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Algebraic Multiplicity Through Transversalization

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

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 < |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.

Keywords

Operator Family Closed Subspace Product Family Transversality Condition Simple Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2007

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