Algebraic Multiplicity Through Transversalization.

Throughout this chapter we will consider $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$, two $$ \mathbb{K} $$-Banach spaces U and V, an open subset $$ \Omega \subset \mathbb{K} $$, a point λ0 ∈ Ω, and a family $$ \mathfrak{L} \in \mathcal{C}^r \left( {\Omega ,\mathcal{L}\left( {U,V} \right)} \right), $$ for some r ∈ ℕ ∪ {∞}, such that $$ \mathfrak{L}_0 : = \mathfrak{L}\left( {\lambda _0 } \right) \in Fred_0 \left( {U,V} \right). $$ When λ0 ∈ Eig$$ \left( \mathfrak{L} \right) $$, the point λ0 is said to be an algebraic eigenvalue of $$ \mathfrak{L} $$ if there exist δ, C > 0 and m ≥ 1 such that, for each 0 < <INNOPIPE>λ − λ0<INNOPIPE> < δ, the operator $$ \mathfrak{L}\left( \lambda \right) $$ is an isomorphism and $$ \left\<INNOPIPE> {\mathfrak{L}\left( \lambda \right)^{ - 1} } \right\<INNOPIPE> \leqslant \frac{C} {{\left<INNOPIPE> {\lambda - \lambda _0 } \right<INNOPIPE>^m }}. $$ The main goal of this chapter is to introduce the concept of algebraic multiplicity of $$ \mathfrak{L} $$ at any algebraic eigenvalue λ0. This algebraic multiplicity will be denoted by $$ \chi \left[ {\mathfrak{L};\lambda _0 } \right] $$, and will be defined through the auxiliary concept of transversal eigenvalue. Such concept will be motivated in Section 4.1 and will be formally defined in Section 4.2. Essentially, λ0 is a transversal eigenvalue of $$ \mathfrak{L} $$ when it is an algebraic eigenvalue for which the perturbed eigenvalues $$ a\left( \lambda \right) \in \sigma \left( {\mathfrak{L}\left( \lambda \right)} \right) $$ from $$ 0 \in \sigma \left( {\mathfrak{L}_0 } \right) $$, as λ moves from λ0, can be determined through standard perturbation techniques; these perturbed eigenvalues a(λ) are those satisfying a(λ0) = 0. This feature will be clarified in Sections 4.1 and 4.4, where we study the behavior of the eigenvalue a(λ) and its associated eigenvector in the special case when 0 is a simple eigenvalue of $$ \mathfrak{L}_0 $$. In such a case, the multiplicity of $$ \mathfrak{L} $$ at λ0 equals the order of the function a at λ0. [ABSTRACT FROM AUTHOR]