New analytical and geometrical aspects of the algebraic multiplicity

A bridge between Vector Bundle Theory and Nonlinear Spectral Theory

“…By [33,Th. 1.2], this concept of multiplicity is consistent, in the sense that it does not depend on the particular choice of the pair P ≡ (P, Q) of L(λ 0 )-projections.…”

“…denotes the open ball of radius ε centered at T ∈ Φ 0 (U ) in L(U ). This operator, introduced by the authors in [33], extends the classical concept of Schur complement of a matrix in the Euclidean space. Precisely, given any block matrix…”

“…In 1988, the classical notions of algebraic multiplicity of eigenvalues of linear operators was generalized to an infinite dimensional Fredholm setting by Esquinas and López-Gómez [14] to characterize the nonlinear eigenvalues of the C r -curves of linear Fredholm operators in the context of local bifurcation theory (see also Esquinas [13] and López-Gómez [29]). More recently, in [33], by adopting a geometrical point of view, the authors established a hidden connection between the generalized algebraic multiplicity and the concept of local intersection index of algebraic varieties, a central device in algebraic geometry. Precisely, it was shown that the algebraic multiplicity of a curve of Fredholm operators of index zero L : [0, 1] → Φ 0 (U ) equals the intersection index of the curve L([0, 1]) ⊂ Φ 0 (U ) with the subspace of singular operators S(U ) ⊂ Φ 0 (U ).…”

“…The main goal of this paper is studying vector bundles via topological K-Theory focusing special attention into the obstruction described by the first Stiefel-Whitney characteristic class, ω 1 . In particular, it will be shown that this obstruction can be completely described from the notion of generalized algebraic multiplicity, and hence, through the concept of intersection index of algebraic varieties, as discussed by the authors in [33]. These connections are established in this paper by combining the Atiyah-Jänich index map (1.1) Ind : [X, Φ 0 (U )] −→ KO(X), with some spectral techniques developed by the authors in a series of recent papers [31,32,33] for continuous maps h : X → Φ 0 (U ).…”

“…In particular, it will be shown that this obstruction can be completely described from the notion of generalized algebraic multiplicity, and hence, through the concept of intersection index of algebraic varieties, as discussed by the authors in [33]. These connections are established in this paper by combining the Atiyah-Jänich index map (1.1) Ind : [X, Φ 0 (U )] −→ KO(X), with some spectral techniques developed by the authors in a series of recent papers [31,32,33] for continuous maps h : X → Φ 0 (U ). In (1.1), [X, Φ 0 (U )] stands for the set of homotopy classes of continuous maps X → Φ 0 (U ), and KO(X) denotes the real reduced K-group of the compact path connected topological space X; the K-group consists of the stable equivalence classes of real vector bundles with base space X.…”