. 38 |
Introduction to operator calculus | p. 41 |
Resolvent operator. Dunford's integral formula | p. 45 |
The spectral mapping theorem | p. 51 |
The exponential matrix | p. 53 |
An example | p. 56 |
Exercises | p. 58 |
Comments on Chapter 2 | p. 62 |
Spectral Projections | |
Estimating the inverse of a matrix | p. 63 |
Vector-valued Laurent series | p. 65 |
The eigenvalues are poles of the resolvent | p. 66 |
Spectral projections | p. 68 |
Exercises | p. 70 |
Comments on Chapter 3 | p. 72 |
Algebraic Multiplicities | |
Summary | p. 76 |
Algebraic Multiplicity Through Transversalization | |
Motivating the concept of transversality | p. 84 |
The concept of transversal eigenvalue | p. 87 |
Algebraic eigenvalues and transversalization | p. 89 |
Perturbation from simple eigenvalues | p. 99 |
Exercises | p. 103 |
Comments on Chapter 4 | p. 105 |
Algebraic Multiplicity Through Polynomial Factorization | |
Derived families and factorization | p. 108 |
Connections between x and ¿ | p. 114 |
Coincidence of the multiplicities x and ¿ | p. 118 |
A formula for the partial ¿-multiplicities | p. 121 |
Removable singularities | p. 122 |
The product formula | p. 123 |
Perturbation from simple eigenvalues revisited | p. 130 |
Exercises | p. 132 |
Comments on Chapter 5 | p. 136 |
Uniqueness of the Algebraic Multiplicity | |
Similarity of rank-one projections | p. 141 |
Proof of Theorem 6.0.1 | p. 142 |
Relaxing the regularity requirements | p. 144 |
A general uniqueness theorem | p. 147 |
Applications. Classical multiplicity formulae | p. 148 |
Exercises | p. 151 |
Comments on Chapter 6 | p. 152 |
Algebraic Multiplicity Through Jordan Chains. Smith Form | |
The concept of Jordan chain | p. 155 |
Canonical sets and ¿-multiplicity | p. 157 |
Invariance by continuous families of isomorphisms | p. 161 |
Local Smith form for C∞ matrix families | p. 167 |
Canonical sets at transversal eigenvalues | p. 171 |
Coincidence of the multiplicities ¿ and ¿ | p. 174 |
Labeling the vectors of the canonical sets | p. 180 |
Characterizing the existence of the Smith form | p. 181 |
Two illustrative examples | p. 189 |
Example 1 | p. 189 |
Example 2 | p. 191 |
Local equivalence of operator families | p. 193 |
Exercises | p. 199 |
Comments on Chapter7 | p. 204 |
Analytic and Classical Families. Stability | |
Isolated eigenvalues | p. 210 |
The structure of the spectrum | p. 211 |
Classic algebraic multiplicity | p. 212 |
Stability of the complex algebraic multiplicity | p. 215 |
Local stability | p. 216 |
Global stability | p. 218 |
Homotopy invariance | p. 220 |
Exercises | p. 220 |
Comments on Chapter8 | p. 222 |
Algebraic Multiplicity Through Logarithmic Residues | |
Finite Laurent developments of L-1 | p. 226 |
The trace operator | p. 229 |
Concept of trace and basic properties | p. 230 |
Traces of the coefficients of the product of two families | p. 231 |
The multiplicity through a logarithmic residue | p. 233 |
Holomorphic and classical families | p. 237 |
Spectral projection | p. 238 |
Exercises | p. 241 |
Comments on Chapter9 | p. 246 |
The Spectral Theorem for Matrix Polynomials | |
Linearization of a matrix polynomial | p. 250 |
Generalized Jordan theorem | p. 252 |
Remarks on scalar polynomials | p. 255 |
Constructing a basis in the phase space | p. 256 |
Exercises | p. 262 |
Comments on Chapter10 | p. 263 |
Further Developments of the Algebraic Multiplicity | |
General Fredholm operator families | p. 265 |
Meromorphic families | p. 266 |
Unbounded operators | p. 267 |
Non-Fredholm operators | p. 269 |
Nonlinear Spectral Theory | |
Summary | p. 272 |
Nonlinear Eigenvalues | |
Bifurcation values. Nonlinear eigenvalues | p. 275 |
A short introduction to the topological degree | p. 278 |
Existence and uniqueness | p. 279 |
Computing the degree. Analytic construction | p. 281 |
The fixed point theorem of Brouwer, Schauder and Tychonoff | p. 283 |
Further properties | p. 284 |
The algebraic multiplicity as an indicator of the change of index | p. 284 |
Characterization of nonlinear eigenvalues | p. 286 |
Comments on Chapter 12 | p. 291 |
Bibliography | p. 295 |
Notation | p. 303 |
Index | p. 307 |
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