ON THE UNIQUENESS OF THE ALGEBRAIC MULTIPLICITY

Abstract

Given a smooth family ${\hbox{\ac L}}$ 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 ${\hbox{\ac L}}$ at a point $x_0$ of the parameter at which the operator ${\hbox{\ac L}}(x_0)$ 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.