ON STABILIZED POINT SPECTRA OF MULTIVALUED SYSTEMS

Alberto SEEGER

ABSTRACT. Let H be an infinite dimensional Hilbert space. Denote by Λ(E, F) the set of all λ ∈ IR for which the multivalued system 0 ∈ (F − λE)(x) admits a nonzero solution x ∈ H. One says that Λ(E, F) is the point spectrum of the pair (E, F). It is well known that Λ(E, F) does not behave in a stable manner with respect to perturbations in the argument (E, F). The purpose of this note is to study the outer-semicontinuous hull (or graph-closure) of the mapping Λ.

1. INTRODUCTION

Let H be an infinite dimensional real Hilbert space with inner product ⟨·, ·⟩ and associated norm || · ||. The chief motivation of this note is the stability analysis of a generalized eigenvalue problem of the form

(1.1)  find λ ∈ IR and a nonzero vector x ∈ H such that 0 ∈ (F − λE)(x) .

The abstract formalism considered in (1.1) covers a large number of equilibrium problems appearing in practice, including variational inequalities and differential systems with multivalued right-hand sides (see [5] and the references therein).

One says that

Λ(E, F) := {λ ∈ IR : 0 ∈ (F − λE)(x) for some x ≠ 0}

is the point spectrum (or set of eigenvalues) of the pair (E, F). Throughout this paper, E is assumed to be an element of the linear space

L(H) := set of bounded linear operators from H to H.

As far as F is concerned, the only assumption made here is that

grF := {(x, y) ∈ H × H : y ∈ F (x)}

is a nonempty closed cone. The class of such multivalued operators is denoted by K(H). The pair (E, F) is thus regarded as an element of the Cartesian product

U(H) := L(H) × K(H) .

2000 Mathematics Subject Classification. 47 H 04 , 47 J 10 , 58 C 40.

Key words and phrases. Eigenvalue, stabilized point spectrum, multivalued operator.

The eigenvalue problem (1.1) is usually solved through an approximation (or discretization) scheme of the form

(1.2)  find λ_{n} ∈ IR and a nonzero vector x_{n} ∈ H such that 0 ∈ (F_{n} − λ_{n}E_{n})(x_{n}) .

Unfortunately, a sequence {λ_{n}}_{n∈IN} constructed in this way may not have a satisfactory limiting behavior. In fact, this sequence may convergence to a number that falls beyond Λ(E, F). From a purely topological point of view, what explains this phenomenon is the lack of outersemicontinuity of the set-valued mapping Λ.

Definition 1.1. (E, F) is said to be a point of outer-semicontinuity of Λ : U(H)\overrightarrow{→}IR if

(1.3)  \begin{matrix}(E_{n}, F_{n}) → (E, F) , |λ_{n} − λ| → 0 \\ λ_{n} ∈ Λ(E_{n}, F_{n}) \end{matrix} \} ⟹ λ ∈ Λ(E, F) .

Of course, for such a definition to make sense, it is necessary to clarify the meaning of convergence in the space U(H). The most natural way of measuring the distance between two elements in L(H) is by using the standard operator norm, that is to say,

||D − E|| := sup_{||x|| ≤1} ||(D − E)(x)|| .

Convergence in K(H) is a more delicate matter. First of all, it should be observed that a multivalued operator can always be identified with its graph. Thus, K(H) is indistinguishable from the class of all nonempty closed cones in H × H. This observation leads us to consider the metric

δ(G, F) := sup_{||(u,v)|| ≤1} | d_{grG}(u, v) − d_{grF} (u, v)| ,

where ||(u, v)|| := \sqrt{||u||^{2} + ||v||^{2}} is the usual norm in the product space H × H, and d_{Q}(z) stands for the distance from z to Q. The Cartesian product U(H) is equipped with any metric yielding the product topology, for instance,

ρ[(D, G), (E, F)] := max{ ||D − E|| , δ(G, F)}.

It is in the sense of the metric ρ that the convergence property mentioned in (1.3) must be understood.

One should say from the outset of the discussion that

osc(Λ) := {(E, F) ∈ U(H) : Λ is outer-semicontinuous at (E, F)}

turns out to be a ”small” portion of U(H), and therefore one must be extremely careful while computing a solution to (1.1) by means of the approximation scheme (1.2). A suitable mathematical tool for discussing this stability issue is the outer-semicontinuous hull (or graph-closure) of Λ.

Definition 1.2. The stabilized point spectrum of (E, F) ∈ U(H) is the set

(1.4)  Σ(E, F) := ⋂_{ε>0} cl [  ⋃_{ρ((D,G),(E,F))≤ε}Λ(D, G)] ,

where “cl” stands for the closure operation in (IR, | · |).

The set on the right-hand side of (1.4) corresponds to the upper Painlevé-Kuratowski limit of Λ(D, G) when the argument (D, G) approaches the reference point (E, F). This fact is indicated by means of the notation

Σ(E, F) = limsup_{(D,G)→(E,F)} Λ(D, G) .

General results concerning the topological nature of the mapping Σ : U(H)\overrightarrow{→}IR can be derived directly from the theory of Painlevé-Kuratowski limits. More specific results on Σ are obtained, of course, by exploiting the particular structure of the mapping Λ.

For an historic background on the stability analysis of point spectra, the reader can consult Conway and Morrel [1]. As far as the multivalued setting is concerned, relevant references are Lavilledieu and Seeger [4], Correa and Gajardo [2], and Seeger [6]. Although our work is heavily inspired from [6], several original aspects of the mapping Σ are presented here for the first time.

2. CHARACTERIZING THE STABILIZED POINT SPECTRUM

The expression (1.4) is the starting point for deriving various theoretical results on Σ, but it has little interest from a computational point of view. Evaluating (1.4) for a particular pair (E, F) may be a very cumbersome task, and therefore it is convenient to develop handy tests for checking whether or not a given real number belongs to the set Σ(E, F).

The next theorem shows that the stability analysis of Λ reduces to that of the partial mapping

F ∈ K(H) \overrightarrow{→} Λ_{E}(F) := Λ(E, F) .

If one fixes the argument E and proceeds to stabilize with respect to the argument F, then one gets

Σ_{E}(F) = limsup_{G→F} Λ_{E}(G) := ⋂_{ε>0} cl [ ⋃_{δ(G,F)≤ε}Λ_{E}(G)] .

The set Σ_{E}(F) is, of course, contained in the joint limit (1.4). The surprising fact is that :

Theorem 2.1. For any pair (E, F) ∈ U(H), one always has Σ(E, F) = Σ_{E}(F) .

Proof. To prove this result, we rely on the auxiliary function Φ defined by

Φ(λ, E, F) := inf_{x≠0} \frac{1}{||(x, λEx)||} d_{grF} (x, λEx) .

The first part of the proof consists in showing that

M(E, F) := {λ ∈ IR : Φ(λ, E, F) = 0} ⊂ Σ_{E}(F).

So, pick up λ in M(E, F) and take any ε > 0. One can find, in such a case, a nonzero vector x such that

d_{grF} (x, λEx) ≤ ε||(x, λEx)|| .

In other words,

(x, λEx) ∈ V_{ε}(grF) ,

where

V_{ε}(Q) := {(u, v) ∈ H × H : d_{Q}(u, v) ≤ ε||(u, v)||}

corresponds to a conic ε-neighborhood of the cone Q ⊂ H × H. Thus, λ ∈ Λ_{E}(F^{ε}), where F^{ε} ∈ K(H) is the operator defined by grF^{ε} := V_{ε}(grF) . But, from

(2.1)  δ(G, F) ≤ ε ⟺ grG ⊂ V_{ε}(grF) and grF ⊂ V_{ε}(gr G) ,

it follows that δ(F^{ε}, F) ≤ ε. Hence,

λ ∈ ⋃_{δ(G,F)≤ε}Λ_{E}(G) .

It suffices now to take the closure operation and then the intersection with respect to ε > 0. The second part of the proof consists in showing that Σ(E, F) is contained in M(E, F). To do this, let us start by examining the set

Γ_{ε} := ⋃_{ρ((D,G),(E,F))≤ε}Λ(D, G) .

Consider an arbitrary element μ in Γ_{ε} . By definition, there are (D, G) ∈ U(H) and x ≠ 0 such that

||D − E|| ≤ ε , δ(G, F) ≤ ε , (x, μDx) ∈ grG .

According to (2.1), one has necessarily (x, μDx) ∈ V_{ε}(grF), and, therefore,

\frac{1}{||(x, μDx)||} d_{grF} (x, μDx) ≤ ε .

Let B_{ε}(E) denote the closed ball with center E and radius ε. What we have shown, then, is that

Γ_{ε} ⊂ {μ ∈ IR : inf_{D∈B_{ε}(E)} Φ(μ, D, F) ≤ ε } .

Pick up now an arbitrary λ in Σ(E, F). According to the above inclusion, one has

lim inf_{μ→λ} inf_{D∈B_{ε}(E)} Φ(μ, D, F) ≤ ε ∀ ε > 0 .

Observe that term on the left-hand side is nonincreasing with respect to ε. By letting ε → 0^{+}, one arrives at

(2.2)  lim inf_{(μ,D)→(λ,E)} Φ(μ, D, F) = 0 .

At this point in time we do not know yet if the function Φ(·, ·, F) is lower-semicontinuous (this will be proven later in Section 3). Anyway, it is possible to get rid of the “liminf” operation taken in (2.2). Indeed, (2.2) amounts to saying that

\frac{d_{grF} (x_{n}, λ_{n}E_{n}x_{n})}{||(x_{n}, λ_{n}E_{n}x_{n})||} → 0

for suitable sequences {(λ_{n}, E_{n})}_{n∈IN} → (λ, E), and {x_{n}}_{n∈IN} ⊂ H \ {0}. By positive homogeneity of F, one can assume that {x_{n}}_{n∈IN} lies in the unit sphere of H. In such a case, the sequence {||(x_{n}, λ_{n}E_{n}x_{n})||}_{n∈IN} remains bounded, and

d_{grF} (x_{n}, λ_{n}E_{n}x_{n}) → 0 .

By taking into account

|d_{grF} (x_{n}, λ_{n}E_{n}x_{n}) − d_{grF} (x_{n}, λEx_{n})| ≤ ||(x_{n}, λ_{n}E_{n}x_{n}) − (x_{n}, λEx_{n})|| ≤ ||λ_{n}E_{n} − λE|| ,

one arrives at

d_{grF} (x_{n}, λEx_{n}) → 0 .

Since 1 ≤ ||(x_{n}, λEx_{n})||, one has

\frac{d_{grF} (x_{n}, λEx_{n})}{||(x_{n}, λEx_{n})||} → 0 ,

showing in this way that Φ(λ, E, F) = 0.

The question of characterizing the elements in Σ(E, F) is answered next. Roughly speaking, the term Φ( . , E, F) introduced in the proof ot Theorem 2.1 can be seen as a sort of characteristic or discriminant function associated to the pair (E, F). Also of interest is the ”reduced” version

Ψ(λ, E, F) := inf_{||x||=1} d_{grF} (x, λEx) .

One can easily check that

\frac{1}{\sqrt{1 + ||λE||^{2}}} Ψ(λ, E, F) ≤ Φ(λ, E, F) ≤ \frac{1}{\sqrt{1 + ν^{2}(λE)}} Ψ(λ, E, F) ≤ Ψ(λ, E, F) ,

with

ν(E) := inf_{||x||=1} ||Ex|| .

Without further ado, we state :

Theorem 2.2. Let (E, F) ∈ U(H). For λ ∈ IR, the following conditions are equivalent :

(a) λ ∈ Σ(E, F) ;

(b) Φ(λ, E, F) = 0 ;

(c) Ψ(λ, E, F) = 0 ;

(d) there are unit vectors {x_{n}}_{n∈IN} in H such that d_{grF} (x_{n}, λEx_{n}) → 0 ;

(e) liminf_{||x||→∞} \frac{1}{||x||} d_{grF} (x, λEx) = 0 .

Proof. The equivalence between (a), (b), (c), and (d) is implicitly contained in the proof of Theorem 2.1. The additional condition (e) is obtained by invoking the positive homogeneity of F.

The stabilized point-spectrum can be described also by using the concept of ε-eigenvalue. The theory of ε-eigenvalues for bounded linear operators is well understood, but its multivalued version has not been yet fully developed. Inspired by Gajardo and Seeger [3], we say that λ ∈ IR is an ε-eigenvalue of (E, F) ∈ U(H) if

∃ (x, y) ∈ grF, with x ≠ 0 , such that ||λEx − y|| ≤ ε||x|| .

The next theorem shows that the stabilized point-spectrum can be recovered by ”shrinking” the set ε-eigenvalues. The notation S_{H} refers to the unit sphere of H.

Theorem 2.3. Let (E, F) ∈ U(H). For λ ∈ IR, the following conditions are equivalent :

(a) λ ∈ Σ(E, F) ;

(b) ∀ ε > 0 , λ is an ε- eigenvalue of (E, F) ;

(c) 0 lies in the closure of (F − λE)(S_{H}) ;

(d) there are unit vectors {x_{n}}_{n∈IN} in H such that d_{F (x_{n})}(λEx_{n}) → 0 ;

(e) liminf_{||x||→∞} \frac{1}{||x||} d_{F (x)}(λEx) = 0 .

Proof. The equivalence between (b) and (d) is obtained by working out the concept of εeigenvalue. One follows the same steps as in [3]. Proving the equivalence between (d), (c), and (e) is rather elementary. That (d) implies (a) follows Theorem 2.2 and the general inequality

d_{grF} (u, v) ≤ d_{F (u)}(v) ∀ (u, v) ∈ H × H .

The implication (a) ⟹ (c) has been proven already by [6] when E is the identity operator. The proof technique developed in [6] can be easily extended to our general framework.

Theorem 2.3 proposes a number of equivalent criteria for checking membership in the stabilized point spectrum. Each one of these conditions has its own merit, but perhaps (c) is the simplest one from a computational point of view. By using the characterization (c) one can derive in a straightforward manner all the calculus rules listed below. In the next corollary, the notation L_{inv}(H) refers to the set of invertible elements of L(H).

Corollary 2.4. Let (E, F) ∈ U(H). Then,

(a) Σ(E, αF) = αΣ(E, F) ∀α ≠ 0 ; (scalar multiplication)

(b) Σ(E, F + μE) = Σ(E, F) + μ ∀μ ∈ IR ; (traslation)

(c) Σ(E, L ∘ F) = Σ(L^{−1} ∘ E, F) ∀L ∈ L_{inv}(H) ; (left composition)

(d) Σ(E, F ∘ R) = Σ(E ∘ R^{−1}, F) ∀R ∈ L_{inv}(H) ; (right composition)

(e) Σ(E, S ∘ F ∘ S^{−1}) = Σ(S^{−1} ∘ E ∘ S, F) ∀S ∈ L_{inv}(H) . (similarity transformation)

\underline{Remark:} Throughout this paper, no invertibility requirement is imposed on the operator E. For the sake of applications, it is important to leave open the possibility of dealing with singular operators as well. According to Corollary 2.4 (c), if E ∈ L(H) happens to be invertible, then one can write Σ(E, F) = Σ(I, E^{−1} ∘ F). In other words, we are back to a situation in which E is simply the identity operator. However, this reduction mechanism is not always as useful as one may expect. For instance, some structural properties of F could be lost after carrying out the left composition by E^{−1}.

Before we close this section, a last comment on Theorem 2.1 is in order. Although this theorem is only of theoretical interest, it provides nevertheless an important piece of information. It essentially says that

joint stabilization of Λ ⟺ \{ \begin{matrix}partial stabilization of Λ \\ with respect to its second argument . \end{matrix}

What about partial stabilization of Λ with respect to its first argument ? The following proposition shows that the set

Σ^{F} (E) := limsup_{D→E} Λ(D, F) := ⋂_{ε>0} cl [ ⋃_{||D−E||≤ε}Λ(D, F)]

is almost the same as the stabilized point spectrum Σ(E, F). If there is any difference at all, it can only concern the ”pathological” value λ = 0.

Proposition 2.5 For any (E, F) ∈ U(H), one has

(2.3)  Σ(E, F)\{0} ⊂ Σ^{F} (E) ⊂ Σ(E, F) .

Proof. Take λ ∈ Σ(E, F)\{0}. In view of Theorem 2.3 (c), there are sequences {v_{n}} → 0 and {x_{n}} ⊂ S_{H} such that

v_{n} + λEx_{n} ∈ F (x_{n}) .

This can be written in the form

λE_{n}x_{n} ∈ F (x_{n}) with E_{n} := E + λ^{−1} v_{n} ⊗ x_{n} .

As usual, a ⊗ b ∈ L(H) denotes the tensor product of a ∈ H and b ∈ H, that is to say,

[a ⊗ b](x) := ⟨b, x⟩ a ∀ x ∈ H .

Observe that λ ∈ Λ(E_{n}, F) and

||E_{n} − E|| = ||λ^{−1} v_{n} ⊗ x_{n}|| ≤ |λ|^{−1}||v_{n}|| → 0 .

This proves the first inclusion in (2.3). The second inclusion is obvious.

3. THE USE OF DISCRIMINANT FUNCTIONS

A wealth of spectral results can be derived by cleverly manipulating the discriminant functions Φ and Ψ. Understanding the topological properties of these functions is, of course, a prerequisite to proceed further. As a first step in this direction, we state:

Propostion 3.1. For arbitrary (λ, E, F) and (μ, D, G) in IR × L(H) × K(H), one has

|Ψ(μ, D, G) − Ψ(λ, E, F)| ≤ ||μD − λE|| + δ(G, F) max\{\sqrt{1 + ||μD||^{2}}, \sqrt{1 + ||λE||^{2}}\} .

In particular,

(a) Ψ : IR × L(H) × K(H) → IR is continuous ;

(b) the partial functions Ψ( · , E, F), Ψ(λ, · , F), and Ψ(λ, E, · ) are Lipschitz continuous, their respective Lipschitz constants being ||E||, |λ|, and \sqrt{1 + ||λE||^{2}} .

Proof. For any unit vector x ∈ H, one can write

d_{grG}(x, μDx) − d_{grF} (x, λEx) ≤ |d_{grG}(x, μDx) − d_{grF} (x, μDx)|
+ |d_{grF} (x, μDx) − d_{grF} (x, λEx)|
≤ δ(G, F) ||(x, μDx)|| + ||μDx − λEx||
≤ δ(G, F) \sqrt{1 + ||μD||^{2}} + ||μD − λE|| .

By passing to the infimum over the unit sphere, one obtains

Ψ(μ, D, G) − Ψ(λ, E, F) ≤ δ(G, F) \sqrt{1 + ||μD||^{2}} + ||μD − λE|| .

The combination of this inequality and its symmetric version

Ψ(λ, E, F) − Ψ(μ, D, G) ≤ δ(F, G) \sqrt{1 + ||λE||^{2}} + ||λE − μD||

yields the announced result.

An interesting application of the reduced discriminant function Ψ is discussed right away. It may happen that a complicate operator F ∈ K(H) has been constructed from simpler ones by means of elementary operations like union, intersection, addition, and so forth. The next theorem provides a rule for computing the stabilized point spectrum of a union.

Theorem 3.2. Assume that w ↦ F_{w} is a continuous map from a compact metric space W to the metric space (K(H), δ). Then, the multivalued operator

x ∈ H \overrightarrow{→} F (x) := ⋃_{w∈W}F_{w}(x)

belongs to K(H), and

(3.1)  Σ(E, F) = ⋃_{w∈W}Σ(E, F_{w}) ∀ E ∈ L(H) .

Proof. Checking membership in K(H) is left as exercise. Observe that F admits the alternative characterization

grF = ⋃_{w∈W}grF_{w} ,

from where one obtains

d_{grF} (u, v) = inf_{w∈W} d_{grF_{w}}(u, v) ∀ (u, v) ∈ H × H .

Hence,

Ψ(λ, E, F) = inf_{||x||=1} inf_{w∈W} d_{grF_{w}}(x, λEx) = min_{w∈W} Ψ(λ, E, F_{w}) .

The above minimum is attained because the index set W is compact and the function

w ∈ W ↦ Ψ(λ, E, F_{w})

is continuous (cf. Proposition 3.1). Now, in view of Theorem 2.2, one has

λ ∈ Σ(E, F) ⟺ Ψ(λ, E, F) = 0 ⟺ ∃ w ∈ W such that Ψ(λ, E, F_{w}) = 0 .

This proves the announced formula.

\underline{Remark:} If one drops the compactness assumption on W, then a suitable definition for F would be

grF = cl [ ⋃_{w∈W}grF_{w} ]

The closure operation is being added to make sure that F remains in K(H). However, the calculus rule (3.1) is no longer true. The correct formula is

Σ(E, F) = ⋂_{ε>0} ⋃_{w∈W}{Ψ( · , E, F_{w}) ≤ ε} ,

where {g ≤ ε} := {λ ∈ IR : g(λ) ≤ ε} stands for the ε-sublevel set of g : IR → IR. Observe that, without compactness, one still has

Σ(E, F) ⊃ cl [ ⋃_{w∈W} ⋂_{ε>0} {Ψ( · , E, F_{w}) ≤ ε}] = cl [ ⋃_{w∈W}Σ(E, F_{w})] ,

but the containment relation may be strict.

We now turn the attention to the discriminant function Φ. Among the many properties of this real-valued function, of special interest is the fact that

Φ(λ, E, F) = inf {d_{grF} (u, v)/||(u, v)|| : (u, v) ∈ gr(λE) , (u, v) ≠ (0, 0)}
= inf {d_{grF} (u, v) : (u, v) ∈ gr(λE) , ||(u, v)|| = 1} ,

that is to say,

(3.2)  Φ(λ, E, F) = gap [grF, S_{H×H} ∩ gr(λE)] ,

where the gap between two nonempty sets P, Q ⊂ H × H is defined by

gap [P, Q] := inf {||p − q|| : p ∈ P , q ∈ Q} .

The gap representation (3.2) shows, in particular, that Φ is nonexpansive with respect to the argument F. In fact, Φ admits the following incremental formula :

Proposition 3.3. For arbitrary (λ, E, F) and (μ, D, G) in IR × L(H) × K(H), one has

|Φ(μ, D, G) − Φ(λ, E, F)| ≤ 2 ||μD − λE|| + δ(G, F) .

Hence,

(a) Φ : IR × L(H) × K(H) → IR is continuous ;

(b) the partial functions Φ( · , E, F), Φ(λ, · , F), and Φ(λ, E, · ) are Lipschitz continuous, their respective Lipschitz constants being 2||E||, 2|λ|, and 1 .

Proof. For any nonempty closed set C ⊂ S_{H×H}, the function F ∈ K(H) ↦ gap[grF, C] is nonexpansive, that is to say,

| gap[grG, C] − gap[grF, C] | ≤ δ(G, F) ∀ G, F ∈ K(H) .

Thus,

|Φ(μ, D, G) − Φ(λ, E, F)| ≤ |Φ(μ, D, G) − Φ(μ, D, F)| + |Φ(μ, D, F) − Φ(λ, E, F)|
≤ δ(G, F) + |Φ(μ, D, F) − Φ(λ, E, F)| .

Due to the general identity

Φ(λ, E, F) = Φ(1, λE, F) ,

it suffices to show that E ∈ L(H) ↦ Φ(1, E, F) is a Lipschitz continuous function with Lipschitz constant 2. Observe that

Φ(1, E, F) = inf_{||x||=1} Φ_{x}(E) ,

with

Φ_{x}(E) := Ψ_{x}(E)/M_{x}(E) , Ψ_{x}(E) := d_{grF} (x, Ex) , M_{x}(E) := ||(x, Ex)|| .

Clearly,

Ψ_{x}(D) − Ψ_{x}(E) = Φ_{x}(D)M_{x}(D) − Φ_{x}(E)M_{x}(E)
= Φ_{x}(D){M_{x}(D) − M_{x}(E)} + {Φ_{x}(D) − Φ_{x}(E)}M_{x}(E) .

Thus,

Φ_{x}(D) − Φ_{x}(E) = \frac{Ψ_{x}(D) − Ψ_{x}(E) − Φ_{x}(D)\{M_{x}(D) − M_{x}(E)\}}{M_{x}(E)} .

Since 1 ≤ M_{x}(E) and Φ_{x}(D) ≤ 1, one arrives at

Φ_{x}(D) − Φ_{x}(E) ≤ |Ψ_{x}(D) − Ψ_{x}(E)| + |M_{x}(D) − M_{x}(E)| ≤ 2||D − E|| ,

from where one gets the announced result.

Gap functionals are usually studied in tandem with excess functionals. Recall that the excess of P over Q is understood as the number

excess [P, Q] := sup {d_{Q}(p) : p ∈ P } .

The underlying theory dealing with these concepts leads us to introduce the complementary term

Θ(λ, E, F) := excess [B_{H×H} ∩ grF, N(λE)] ,

where N(λE) corresponds to the orthogonal space of gr(λE), and B_{H×H} denotes the closed unit ball in H × H. A routine computation shows that

[d_{N(λE)}(u, v)]^{2} = ⟨u + λE^{∗}v, [I + λ^{2}E^{∗}E]^{−1}(u + λE^{∗}v)⟩ ,

with E^{∗} denoting the adjoint operator of E. A more explicit representation of the complementary function Θ : IR × L(H) × K(H) → IR is then

(3.3)  Θ(λ, E, F) = sup_{(u,v)∈B_{H×H}∩grF} \sqrt{⟨u + λE^{∗}v, [I + λ^{2}E^{∗}E]^{−1}(u + λE^{∗}v)⟩} .

The introduction of this function seems rather artificial, but the next theorem sets everything straight :

Theorem 3.4. The stabilized point spectrum of (E, F) ∈ U(H) is given by

(3.4)  Σ(E, F) = {λ ∈ IR : Θ(λ, E, F) = 1} .

Proof. Suppose that grF ≠ {(0, 0)}, otherwise there is nothing to prove. By positive homogeneity of d_{N(λE)}(·, ·), one can write

Θ(λ, E, F) = excess [S_{H×H} ∩ grF, N(λE)] .

For each (λ, E, F) in IR × L(H) × K(H), define

Δ(λ, E, F) := gap [S_{H×H} ∩ grF, gr(λE)] .

This expression looks similar to (3.2), but it is not the same. By using the Pythagorean identity

[d_{gr(λE)}(u, v)]^{2} + [d_{N(λE)}(u, v)]^{2} = ||(u, v)||^{2}

one gets

[Δ(λ, E, F)]^{2} = 1 − sup{[d_{N(λE)}(u, v)]^{2} : (u, v) ∈ S_{H×H} ∩ grF } ,

and therefore

[Δ(λ, E, F)]^{2} + [Θ(λ, E, F)]^{2} = 1 .

Due to Theorem 2.2, what remains to be shown is that

Δ(λ, E, F) = 0 ⟺ Φ(λ, E, F) = 0 .

But this follows from an abstract result establishing that

gap [S_{X} ∩ P, Q] = 0 ⟺ gap [P, S_{X} ∩ Q] = 0 ,

whenever P and Q are closed cones in some normed space X.

\underline{Remark:} The characterization (3.4) was anticipated by [6] in the particular case in which E is the identity operator. The proof technique employed in [6] is different from ours, the underlying ingredients being drawn from the theory of conic ε-neighborhoods.

In view of Theorem 3.4 and the general relation

0 ≤ Θ(λ, E, F) ≤ 1 ∀ λ ∈ IR ,

one may interpretate Θ( · , E, F) as a sort of ”membership function” for the set Σ(E, F). Below we state a perturbation result in line with Proposition 3.3. As can be seen, the complementary function Θ has a very nice Lipschitz behavior in each of its arguments.

Proposition 3.5. For arbitrary (λ, E, F) and (μ, D, G) in IR × L(H) × K(H), one has

|Θ(μ, D, G) − Θ(λ, E, F)| ≤ ||μD − λE|| + δ(G, F) .

In particular,

(a) Θ : IR × L(H) × K(H) → IR is continuous ;

(b) the partial functions Θ( · , E, F), Θ(λ, · , F), and Θ(λ, E, · ) are Lipschitz continuous, their respective Lipschitz constants being ||E||, |λ|, and 1 .

Proof. One can easily prove that

sup_{||(u,v)|| ≤1} | d_{grD}(u, v) − d_{grE}(u, v)| ≤ ||D − E|| ∀ D, E ∈ L(H) .

By applying the Walkup-Wets isometry theorem [7], one gets

sup_{||(u,v)|| ≤1} | d_{N(D)}(u, v) − d_{N(E)}(u, v)| ≤ ||D − E|| ∀ D, E ∈ L(H) .

In particular,

d_{N(μD)}(u, v) ≤ ||μD − λE|| + d_{N(λE)}(u, v) ∀ (u, v) ∈ B_{H×H} .

Thus,

Θ(μ, D, G) ≤ ||μD − λE|| + sup{d_{N(λE)}(u, v) : (u, v) ∈ B_{H×H} ∩ grG} .

We now invoke a general property related to the metric δ, namely

B_{H×H} ∩ grG ⊂ B_{H×H} ∩ grF + rB_{H×H} ∀ r > δ(G, F) .

The use of this inclusion leads us to the inequality

Θ(μ, D, G) ≤ ||μD − λE|| +
sup{d_{N(λE)}((u_{1}, v_{1}) + r(u_{2}, v_{2})) : (u_{1}, v_{1}) ∈ B_{H×H} ∩ grF , (u_{2}, v_{2}) ∈ B_{H×H}} .

Since the distance to a linear space is a sublinear function, one obtains

Θ(μ, D, G) ≤ ||μD − λE|| + Θ(λ, E, F) + r sup{d_{N(λE)}(u_{2}, v_{2}) : (u_{2}, v_{2}) ∈ B_{H×H}} .

It suffices now to let r → δ(G, F), and check that the above supremum does not exceed 1 .

\underline{Remark:} The calculus rule (3.1) established in Theorem 3.2 can be proved also by using the representation formula (3.4). It is enough to observe that

Θ(λ, E, F) = max_{w∈W} Θ(λ, E, F_{w}) ,

the maximum being attained because W is compact and the function w ∈ W ↦ Θ(λ, E, F_{w}) is continuous (cf. Proposition 3.5).

Another interesting application of Θ has to do with the issue of detecting points of outersemicontinuity for the spectral mapping Λ. What we want to do now is trying to recognize some pairs (E, F) ∈ U(H) for which the equality Σ(E, F) = Λ(E, F) holds. To begin with, we state a technical lemma involving the set

Γ(E, F) := {λ ∈ IR : d_{gr(λE)}(·, ·) attains its supremum over B_{H×H} ∩ grF }

which appears implicitly in the definition of the complementary function Θ.

Lemma 3.6 For any (E, F) ∈ U(H), one has

(3.5)  Σ(E, F) ∩ Γ(E, F) ⊂ Λ(E, F) ⊂ Σ(E, F) .

Proof. The second inclusion in (3.5) is obvious. It has been stated here only for the sake of convenience. The proof of the first inclusion follows essentially the same steps as in [6, Corollary 4.3]. Take any λ ∈ Σ(E, F) ∩ Γ(E, F). By Theorem 3.4, we know that

1 = Θ(λ, E, F) = d_{gr(λE)}(u_{0}, v_{0}) for some (u_{0}, v_{0}) ∈ B_{H×H} ∩ grF .

Without loss of generality, one may suppose that (u_{0}, v_{0}) ∈ S_{H×H}. The reader can check that (u_{0}, v_{0}) solves the variational problem

maximize {⟨u + λE^{∗}v, [I + λ^{2}E^{∗}E]^{−1}(u + λE^{∗}v)⟩ : ||u||^{2} + ||v||^{2} = 1 } .

By applying the standard technique of Lagrange multipliers, one deduces that v_{0} = λEu_{0} with u_{0} ≠ 0. This shows that λ ∈ Λ(E, F).

As a consequence of (3.5), one sees that (E, F) is a point of outer-semicontinuity of Λ whenever Γ(E, F) = IR. In particular,

Proposition 3.7 (cf.[6]) If the graph of F ∈ K(H) is locally compact, then

Σ(E, F) = Λ(E, F) ∀ E ∈ L(H) .

As a general rule, local compactness of grF is a fairly strong hypothesis. Remember that we are working with multivalued operators defined on infinite dimensional Hilbert spaces.

4. EXTENSION TO THE POLYNOMIAL EIGENVALUE PROBLEM

Polynomial eigenvalue problems arise in connection with the analysis of higher-order differential systems. To fix the ideas, consider the model

(4.1)  E_{1}z′(t) + E_{2}z′′(t) + ⋯ + E_{m}z^{(m)}(t) ∈ F (z(t)) ,

with multivalued right-hand side F ∈ K(H). The operators E_{1}, ⋯, E_{m} ∈ L(H) are introduced to incorporate more flexibility to the model, but they do not play a prominent role. The complexity of (4.1) derives essentially from the multivaluedness of F.

Since z(·) = 0 is clearly a solution to the above inclusion, it is natural to ask whether or not a system like (4.1) admits nontrivial trajectories. One way of constructing nontrivial trajectories is by solving the polynomial eigenvalue problem

(4.2)  find λ ∈ IR and x ∈ H\{0} such that [λE_{1} + λ^{2}E_{2} + ⋯ + λ^{m}E_{m}]x ∈ F (x) .

Once a pair (λ, x) has been found, it is enough to set z(t) = exp(λt) x .

\underline{Remark:} Solutions of the exponential type do not exhaust, however, all the possibilities. We are not detecting here the possible solutions arising from the presence of ”complex” eigenvalues. Dealing with complex eigenvalues in a multivalued setting is a delicate matter that will be treated elsewhere.

Usually, a system like (4.1) is converted to a bigger system of first-order (m=1), remaining in this way within the context of the standard eigenvalue problem (1.1). We prefer, however, to handle the system (4.1) in its original form. In fact, the theory of stabilized point spectra developed in Sections 2 and 3 extends without extra effort from the standard case to the polynomial case. A minor notational adjustment is all what is needed. For convenience, we write

λ ⋄ E := λE_{1} + λ^{2}E_{2} + ⋯ + λ^{m}E_{m} .

Some authors refer to λ ⋄ E ∈ L(H) as the operator pencil associated to the m-tuple

E := (E_{1}, ⋯, E_{m}) ∈ [L(H)]^{m} .

The stabilized version of

Λ_{m}(E, F) := {λ ∈ IR : 0 ∈ (F − λ ⋄ E)(x) for some x ≠ 0}

corresponds to

Σ_{m}(E, F) = limsup_{(D,G)→(E,F)} Λ_{m}(D, G) ,

where the convergence (D, G) → (E, F) takes place in the metric space [L(H)]^{m} × K(H). Rules for checking membership in Σ_{m}(E, F) are listed below. The proof of the next theorem is omitted since everything follows directly from Sections 2 and 3.

Theorem 4.1 Let (E, F) ∈ [L(H)]^{m} × K(H). For λ ∈ IR, the following conditions are equivalent :

(a) λ ∈ Σ_{m}(E, F) ;

(b) Φ(1, λ ⋄ E, F) = 0 ;

(c) Ψ(1, λ ⋄ E, F) = 0 ;

(d) Θ(1, λ ⋄ E, F) = 1 ;

(e) 0 lies in the closure of (F − λ ⋄ E)(S_{H}) ;

(f) liminf_{||x||→∞} \frac{1}{||x||} d_{grF} (x, [λ ⋄ E]x) = 0 ;

(g) liminf_{||x||→∞} \frac{1}{||x||} d_{F (x)}([λ ⋄ E]x) = 0 ;

(h) there are unit vectors {x_{n}}_{n∈IN} in H such that d_{grF} (x_{n}, [λ ⋄ E]x_{n}) → 0 ;

(i) there are unit vectors {x_{n}}_{n∈IN} in H such that d_{F (x_{n})}([λ ⋄ E]x_{n}) → 0 .

Although the polynomial case bears many similarities with respect to the first-order case, there are also some important differences that one should be aware of. For instance, we know that

λ ∈ IR ↦ Ψ(1, λE, F)

is Lipschitz continuous with Lipschitz constant ||E||. This function serves therefore to estimate the distance

d_{Σ(E,F)}(λ) := inf{ |λ − μ| : μ ∈ Σ(E, F)}

from an arbitrary λ ∈ IR to the set Σ(E, F). Indeed, by combining Theorem 2.2 and Proposition 3.1, one gets

\frac{Ψ(1, λE, F)}{||E||} ≤ d_{Σ(E,F)}(λ) ∀λ ∈ IR ,

where (E, F) ∈ U(H) is any pair such that E ≠ 0. In constrast with the first-order case, the function

λ ∈ IR ↦ Ψ(1, λ ⋄ E, F)

does not need to behave in a Lipschitz manner. This fact rules out the possibility of obtaining an analogous estimate for the distance to Σ_{m}(E, F). Such a remark applies also to the discriminant function Φ and the complementary function Θ.

5. BY WAY OF CONCLUSION

The stabilized point spectral mapping Σ has been defined in Section 1 through the expression (1.4). With such a definition,

grΣ := {(E, F, λ) ∈ U(H) × IR : λ ∈ Σ(E, F)}

corresponds precisely to the topological closure of

grΛ := {(E, F, λ) ∈ U(H) × IR : λ ∈ Λ(E, F)}

in the metric space (U(H) × IR, ρ × | · |). The expression (1.4) is, however, of limited interest when it comes to evaluate Σ(E, F) for a particular pair (E, F). One of the main goals of this paper was deriving alternative characterizations for the set Σ(E, F). This issue has been settled with the help of Theorems 2.2 and 2.3.

A fundamental role has been played by the complementary function Θ, as well as by the discriminant function Φ and its reduced version Ψ. It has been shown that

λ ∈ Σ(E, F) ⟺ Θ(λ, E, F) = 1 ⟺ Φ(λ, E, F) = 0 ⟺ Ψ(λ, E, F) = 0 .

All these functions are (jointly) continuous and behave in a Lipschitz continuous manner with respect to each of their arguments. Such properties turn out to be relevant for various purposes. For instance, the calculus rule established in Theorem 3.2 exploits the continuity of Ψ in its last argument.

\underline{Acknowledgements :} This work was completed while the author was visiting IMPA at Rio de Janeiro. Thanks are due to Professor Alfredo Iusem for his hospitality and for many fruitful discussions on the subject of this paper. The author acknowledges also the financial support from the France-Brazil Exchange Program in Mathematics.

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[3] P. Gajardo and A. Seeger, Epsilon-eigenvalues of multivalued operators, Preprint No. 30, Dep. of Mathematics, Univ. of Avignon, 2001.

[4] P. Lavilledieu and A. Seeger, Eigenvalue stability for multivalued operators, Topol. Meth. in Nonlinear Analysis, 15 (2000), 115-128.

[5] P. Lavilledieu and A. Seeger, Existence de valeurs propres pour les systèmes multivoques: résultats anciens et nouveaux, Ann. Sci. Math. Québec, 25 (2001), 47-70.

[6] A. Seeger, Stabilization of the point spectral mapping, Preprint No. 32, Dep. of Mathematics, Univ. of Avignon, 2001, submitted.

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A. SEEGER

University of Avignon, Department of Mathematics, 33, rue Louis Pasteur, 84000 Avignon, France e-mail : alberto.seeger@univ-avignon.fr