Algebraic Multiplicity Through Transversalization

Part of the Operator Theory: Advances and Applications book series (OT, volume 177)


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 < |λλ0| < 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.


Operator Family Closed Subspace Product Family Transversality Condition Simple Eigenvalue 
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© Birkhäuser Verlag AG 2007

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