Algebraic Multiplicity of Eigenvalues of Linear Operators

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Springer Science & Business Media, Jun 22, 2007 - Mathematics - 310 pages
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This 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
Bibliography
295
189
298
Notation
303
212
307
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