Algebraic Multiplicity Through Transversalization

Abstract

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 < |λ − λ₀| < δ, the operator $\mathfrak{L}\left( \lambda \right)$ is an isomorphism and

$$ \left\| {\mathfrak{L}\left( \lambda \right)^{ - 1} } \right\| \leqslant \frac{C} {{\left| {\lambda - \lambda _0 } \right|^m }}. $$

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.