Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.


Introduction
Nonlinear spectral theory is a research field of increasing interest, which finds application to properties of the structure of the solution set of differential equations, see e.g. [1,14]. In this context a nontrivial question consists in studying nonlinear perturbations of linear problems and in investigating the so-called "persistence" of eigenvalues and eigenvectors.
More precisely, let G and H denote two real Hilbert spaces. By a "perturbed eigenvalue problem" we mean a system of the following type: where s, λ are real parameters, L, C : G → H are bounded linear operators, S denotes the unit sphere of G, and N : S → H is a nonlinear map. We call solution of (1.1) a triple (s, λ, x) ∈ R×R×S satisfying the above system. The element x ∈ S is then said a unit eigenvector corresponding to the eigenpair (s, λ) of (4.1), and the set of solutions of (1.1) will be denoted by Σ ⊆ R×R×S.
To investigate the topological properties of Σ, we consider (1.1) as a (nonlinear) perturbation of the eigenvalue problem where we assume that the operator L − λC ∈ L(G, H) is invertible for some λ ∈ R. When λ ∈ R is such that Ker(L − λC) is nontrivial, we call λ an eigenvalue of the equation L = λC or, equivalently, of problem (1.2). A solution (λ, x) of (1.2) will be called an eigenpoint ; in this case λ and x are, respectively, an eigenvalue and a unit eigenvector of the equation Lx = λCx. Let (λ * , x * ) be an eigenpoint of (1.2) and suppose that the following conditions hold: (H1) C is a compact operator, (H2) Ker(L − λ * C) is odd dimensional, (H3) Img(L − λ * C) ∩ C (Ker(L − λ * C)) = {0}.
The proof of Theorem 4.4, which can be thought of as a Rabinowitz-type global continuation result [21], is based on a preliminary study of the "unperturbed" problem (1.2). In particular, notice that the eigenpoints of (1.2) coincide with the solutions of the equation where ψ is the H-valued function (λ, x) → Lx − λCx defined on the cylinder R×S, which is a smooth 1-codimensional submanifold of the Hilbert space R×G. A crucial point is then to evaluate the topological degree of the map ψ. Since the domain of ψ is a manifold, we cannot apply the classical Leary-Schauder degree. Instead, we use a notion of topological degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds, developed by two authors of this paper, and whose construction and properties are summarized in Section 3 for the reader's convenience. Such a notion of degree has been introduced in [8] (see also [7,9,10] for additional details).
Taking advantage of the odd multiplicity assumption (H2), of condition (H1) on the compactness of C, and of the transversality condition (H3), we are then able to apply a result of [6] concerning the case of simple eigenvalues. Precisely, call λ * ∈ R a simple eigenvalue of (1.2) if there exists x * ∈ S such that Ker(L − λ * C) = Rx * and H = Img(L − λ * C) ⊕ RCx * . In [6] we proved that • if λ * is a simple eigenvalue of (1.2) and x * and −x * are the two corresponding unit eigenvectors, then the "twin" eigenpoints p * = (λ * , x * ) and p * = (λ * , −x * ) are isolated zeros of ψ. Moreover, under the assumption that the operator C is compact, they give the same contribution to the bf -degree, which is either 1 or −1, depending on the orientation of ψ.
Such an assertion generalizes, to the infinite dimensional case, an analogous result in [5] concerning a "classical eigenvalue problem" in R k . Let us point out that the result in [5] is based on the notion of Brouwer degree for maps between finite dimensional oriented manifolds, whereas, as already stressed, the extension to the infinite-dimensional setting of [6] requires a degree for Fredholm maps of index zero acting between Banach manifolds, as the one introduced in [8]. To apply this degree we need the unit sphere S to be a smooth manifold: for this reason, we restrict our study to Hilbert spaces instead of the more general Banach environment.
The study of the local [2,13,[15][16][17][18][19] as well as global [3][4][5][6][7] persistence property when the eigenvalue λ * is not necessarily simple has been performed in recent papers by the authors, also in collaboration with R. Chiappinelli. In particular a first pioneering result in this sense is due to Chiappinelli [12], who proved the existence of the local persistence of eigenvalues and eigenvectors, in Hilbert spaces, in the case of a simple isolated eigenvalue.
Among others, let us quote our paper [3] in which we tackled a problem very similar to the one we consider here. The main result of [3] regards, roughly speaking, the global persistence property of the eigenpairs (s, λ) of (1.1), in the sλ-plane, under the odd multiplicity assumption. Thus, the result we obtain here on the global persistence of the solutions (s, λ, x) of (1.1) was, in some sense, implicitly conjectured in [3].
The present paper generalizes the "global persistence" property of solution triples which, either in finite-dimensional or infinite-dimensional case, has been studied in [4][5][6][7] in the case of a simple eigenvalue. Since it is known that the persistence property need not hold if λ * is an eigenvalue of even multiplicity, it is natural to investigate the odd-multiplicity case. However such an extension is not trivial and is based on advanced degree-theoretical tools.
We close the paper with some illustrating examples showing, in particular, that the odd dimensionality of Ker(L − λ * C) cannot be removed, the other assumptions remaining valid.

Preliminaries
In this section we recall some notions that will be used in the sequel. We mainly summarize some concepts which are needed for the construction of the topological degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds introduced in [8], here called bf -degree to distinguish it from the Leray-Schauder degree, called LS-degree (see [7,9,10] for additional details).
It is necessary to begin by focusing on the preliminary concept of orientation for Fredholm maps of index zero between manifolds. The starting point is an algebraic notion of orientation for Fredholm linear operators of index zero.
Consider two real Banach spaces E and F and denote by L(E, F ) the space of the bounded linear operators from E into F with the usual operator norm. If E = F , we write L(E) instead of L(E, E). By Iso(E, F ) we mean the subset of L(E, F ) of the invertible operators, and we write GL(E) instead of Iso(E, E). The subspace of L(E, F ) of the compact operators will be denoted by K(E, F ), or simply by K(E) when F = E. Finally, F (E, F ) will stand for the vector subspace of L(E, F ) of the operators having finite dimensional image (recall that, in the infinite dimensional context, F (E, F ) is not closed in L(E, F )). We shall write F (E) when F = E.
Recall that an operator T ∈ L(E, F ) is said to be Fredholm (see e.g. [23]) if its kernel, Ker T , and its cokernel, coKer T = F/T (E), are both finite dimensional. The index of a Fredholm operator T is the integer In particular, any invertible linear operator is Fredholm of index zero. Observe also that, if T ∈ L(R k , R s ), then ind T = k − s.
The subset of L(E, F ) of the Fredholm operators will be denoted by Φ(E, F ); while Φ n (E, F ) will stand for the set {T ∈ Φ(E, F ) : ind T = n}. By Φ(E) and Φ n (E) we will designate, respectively, Φ(E, E) and Φ n (E, E).
We recall some important properties of Fredholm operators.
The composition of Fredholm operators is Fredholm and its index is the sum of the indices of all the composite operators.
Let T ∈ L(E) be given. If I − T ∈ F (E), where I ∈ L(E) is the identity, we say that T is an admissible operator (for the determinant). The symbol A(E) will stand for the affine subspace of L(E) of the admissible operators.
It is known (see [20]) that the determinant of an operator T ∈ A(E) is well defined as follows: det T := det T |Ê, where T |Ê is the restriction (as domain and as codomain) to any finite dimensional subspaceÊ of E containing Img(I − T ), with the understanding that det T |Ê = 1 ifÊ = {0}. As one can check, the function det : A(E) → R inherits most of the properties of the classical determinant. For more details, see e.g. [11].
Let T ∈ Φ 0 (E, F ) be given. As in [7], we will say that an operator Observe in particular that any T ∈ Iso(E, F ) has a natural companion: that is, the zero operator 0 ∈ L(E, F ). This fact was crucial in [8] for the construction of the bf -degree.
Given T ∈ Φ 0 (E, F ), we denote by C(T ) the (nonempty) subset of F (E, F ) of all the companions of T . The following definition establishes a partition of C(T ) in two equivalence classes and is a key step for the definition of orientation given in [8].
Observe that the oriented composition is associative and, consequently, this notion can be extended to the composition of three (or more) oriented operators.
Definition 2.5 (Sign of an oriented operator). Let T ∈ Φ 0 (E, F ) be an oriented operator. Its sign is the integer if T is invertible and naturally oriented, −1 if T is invertible and not naturally oriented, 0 if T is not invertible.
A crucial fact in the definition of oriented map and the consequent construction of the bf -degree is that • the orientation of any operator T * ∈ Φ 0 (E, F ) induces an orientation of the operators in a neighborhood of T * . In fact, since Iso(E, F ) is open in L(E, F ), for any companion K of T * we have that T + K is invertible when T is sufficiently close to T * . Thus, because of property (F3) of the Fredholm operators, any such T belongs to Φ 0 (E, F ). Consequently, K is as well a companion of T . Definition 2.6. Let Γ : X → Φ 0 (E, F ) be a continuous map defined on a metric space X. A pre-orientation of Γ is a function that to any x ∈ X assigns an orientation ω(x) of Γ(x). A pre-orientation (of Γ) is an orientation if it is continuous, in the sense that, given any x * ∈ X, there exist K ∈ ω(x * ) and a neighborhood W of x * such that K ∈ ω(x) for all x ∈ W . The map Γ is said to be orientable if it admits an orientation, and oriented if an orientation has been chosen. In particular, a subset Y of Φ 0 (E, F ) is orientable or oriented if so is the inclusion map Y ֒→ Φ 0 (E, F ).
Observe that the setΦ 0 (E, F ) of the oriented operators of Φ 0 (E, F ) has a natural topology, and the natural projection π :Φ 0 (E, F ) → Φ 0 (E, F ) is a 2-fold covering space (see [9] for details). Therefore, an orientation of a map Γ as in Definition 2.6 could be regarded as a liftingΓ of Γ. This implies that, if the domain X of Γ is simply connected and locally path connected, then Γ is orientable.
Let f : U → F be a C 1 -map defined on an open subset of E, and denote by df x ∈ L(E, F ) the Fréchet differential of f at a point x ∈ U .
We recall that f is said to be Fredholm of index n, called Φ n -map and hereafter also denoted by Therefore, if f ∈ Φ 0 , Definition 2.6 and the continuity of the differential map df : U → Φ 0 (E, F ) suggest the following Definition 2.7 (Orientation of a Φ 0 -map in Banach spaces). Let U be an open subset of E and f : U → F a Fredholm map of index zero. A pre-orientation or an orientation of f are, respectively, a pre-orientation or an orientation of df , according to Definition 2.6. The map f is said to be orientable if it admits an orientation, and oriented if an orientation has been chosen.
Remark 2.8. A very special Φ 0 -map is given by an operator T ∈ Φ 0 (E, F ). Thus, for T there are two different notions of orientations: the algebraic one and that in which T is seen as a C 1 -map, according to Definitions 2.2 and 2.7, respectively. In each case T admits exactly two orientations (in the second one this is due to the connectedness of the domain E). Hereafter, we shall tacitly assume that the two notions agree. Namely, T has an algebraic orientation ω if and only if its differential dT x :ẋ → Tẋ has the ω orientation for all x ∈ E.
Let us summarize how the notion of orientation can be given for maps acting between real Banach manifolds. In the sequel, by manifold we shall mean, for short, a smooth Banach manifold embedded in a real Banach space.
Given a manifold M and a point x ∈ M, the tangent space of M at x will be denoted by T x M. If M is embedded in a Banach space E, T x M will be identified with a closed subspace of E, for example by regarding any tangent vector of T x M as the derivative γ ′ (0) of a smooth curve γ : Assume that f : M → N is a C 1 -map between two manifolds, respectively embedded in E and F and modelled on E and F . As in the flat case, f is said to be for any x ∈ M (see [22]).
Given f ∈ Φ 0 , suppose that to any x ∈ M it is assigned an orientation ω(x) of df x (also called orientation of f at x). As above, the function ω is called a preorientation of f , and an orientation if it is continuous, in a sense to be specified (see Definition 2.10).
Definition 2.9. The pre-oriented composition of two (or more) pre-oriented maps between manifolds is given by assigning, at any point x of the domain of the composite map, the composition of the orientations (according to Definition 2.4) of the differentials in the chain representing the differential at x of the composite map.
Assume that f : M → N is a C 1 -diffeomorphism. Thus, for any x ∈ M, we may take as ω(x) the natural orientation of df x (recall Definition 2.3). This preorientation of f turns out to be continuous according to Definition 2.10 below (it is, in some sense, constant). From now on, unless otherwise stated, • any diffeomorphism will be considered oriented with the natural orientation.
In particular, in a composition of pre-oriented maps, all charts and parametrizations of a manifold will be tacitly assumed to be naturally oriented.
Definition 2.10 (Orientation of a Φ 0 -map between manifolds). Let f : M → N be a Φ 0 -map between two manifolds modelled on E and F , respectively. A preorientation of f is an orientation if it is continuous in the sense that, given any two charts, ϕ : is an oriented map according to Definition 2.7. The map f is said to be orientable if it admits an orientation, and oriented if an orientation has been chosen.
For example any local diffeomorphism f : M → N admits the natural orientation, given by assigning the natural orientation to the operator df x , for any x ∈ M (see Definition 2.3).
In contrast, a very simple example of non-orientable Φ 0 -map is given by a constant map from the 2-dimensional projective space into R 2 (see [9]). Notation 2.11. Let D be a subset of the product X × Y of two metric spaces.
Similarly to the case of a single map, one can define a notion of orientation of a continuous family of Φ 0 -maps depending on a parameter s ∈ [0, 1]. To be precise, one has the following is an orientation, according to Definition 2.6.
The homotopy h is said to be orientable if it admits an orientation, and oriented if an orientation has been chosen.
If a Φ 0 -homotopy h has an orientation ω, then any partial map h s = h(s, ·) has a compatible orientation ω(s, ·). Conversely, one has the following Proposition 2.13 ( [8,9]). Let h : [0, 1]×M → N be a Φ 0 -homotopy, and assume that one of its partial maps, say h s , has an orientation. Then, there exists and is unique an orientation of h which is compatible with that of h s . In particular, if two maps from M to N are Φ 0 -homotopic, then they are both orientable or both non-orientable.
As a consequence of Proposition 2.13, one gets that any C 1 -map f : M → M which is Φ 0 -homotopic to the identity is orientable, since so is the identity (even when M is finite dimensional and not orientable).
The bf -degree, introduced in [8], satisfies the three fundamental properties listed below: Normalization, Additivity and Homotopy Invariance. In [10], by means of an axiomatic approach, it is proved that the bf -degree is the only possible integervalued function that satisfies these three properties.
More in detail, the bf -degree is defined in a class of admissible triples. Given an oriented Φ 0 -map f : M → N , an open (possibly empty) subset U of M, and a target value y ∈ N , the triple (f, U, y) is said to be admissible for the bf -degree provided that U ∩ f −1 (y) is compact. From the axiomatic point of view, the bfdegree is an integer-valued function, deg bf , defined on the class of all the admissible triples, that satisfies the following three fundamental properties. • (Additivity) Let (f, U, y) be an admissible triple. If U 1 and • (Homotopy Invariance) Let h : [0, 1]×M → N be an oriented Φ 0 -homotopy, and γ : [0, 1] → N a continuous path. If the set Other useful properties are deduced from the fundamental ones (see [10] for details). Here we mention some of them. deg bf (f, U, y) = deg bf (g • f, U, g(y)).

Some further notation and definitions are in order.
Notation 2.14. Hereafter we will use the shorthand notation deg bf Regarding Definition 2.16, we observe that the finite union of isolated subsets of f −1 (0) is still an isolated subset. Moreover, from the excision and the additivity properties of the bf -degree one gets that the contribution to the bf -degree of this union is the sum of the single contributions of these subsets.
3. The eigenvalue problem and the associated topological degree Let, hereafter, G and H denote two real Hilbert spaces and consider the eigenvalue problem where λ is a real parameter, L, C : G → H are bounded linear operators, and S denotes the unit sphere of G. To prevent the problem from being meaningless, • we will always assume that the operator L − λC ∈ L(G, H) is invertible for some λ ∈ R. When λ ∈ R is such that Ker(L−λC) is nontrivial, then λ is called an eigenvalue of the equation L = λC or, equivalently, of problem (3.1).
A solution (λ, x) of (3.1) will also be called an eigenpoint. In this case λ and x are, respectively, an eigenvalue and a unit eigenvector of the equation Lx = λCx.
Notice that the eigenpoints are the solutions of the equation where ψ is the H-valued function (λ, x) → Lx − λCx defined on the cylinder R×S, which is a smooth 1-codimensional submanifold of the Hilbert space R×G. By S we will denote the set of the eigenpoints of (3.1). Therefore, given any λ ∈ R, the λ-slice S λ = {x ∈ S : (λ, x) ∈ S} of S coincides with S ∩ Ker(L − λC).
Thus, S λ is nonempty if and only if λ is an eigenvalue of problem (3.1). In this case S λ will be called the eigensphere of (3.1) corresponding to λ or, simply, the λ-eigensphere. Observe that S λ is a sphere whose dimension equals that of Ker(L − λC) minus one. The nonempty subset {λ}×S λ of the cylinder R×S will be called an eigenset of (3.1).
Remark 3.1. The assumption that L − λC is invertible for some λ ∈ R implies that, for any λ ∈ R, the restriction of C to the (possibly trivial) kernel of L − λC is injective.
Remark 3.1 can be proved arguing by contradiction. In fact, assume that the assertion is false. Then, there are λ * ∈ R and a nonzero vector This implies that, for any λ, the operator L−λC is non-injective and, consequently, non-invertible, in contrast to the assumption. In fact, for any λ, one has

Remark 3.2.
If the operator C is compact, then, from the assumption that L − λC is invertible for someλ ∈ R, it follows that L − λC is Fredholm of index zero for any λ ∈ R and, consequently, the set of the eigenvalues of problem (3.1) is discrete. Moreover, Ker(L − λC) is always finite dimensional, and so is the intersection Consequently, if this intersection is the singleton {0}, taking into account Remark 3.1 and the fact that L − λC ∈ Φ 0 (G, H), one has H = Img(L − λC) ⊕ C(Ker(L − λC)).
To prove Remark 3.2 notice that, if L −λC is invertible, then it is trivially Fredholm of index zero. Now, given any λ ∈ R, one has Thus, because of the compactness of C, from property (F3) of Fredholm operators, one gets that L − λC is also Fredholm of index zero. Finally, the set of the eigenvalues of problem (3.1) is discrete since so is, according to the spectral theory of linear operators, the set of the characteristic values of (L −λC) −1 C.

Because of Remark 3.2,
• from now until the end of this section we assume that the operator C is compact.
Observe that the function ψ defined above is the restriction to R × S of the nonlinear smooth map According to Remark 3.2, any partial map ψ λ : G → H of ψ is Fredholm of index zero. Since the map σ : R× G → G given by σ(λ, x) = x is clearly Φ 1 , the same holds true, because of the property (F2) of Fredholm operators, for the composition ψ = ψ λ • σ. Consequently, again because of property (F2), one has that the restriction ψ of ψ to the 1-codimensional submanifold R×S of R×G is Φ 0 .
Notice that, if dim G = 1, the cylinder R×S is disconnected: it is the union of two horizontal lines, R×{−1} and R×{1}. Because of this, to make some statements simpler, • from now on, unless otherwise stated, we assume that the dimension of the space G is greater than 1. In this case the cylinder R × S is connected, and simply connected if dim G > 2. It is actually contractible if G is infinite dimensional. Therefore, the Φ 0 -map ψ, defined above, is orientable and admits exactly two orientations. We choose one of them and • hereafter we assume that ψ is oriented. Remark 3.3. Letλ ∈ R be such that L −λC is invertible and let Z : H → G denote its inverse. Then, given any λ ∈ R, the two equations are equivalent (I being the identity on G). Therefore, if B denotes the unit ball of G, the Leray-Schauder degree with target 0 ∈ G, deg LS (η λ , B), of the compact vector field η λ is well defined whenever λ is not an eigenvalue of the equation Lx = λCx.
Observe that, as a consequence of the homotopy invariance property of the Leray-Schauder degree, the function λ → deg LS (η λ , B) is constant on any interval in which it is defined. Moreover, in these intervals, deg LS (η λ , B) is either 1 or −1, since the equation η λ (x) = 0 has only one solution: the regular point 0 ∈ G.
Remark 3.4. Let U be an isolating neighborhood of a compact subset of the set S of the eigenpoints of (3.1), and let Z : H → G be as in Remark 3.3. Then deg bf (ψ, U ) = deg bf (η, U ), provided that the map η = Zψ is the oriented composition obtained by considering Z as a naturally oriented diffeomorphism.
Concerning possible relations between the LS-degree of η λ and the bf -degree of ψ (or, equivalently, of η = Zψ), we believe that the following is true (but up to now we were unable to prove or disprove). In support of the above conjecture we observe that both the conditions imply the existence of at least one eigenpoint p * = (λ * , x * ) ∈ U . The first one because of the existence property of the bf -degree and the last one due to the homotopy invariance property of the LS-degree. Definition 3.6. An eigenpoint (λ * , x * ) of (3.1) is said to be simple provided that the operator T = L − λ * C is Fredholm of index zero and satisfies the conditions: (1) Ker T = Rx * , We point out that, if an eigenpoint p * = (λ * , x * ) is simple, then the corresponding eigenset {λ * }×S λ * is disconnected. In fact, it has only two elements: p * and its twin eigenpointp * = (λ * , −x * ), which is as well simple.
The following theorem obtained in [7] was essential in the proofs of some results in [7] concerning perturbations of (3.1), as problem (4.1) in the next section.
Theorem 3.7. In addition to the compactness of C, assume that p * = (λ * , x * ) and p * = (λ * , −x * ) are two simple twin eigenpoints of (3.1). Then, the contributions of p andp to the bf -degree of ψ are equal: they are both either 1 or −1 depending on the orientation of ψ. Consequently, if U is an isolating neighborhood of the eigenset {λ * }×S λ * , one has deg bf (ψ, U ) = ±2.
We close this section strictly devoted to the unperturbed eigenvalue problem (3.1) with a consequence of Theorem 3.7, which will be crucial in the proof of our main result (Theorem 4.4 in Section 4). Proof. Because of the assumption Img T ∩ C(Ker T ) = {0}, as well as the fact that T is Fredholm of index zero, we can split the spaces G and H as follows: With these splittings, T and C can be represented in block matrix form as follows: The operators T 11 : G 1 → H 1 and C 22 : G 2 → H 2 are isomorphisms (the second one because of Remark 3.1), while C 11 : G 1 → H 1 and C 21 : G 1 → H 2 are, respectively, compact and finite dimensional. We can equivalently regard the equation ψ(λ, x) = 0 as Zψ(λ, x) = 0, where Z : H → G is an isomorphism. We choose Z as follows: Given any λ ∈ R, the operator ψ λ = L − λC ∈ L(G, H) can be written as T − (λ − λ * )C. Therefore, putting η = Zψ : R×G → G, the partial map η λ : G → G (see Notation 2.11) can be represented as where I is the identity on G = G 1 ⊕ G 2 and C = ZC (observe that C 22 coincides with the identity I 22 ∈ L(G 2 )). This shows that, given any λ ∈ R, the endomorphism η λ : G → G is a compact vector field. Therefore, its Leray-Schauder degree on the unit ball B of G is well defined whenever λ is not an eigenvalue of the equation Lx = λCx, and this happens when λ is close to, but different from, λ * . Since G 2 is odd dimensional and, because of assumption (H3), the geometric and algebraic multiplicities of λ * coincide, the function λ → deg LS (η λ , B) has a sign-jump crossing λ * . Therefore, if Conjecture 3.5 were true, we would have done. So we need to proceed differently.
First of all we point out that since otherwise the equation Lx = λCx would have eigenvalues different from λ * in the interval [α, β]. Now, given ε > 0 such that (λ * − ε, λ * + ε) ⊂ (α, β), we choose a linear operator A ε ∈ L(G 2 ) with the following properties: • in the operator norm, the distance between A ε and λ * I 22 is less than ε, • the eigenvalues of A ε are real and simple, • any eigenvalue λ of A ε is such that |λ − λ * | < ε.
Let the isomorphism Z be naturally oriented and let the restriction η of η to the manifold R×S be oriented according to the composition Zψ. Thus, because of the topological invariance property of the bf -degree, we get deg bf (η, U ) = deg bf (ψ, U ).
Let us prove that this is true for the left boundary of U ; that is, for λ = α. The argument for λ = β will be the same.
We need to show that (if ε is small) the linear operator A t = tη ε α + (1 − t)η α of L(G) is invertible for any t ∈ [0, 1]. In fact, since A 0 = η α is invertible, and the set of the invertible operators of L(G) is open, this holds true for all A t provided that ε is sufficiently small.

The perturbed eigenvalue problem and global continuation
Here, as in Section 3, G and H denote two real Hilbert spaces, L, C : G → H are bounded linear operators, S is the unit sphere of G and, as in problem (3.1), the operator L − λC is invertible for some λ ∈ R.
Consider the perturbed eigenvalue problem where N : S → H is a C 1 compact map and s is a real parameter.
A solution of (4.1) is a triple (s, λ, x) ∈ R×R×S satisfying (4.1). The element x ∈ S is a unit eigenvector corresponding to the eigenpair (s, λ).
The set of solutions of (4.1) will be denoted by Σ and E is the subset of R 2 of the eigenpairs. Notice that E is the projection of Σ into the sλ-plane and the s = 0 slice Σ 0 of Σ is the same as the set S = ψ −1 (0) of the eigenpoints of (3.1), where ψ has been defined in the previous section.
A solution (s, λ, x) of (4.1) is regarded as trivial if s = 0. In this case p = (λ, x) is the corresponding eigenpoint of problem (3.1). When p is simple, the triple (0, λ, x) ∈ Σ will be as well said to be simple. A nonempty subset of Σ of the type {0}×{λ}×S λ will be called a solution-sphere.
We say that a bifurcation point q * = (0, λ * , x * ) is global (in the sense of Rabinowitz [21]) if in the set of nontrivial solutions there exists a connected component, called global (bifurcating) branch, whose closure in Σ contains q * and it is either unbounded or includes a trivial solution q * = (0, λ * , x * ) with λ * = λ * . In the second case q * is as well a global bifurcation point.
A meaningful case is when a bifurcation point q * = (0, λ * , x * ) belongs to a connected solution-sphere {0} × {λ * } × S λ * . In this case the dimension of S λ * is positive and we will simply say that x * is a bifurcation point. In fact, 0 and λ * being known, x * can be regarded as an alias of q * .
For a necessary condition as well as some sufficient conditions for a point x * of a connected eigensphere to be a bifurcation point see [15]. Other results regarding the existence of bifurcation points belonging to even-dimensional eigenspheres can be found in [2-7, 16, 17, 19].
As already pointed out, if the operator C is compact, then ψ : R × S → H is Fredholm of index zero, and this is crucial for the global results regarding the perturbed eigenvalue problem (4.1). Because of this, • from now on, unless otherwise stated, we will tacitly assume that the linear operator C is compact. We define the C 1 -map in which ψ : R×S → H, as in Section 3, is given by ψ(λ, x) = Lx − λCx. Therefore the set (ψ + ) −1 (0) of the zeros of ψ + coincides with Σ.
As shown in [7], because of the compactness of C and N , one gets that • ψ + is proper on any bounded and closed subset of its domain. Consequently, any bounded connected component of Σ is compact. This fact will be useful later.
Notice that ψ + is the restriction to the manifold R×R×S of the nonlinear map where ψ is as in Section 3 and N is the positively homogeneous extension of N .
The following result of [7] is crucial for proving the existence of global bifurcation points. be its 0-slice. If deg bf (ψ, Ω 0 ) is well defined and nonzero, then Ω contains a connected set of nontrivial solutions whose closure in Ω is non-compact and meets at least one trivial solution of (4.1). Corollary 4.2 below, which was deduced in [7] from Theorem 4.1, asserts that the contribution to the bf -degree of the 0-slice of any compact (connected) component of Σ is null. We will need this basic property later. The following result, obtained in [7,Theorem 4.5], regards the existence of a global branch of solutions emanating from a trivial solution of problem (4.1) which corresponds to a simple eigenpoint of (3.1). Theorem 4.3. If (λ * , x * ) is a simple eigenpoint of problem (3.1), then, in the set Σ of the solutions of (4.1), the connected component containing (0, λ * , x * ) is either unbounded or includes a trivial solution (0, λ * , x * ) with λ * = λ * .
We are now ready to prove our main result, which extends Theorem 4.3 and provides a global version of Theorem 3.9 in [19], the latter concerning the existence of local bifurcation points belonging to even dimensional eigenspheres.
Proof. Because of the compactness of C, according to Remark 3.2, the operator L − λC is Fredholm of index zero for all λ ∈ R. Moreover, the set of the eigenvalues of problem (3.1) is discrete. Consequently, the eigenset {λ * }×S λ * , which is compact and nonempty, is relatively open in the set S of the eigenpoints. Thus, it admits an isolating neighborhood U ⊂ R×S and, therefore, deg bf (ψ, U ) is well defined.
Denote by D the connected component of Σ containing (0, λ * , x * ). We may assume that D is bounded. Thus, it is actually compact, since ψ + is proper on any bounded and closed subset of R×R×S. We need to prove that D contains a trivial solution (0, λ * , x * ) with λ * = λ * .
Assume, by contradiction, that this is not the case. Then the 0-slice D 0 of D is contained in the eigenset {λ * }×S λ * . We will show that this contradicts Corollary 4.2. We distinguish two cases: n = 1 and n > 1, where n is the dimension of Ker T .

Some illustrating examples
In this section we provide three examples in ℓ 2 concerning Theorem 4.4. The dimensions of Ker T (where T = L − λ * C) are, respectively, 3, 2, and 1. The second example, in which Ker T is two dimensional, shows that in Theorem 4.4, as well as in Remark 4.5, the hypothesis of the odd dimensionality of Ker T cannot be removed, the other assumptions remaining valid.
Given any compact (possibly nonlinear) map N : ℓ 2 → ℓ 2 of class C 1 , consider the perturbed eigenvalue problem where S is the unit sphere of ℓ 2 . As before, we denote by Σ the set of solutions (s, λ, x) of (5.1).
Observe that, for any k ∈ N, k ≥ 1, λ * = 0 is an eigenvalue of the unperturbed equation T k x = λCx and the condition Img T k ∩ C(Ker T k ) = {0} is satisfied. Therefore, according to Theorem 4.4, given any positive odd integer k, any compact perturbing map N : ℓ 2 → ℓ 2 of class C 1 , and any x * ∈ S ∩ Ker T k , the connected component of Σ containing (0, 0, x * ) is either unbounded or encounters a trivial solution (0, λ * , x * ) with λ * = 0.
According to Remark 4.5, there exists at least one bifurcation pointx ∈ S λ * . Actually, in this case one gets exactly two (global) bifurcation points. This is due to the fact that D \ D * has two disjoint "twin" branches whose closures meet the solution-sphere D * . The branches can be parametrized with θ ∈ (0, 2π) as follows: Then, if the following limits exist: lim θ→0 q(θ) and lim θ→0q (θ), we get the bifurcation points (as elements of D * ). Equivalently, to find the aliases of these points (that is, the corresponding elements in the eigensphere S λ * ) we compute ± lim Example 5.2 (k = 2). The eigenvalues of the unperturbed equation T 2 x = λCx are 0, 3, 4, 5, 6, . . . The first one, λ * = 0, has geometric and algebraic multiplicity 2 and all the others are simple.
In conclusion, in Theorem 4.4 and Remark 4.5, the assumption that Ker T is odd dimensional cannot be removed. In addition to the horizontal lines, the set of the eigenpairs has two bounded components: an ellipse with center (0, 1) and half-axes 1/ √ 2 and 1, therefore containing (0, 0) and (0, 2); and, as in Example 5.2, an ellipse joining (0, 3) with (0, 4), with center (0, 7/2) and half-axes 1/ √ 48 and 1/2. Finally, one can check that, in accord with Theorem 4.4, given any one of the two points of the 0-dimensional solution-sphere {0}×{0}×S 0 , its connected component in Σ is bounded and contains a point of {0}×{2}×S 2 . This agrees with Theorem 4.4.