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Throughout this chapter, the field $$ \mathbb{K} $$ will always be the real field ℝ; we consider a real Banach space U, an open interval ℭ ⊂ ∝, a neighborhood $$ \mathcal{U} $$ of 0 ∈ U, an integer number r ≥ 0, a family $$ \mathfrak{L} $$∈Cr(Ω,$$ \mathcal{L} $$(U)), and a nonlinear map $$ \mathfrak{N} $$∈ C(Ω × $$ \mathcal{U} $$, U) satisfying the following conditions: (AL)$$ \mathfrak{L} $$(λ) ™ IU∈ K(U) for every λ ∈ Ω, i.e., $$ \mathfrak{L} $$(λ) is a compact perturbation of the identity map. (AN)$$ \mathfrak{N} $$ is compact, i.e., the image by $$ \mathfrak{N} $$ of any bounded set of Ω × $$ \mathcal{U} $$ is relatively compact in U. Also, for every compact K ⊂ Ω, $$ \mathop {\lim }\limits_{u \to 0} \mathop {\sup }\limits_{\lambda \in K} \frac{{\left\ {\mathfrak{N}\left( {\lambda ,u} \right)} \right\}} {{\left\ u \right\}} = 0. $$. From now on, we consider the operator $$ \mathfrak{F} \in \mathcal{C}\left( {\Omega \times \mathcal{U},U} \right) $$ defined as 12.1$$ \mathfrak{F}\left( {\lambda ,u} \right): = \mathfrak{L}\left( \lambda \right)u + \mathfrak{N}\left( {\lambda ,u} \right), $$ and the associated equation 12.2$$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {\left( {\lambda ,u} \right) \in \Omega } \\ \end{array} \times \mathcal{U}. $$ By Assumptions (AL) and (AN), it is apparent that $$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {D_u \mathfrak{F}\left( {\lambda ,u} \right) = \mathfrak{L}\left( \lambda \right),} & \lambda \\ \end{array} \in \Omega , $$ and, hence, (12.2) can be thought of as a nonlinear perturbation around (λ, 0) of the linear equation 12.3$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right)u = 0,} & {\lambda \in \Omega ,} & u \\ \end{array} \in U. $$ Equation (12.2) can be expressed as a fixed-point equation for a compact operator. Indeed, $$ \mathfrak{F}\left( {\lambda ,u} \right) $$ = 0 if and only if $$ u = \left[ {I_U - \mathfrak{L}\left( \lambda \right)} \right]u - \mathfrak{N}\left( {\lambda ,u} \right). $$. [ABSTRACT FROM AUTHOR]

This chapter studies the polynomial families of the form 10.1$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right) = \sum\limits_{j = 0}^\ell {A_j \lambda ^j } ,} & {\lambda \in \mathbb{C}} \\ \end{array} , $$ where ℓ ∈ ℕ and $$ \begin{array}{*{20}c} {A_0 , \ldots ,A_\ell \in \mathcal{M}_N \left( \mathbb{C} \right),} & {A_\ell \ne 0,} \\ \end{array} $$ for some N ∈ ℕ*. These families are called matrix polynomials in most of the available literature. More precisely, the family $$ \mathfrak{L} $$ defined in (10.1) is said to be a matrix polynomial of order N and degree ℓ. The main goal of this chapter is to obtain a spectral theorem for matrix polynomials, respecting the spirit of the Jordan Theorem 1.2.1. [ABSTRACT FROM AUTHOR]

This chapter exposes briefly some further developments of the theory of multiplicity whose treatment lies outside the general scope of this book. Consequently, it possesses an expository and bibliographic character. [ABSTRACT FROM AUTHOR]

The stability results of Section 8.4 can be regarded as infinite-dimensional versions of the classic Rouché theorem. A closely related topic in complex function theory is the so-called argument principle, otherwise known as the logarithmic residue theorem, which has been established by Theorem 3.4.1 (for classical families) and Corollary 6.5.2 in a finite-dimensional setting. [ABSTRACT FROM AUTHOR]

This chapter focuses attention on the analytic operator families. After studying some universal spectral properties of these families, this chapter deals with the classical families $$ \mathfrak{L}^A $$ of the form 8.1$$ \begin{array}{*{20}c} {\mathfrak{L}^A \left( \lambda \right): = \lambda I_U - A,} & {\lambda \in \mathbb{K}} \\ \end{array} , $$ for a given A ∈$$ A \in \mathcal{L} $$(U), in order to show that, in this particular case, the classic concepts of algebraic ascent and multiplicity equal the generalized concepts introduced in the previous four chapters. Consequently, the algebraic multiplicity analyzed in this book, from a series of different perspectives, is indeed a generalization of the classic concept of algebraic multiplicity. [ABSTRACT FROM AUTHOR]

This chapter studies the algebraic multiplicity according to the theory of Jordan chains. The concept of Jordan chain extends the classic concept of chain of generalized eigenvectors, already studied in Section 1.3. It will provide us with a further approach to the algebraic multiplicities χ and µ introduced and analyzed in Chapters 4 and 5, respectively, whose axiomatization has already been accomplished through the uniqueness theorems included in Chapter 6. [ABSTRACT FROM AUTHOR]

Throughout this chapter, given $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$, two non-zero $$ \mathbb{K} $$-Banach spaces U, V, and $$ \lambda _0 \in \mathbb{K} $$, we denote by $$ \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ the set of all families $$ \mathfrak{L} $$ of class C∞ in a neighborhood of λ0 with values in $$ \mathcal{L}\left( {U,V} \right) $$ such that $$ \mathfrak{L}_0 : = \mathfrak{L}\left( {\lambda _0 } \right) \in Fred_0 \left( {U,V} \right); $$ the neighborhood may depend on $$ \mathfrak{L} $$. We also set $$ \mathcal{S}_{\lambda _0 }^\infty \left( U \right): = \mathcal{S}_{\lambda _0 }^\infty \left( {U,U} \right). $$ Clearly, $$ \mathfrak{L}\mathfrak{M} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ if $$ \mathfrak{L}^{ - 1} \in \mathcal{S}_{\lambda _0 }^\infty \left( {W,V} \right) $$ and $$ \mathfrak{M} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,W} \right) $$, where W is another $$ \mathbb{K} $$-Banach space. Moreover, $$ \mathfrak{L}^{ - 1} \in \mathcal{S}_{\lambda _0 }^\infty \left( {V,U} \right) $$ whenever $$ \mathfrak{L} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ and $$ \mathfrak{L}_0 \in Iso\left( {U,V} \right) $$ Iso(U, V). [ABSTRACT FROM AUTHOR]

This chapter describes an equivalent approach to the concept of multiplicity $$ \chi \left[ {\mathfrak{L};\lambda _0 } \right] $$ introduced in Chapter 4; in this occasion by means of an appropriate polynomial factorization of $$ \mathfrak{L} $$ at λ0. However, at first glance these approaches are seemingly completely different. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This chapter considers $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$ and uses the Dunford integral formula to construct the spectral projections of ℂN onto N[(A − λI)ν(λ)] for every λ ∈ σ(A). It also shows that, for each λ ∈ σ(A), the algebraic ascent ν(λ) equals the order of λ as a pole of the associated resolvent operator 3.1$$ \begin{array}{*{20}c} {\mathcal{R}\left( {z;A} \right): = \left( {zI - A} \right)^{ - 1} ,} & {z \in \rho \left( A \right): = \mathbb{C}\backslash \sigma \left( A \right)} \\ \end{array} . $$ Precisely, this chapter is structured as follows. Section 3.1 gives a universal estimate for the norm of the inverse of a matrix in terms of its determinant and its norm. From this estimate it will become apparent that the eigenvalues of A are poles of the resolvent operator (3.1). The necessary analysis to show this feature will be carried out in Section 3.3. Section 3.2 gives a result on Laurent series valid for vector-valued holomorphic functions. Finally, Section 3.4 constructs the spectral projections associated with the direct sum decomposition (1.6), whose validity was established by the Jordan Theorem 1.2.1. [ABSTRACT FROM AUTHOR]

In this chapter we prove the Jordan theorem, a pivotal result in mathematics, which establishes that, for every $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$, the space ℂN decomposes as the direct sum of the ascent generalized eigenspaces associated with each of the eigenvalues of A. Then, by choosing an appropriate basis in each of the ascent generalized eigenspaces, the Jordan canonical form of A is constructed. These bases are chosen in order to attain a similar matrix to A with a maximum number of zeros. [ABSTRACT FROM AUTHOR]

In Section 1.2 we defined P(A) when P is a polynomial and $$ A \in \mathcal{M}_N \left( \mathbb{K} \right) $$ with $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$. More generally, one of the main goals of operator calculus consists in giving an appropriate definition of f(A) when $$ f:\mathbb{K} \to \mathbb{K} $$ is an arbitrary function, as well as in studying the most important analytical properties of f(A). This chapter covers these issues for the special, but important, case when f is a certain holomorphic function and $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$. [ABSTRACT FROM AUTHOR]