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

λ

0

∈ 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

λ

0

∈ Eig

$$ \left( \mathfrak{L} \right) $$

, the point

λ

0

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

−

λ

0

| <

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

λ

0

. 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,

λ

0

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

λ

0

, can be determined through standard perturbation techniques; these perturbed eigenvalues

a

(

λ

) are those satisfying

a

(

λ

0

) = 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

λ

0

equals the order of the function

a

at

λ

0

.