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On the Uniqueness of the Algebraic Multiplicity
Author(s) -
Mora-Corral C.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300471x
Subject(s) - multiplicity (mathematics) , uniqueness , mathematics , algebraic number , pure mathematics , algebra over a field , mathematical analysis
Given a smooth family £ of real or complex variable taking values within the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of the concept of algebraic multiplicity of the family £ at a point x 0 of the parameter at which the operator L ( x 0 ) becomes non‐invertible. The purpose of the paper is to show that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction. The main technical tools to obtain this uniqueness result are a Lyapunov–Schmidt reduction, the local Smith form and a new factorization result for general families at non‐algebraic eigenvalues obtained in the paper.