## 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. |

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### Contents

Summary | 2 |

Operator Calculus | 36 |

Spectral Projections | 63 |

Summary 76 | 75 |

Algebraic Multiplicity Through Transversalization | 82 |

Algebraic Multiplicity Through Polynomial Factorization | 107 |

Uniqueness of the Algebraic Multiplicity | 139 |

7 | 153 |

The Spectral Theorem for Matrix Polynomials | 248 |

Further Developments of the Algebraic Multiplicity | 265 |

Summary | 272 |

295 | |

298 | |

Notation | 303 |

307 | |

### 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

A G Q A I A0 A0 G according to Theorem algebraic eigenvalue algebraic multiplicity analytic Axioms A1 Banach space basis bifurcation theory canonical set chains of length Chapter classic compact compact operators concludes the proof Consequently construction deﬁned Deg(f,D denote direct sum eigenvectors equation equivalent Exercise exist an open ﬁnd ﬁnite finite-dimensional Fredholm operators G Fred0(U G Iso(U g j g p G N U Gohberg hence holomorphic functions identity integer integer 1 g introduced invertible isomorphism Jordan canonical form Jordan chains Jordan theorem Lancaster & Rodman Laurent Lemma linear operator matrix polynomial MN(C Mora-Corral Moreover neighborhood of A0 Nonlinear Analysis nonlinear eigenvalue norm open neighborhood open set operator families perturbation proof of Theorem Proposition prove satisﬁes satisfying Section set of Jordan Smith form spectral theorem subspace Suppose theory topological degree unique vector zero