Algebraic Multiplicity of Eigenvalues of Linear OperatorsThis book analyzes the existence and uniqueness of a generalized algebraic m- tiplicity for a general one-parameter family L of bounded linear operators with Fredholm index zero at a value of the parameter ? whereL(? ) is non-invertible. 0 0 Precisely, given K?{R,C}, two Banach spaces U and V over K, an open subset ? ? K,andapoint ? ? ?, our admissible operator families are the maps 0 r L?C (? ,L(U,V)) (1) for some r? N, such that L(? )? Fred (U,V); 0 0 hereL(U,V) stands for the space of linear continuous operatorsfrom U to V,and Fred (U,V) is its subset consisting of all Fredholm operators of index zero. From 0 the point of view of its novelty, the main achievements of this book are reached in case K = R, since in the case K = C and r = 1, most of its contents are classic, except for the axiomatization theorem of the multiplicity. |
Contents
Finitedimensional Classic Spectral Theory | 2 |
Spectral Projections | 3 |
Algebraic Multiplicity Through Jordan Chains Smith Form | 7 |
The Spectral Theorem for Matrix Polynomials | 10 |
6 | 31 |
7 | 58 |
Summary | 75 |
Algebraic Multiplicity Through Transversalization | 82 |
Algebraic Multiplicity Through Logarithmic Residues | 155 |
63 | 204 |
Further Developments of the Algebraic Multiplicity | 265 |
Summary | 272 |
| 295 | |
| 299 | |
Notation | 303 |
| 307 | |
Algebraic Multiplicity Through Polynomial Factorization | 107 |
Uniqueness of the Algebraic Multiplicity | 139 |
Other editions - View all
Algebraic Multiplicity of Eigenvalues of Linear Operators Julián López-Gómez,Carlos Mora-Corral No preview available - 2009 |
Common terms and phrases
according to Theorem Alg(L algebraic eigenvalue algebraic multiplicity Axioms A1 Banach space basis canonical set Cauchy contour Cauchy integral formula chains of length Chapter compact concludes the proof Consequently construction defined Definition denote derived families diag dim N[Lo direct sum Eig(L eigenvectors equivalent Exercise exist an open finite finite-dimensional Fredholm operators Fredo Gohberg hence Hermitian matrix holomorphic functions identity implies integer introduced invertible Iso(U isomorphism Jordan blocks Jordan canonical form Jordan chains K-Banach k-transversal eigenvalue L(CN Laurent Laurent series Lemma linear operator López-Gómez m[Cp MN(C Moreover nonlinear eigenvalue norm open neighborhood operator families perturbation product formula proof of Theorem Proposition prove R[Lo rank satisfies Section sequence set of Jordan shows Smith form spectral spectral theory subset subspace Suppose theory transversal U₁ unique vector zero ΛΕΩ λο Σπί