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Algebraic Multiplicity of Eigenvalues of Linear Operators

:	Julián López-Gómez, Carlos Mora-Corral
:	Algebraic Multiplicity of Eigenvalues of Linear Operators
:	Birkhäuser Basel
:	9783764384012
:	1
:	CHF 85.80
:

:	Analysis
:	English

:	310
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

This book brings together all available results about the theory of algebraic multiplicities. It first offers a classic course on finite-dimensional spectral theory and then presents the most general results available about the existence and uniqueness of algebraic multiplicities for real non-analytic operator matrices and families. Coverage next transfers these results from linear to nonlinear analysis.

	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