Algebraic Multiplicity of Eigenvalues of Linear Operators

by Lopez-Gomez, J.; Mora-corral, C.

• ISBN13: 9783764384005
• ISBN10: 376438400X
• Format: Hardcover
• Copyright: 2007-10-04
• Publisher: Birkhauser

Summary

This book brings together all available results about the theory of algebraic multiplicities, from the most classic results, like the Jordan Theorem, to the most recent developments, like the uniqueness theorem and the construction of the multiplicity for non-analytic families. Part I (first three chapters) is a classic course on finite-dimensional spectral theory, Part II (the next eight chapters) presents the most general results available about the existence and uniqueness of algebraic multiplicities for real non-analytic operator matrices and families, and Part III (last chapter) transfers these results from linear to nonlinear analysis. The text is as self-contained as possible and suitable for students at the advanced undergraduate or beginning graduate level.

Table of Contents

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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