Global folds between Banach spaces as perturbations

0.23 MB·

by  Marta Calanchi

Global folds between Banach spaces
as perturbations
Marta Calanchi, Carlos Tomei and Andr´e Zaccur
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2 Abstract

n
a Global folds between Banach spaces are obtained from a simple geo-
J metric construction: a Fredholm operator T of index zero with one dimen-

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sional kernel is perturbed by a compatible nonlinear term P. The scheme
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encapsulates most of the known examples and suggests new ones. Con-

] crete examples rely on the positivity of an eigenfunction. For the standard
P
Nemitskii case P(u) = f(u) (but P might be nonlocal, non-variational),
A
T might be the Laplacian with different boundary conditions, as in the
.
h
Ambrosetti-Prodi theorem, or the Schro¨dinger operators associated with
t
a the quantum harmonic oscillator or the Hydrogen atom, a spectral frac-
m
tional Laplacian, a (nonsymmetric) Markov operator. For self-adjoint op-
[
erators, we use results on the nondegeneracy of the ground state. On

1 Banach spaces, a similar role is played by a recent extension by Zhang of
v
the Krein-Rutman theorem.
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3 Keywords: Ambrosetti-Prodi theorem, folds, Krein-Rutmantheorem, positivity
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preserving semigroups.
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1. MSC-class: 35J65, 47H15, 47H30, 58K05.
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1 Introduction
:
v
i
X
A continuous function F : X → Y between real Banach spaces X and Y is a
r global foldifthere isaBanachspace W andhomeomorphisms ζ : X → W⊕R and
a
ξ : Y → W ⊕R for which F˜(w,t) = ξ◦F ◦ζ−1(w,t) = (w,−|t|). The celebrated
Ambrosetti-Prodi theorem [1], refined and reinterpreted by subsequent work of
Micheletti and Manes ([16]), Berger and Podolak ([5]) and Berger and Church
([6]), describes a class of functions given by nonlinear differential operators which
are global folds (the result is presented below in this introduction).
More than a theorem, these works present a strategy to identify folds. In
a similar vein, Church and Timourian ([11]) introduced other characterizations,
leading to new examples and simpler arguments in known situations. Difficulties
frequently arise when verifying the necessary hypotheses. The same authors also
1
provided sufficient conditions for a global fold, closer in spirit to Berger and
Podolak ([5]), that are easier to check.
We take a similar point of view. Start with a Fredholm operator T : X → Y
of index 0, dimkerT = 1, which by simple linear changes of variable becomes
Ta : W ⊕R → W ⊕R, Ta(w,t) = (w,0) ,
where W = RanT. For each w ∈ W fixed, the image under Ta of a vertical
0
line {(w ,t),t ∈ R} is the point (w ,0). Consider a unimodal function ha(w ,.)
0 0 0
whose domain splits in two intervals on which it is first strictly increasing and
then strictly decreasing, and suppose that ha(w ,t) → −∞ as |t| → ∞. Clearly
0
(w,t) ∈ W ⊕R 7→ (w,ha(w,t)) ∈ W ⊕R
is a global fold. And a homeomorphism on the domain which keeps invariant
each horizontal plane, Ψa(w,t) = (Fa(w),t) preserves the fold structure. As we
t
shall see, the familiar global folds described in [11] are of the form Fa(w,t) =
(Fa(w),ha(Fa(w),t)). This geometric approach leads also to algorithms for the
t t
numerical solution of a class of nonlinear differential equations ([24], [7]).
This approach motivates Theorem A, implicit in [23]. A continuous (resp.
Lipschitz, Ck) map F : X → Y admits an LS-decomposition ( [5], [8],[17],[18])
if there is a (Lipschitz, Ck) homeomorphism Φ : Y → X and a continuous
(Lipschitz, Ck) ha : W ⊕R → R such that Fa = F ◦Φ is a rank one nonlinear
Y
perturbation of the identity,
Fa : Y = (W ⊕R) → W ⊕R, (z,t) 7→ (z,ha(z,t)).
Y Y
Clearly, F is a global fold if and only if Fa is. Singularity theorists say that Fa
provides adapted coordinates to F.
Theorem A. Suppose F : X → Y is a continuous map. If F satisfies the
hypotheses below, it is either a homeomorphism or a global fold.
(LS) F admits an LS-decomposition.
(PR) F is proper.
(NT) No point of Y has three preimages under F.
Theorem A implies Theorems 1 and 2 below, from which we obtain more
examples ofglobalfolds, withsimpler arguments. Weconsider F(u) = Tu−P(u),
where T is a linear operator and P a nonlinear perturbation. The Ambrosetti-
Prodi theorem is the case T = −∆−λ I with Dirichlet boundary conditions on
m
a smooth bounded domain, λ is its smallest eigenvalue, and P is the operator
m
P(u) = f(u)−λ u for a strictly convex C2 function f : R → R such that f′(R)
m
intersects the spectrum of T only at λ . By taking appropriate translations, we
m
may consider relevant eigenvalues of T to be different from zero.
2
We sacrifice generality and concentrate on two classes of amenable operators.
An eigenvalue λ ∈ C of T : X → Y is elementary if
(I) T − λI : X → Y is a Fredholm operator of index zero and ker(T − λI) is
spanned by the eigenvector φ.
(II) There is no w ∈ X for which (T −λI)w = φ.
(III) λ is an isolated point of σ(T).
Let H = L2(M,dµ) for a σ-finite measure space (M,µ). A self-adjoint oper-
ator T : D ⊂ H → H is m-amenable if
(m-a) λ = min σ(T) is an elementary eigenvalue associated with a normalized
m
eigenvector φ > 0 a.e.
m
(m-b) For all t > 0, the operators e−tT : H → H are positivity improving: for
any nonzero g ≥ 0, e−tTg > 0 a.e.
Self-adjoint operators with positive ground state φ are the archetypical ex-
m
amples of m-amenability. Also, m-amenability will still hold for the Jacobians
DF(u) = T −DP(u) for m-compatible perturbations P(u) : H → H, defined in
Section 3. And such perturbations lead to folds.
Theorem 1. Let T : D → H be an m-amenable operator and P : D → H and
be an m-compatible C1 function. Then F = T −P : D → H is a global fold.
An operator T : X → X with spectral radius r(T) is M-amenable if
(M-a) r(T) is an elementary eigenvalue of T and σ(T)∩{z ∈ C,|z| = r} = r(T).
(M-b) There is a cone K ⊂ X with interior K˚ 6= ∅ for which T(K \{0}) ⊂ K˚.
Markov operators are related to M-amenability. The usual fact that φ ∈ K˚
M
under small M-compatible perturbations P(u), defined in Section 4, is proved by
an extension of the Krein-Rutman theorem due to Zhang [29].
Theorem 2. Let T : X → X be an M-amenable operator and P : X → X and
be an M-compatible C1 function. Then there is R ∈ R so that if, for all u ∈ X,
T
we have kDP(u)−r(T) Ik < R then F = T −P : X → X is a global fold.
T
Amenability of T and of a translation T +R I, for a large R, are essentially
equivalent. As the attentive reader will notice along the text, this fact essentially
masks therelevance ofthis featureintheoriginal Ambrosetti-prodi context, when
T is the Laplacian on a bounded set and P(u) = f(u) is a Nemitskii operators ,
where f : R → R has bounded derivative.
In the self-adjoint case, the hypotheses needed for our arguments are given
by spectral data of T. For M-amenable operators, however, we are limited to a
perturbation result: the nonlinear term P −r(T) I (or better, its Jacobian) has
to be sufficiently small.
3
Most arguments are geometric and do not depend on the locality of amenable
or compatible operators. We work with positive functions in the Hilbert context
and privileged cones in Banach spaces, so as to preserve familiar procedures.
Smoothness is a key point in this text. In Section 2 we introduce hypotheses
yielding (LS) and (PR) for a Lipschitz map F (providing details for arguments
in [26], [8]), and the proofs of Theorems 1 and 2 then depend only on the veri-
fication of (NT). For Lipschitz compatible perturbations given by maps close to
Nemitskii operators, a proof of (NT) is presented in Section 5.3. For more general
compatible perturbations, we require that the maps F be C1, and consider spec-
tral properties of the Jacobians DF(u) and some averages, the mean Jacobians
MF(u,v), defined in Section 3. For m-compatible operators, (NT) is obtained
in Section 3 from the study of the smallest eigenvalue of mean Jacobians. In the
M-compatible case instead it is the spectral radius which has to be controlled
and we prove (NT) in Section 4.
As we shall see among the examples in Section 5.1, some amenable oper-
ators are the Laplacian itself (with Dirichlet, Neumann or periodic boundary
conditions), the hydrogen atom Tv = −∆v −v/r in R3, the quantum harmonic
oscillator Tv(x) = −v′′(x)+x2v(x), spectral fractional Laplacians in bounded do-
mains, integral operators with positive kernels. We consider only perturbations
T − P(u) but nonautonomous maps T − P(x,u) may be handled with minor
modifications. Section 5.2 comments on the rarity of global folds associated with
compatible Nemitskii operators, a fact shown in [9] for the Laplacian. Section
5.3 shows an equivariant version of the Ambrosetti-Prodi theorem: in this case,
the associated compatible operator is nonlocal.
For a C2 function F, the differentiability of the eigenvalues of the Jacobians
permits additional simplifications, as shown in Corollary 1. The result is then
used inSection5.5, which ina sense isanideal situation. Wedescribe amapfrom
L2 to L2 which restricts appropriately to one from C0 to C0 with the following
properties. There is sufficient (spectral) information of the self-adjoint kind to
derive (LS) and (PR) on the L2 scenario. We then transplant these facts to an
associated map in C0, on which the hypotheses leading to (NT) are especially
weak, and end up with global folds in both scenarios.
This text provides the proofs of most of the results stated in [8].
2 Hypotheses (LS) and (PR), fibers and heights
Thefirst two hypotheses ofTheorem A,(LS)and(PR),admit aunifiedtreatment
for both kinds of amenable operators. There are other approaches to (LS) ([22],
[17], also Section 5.5), which are not relevant in Sections 3 and 4.
For m-admissibile operators T : D → H, the original Banach spaces are
the real Hilbert space Y = H = L2(M,dµ) and X = D ⊂ H, the domain of
self-adjointness of T (with norm kuk = kuk + kTuk). Let V be the subspace
D
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spanned by φ , the eigenvector associated with the simple eigenvalue λ . Since
m m
T is m-admissible, W = Ran(T − λ I) is a closed subspace of codimension
H m
one. Then W = W ∩D induce orthogonal decompositions
D H
D = W ⊕V , H = W ⊕V
D H
andanorthogonalprojectionΠ : H → W . TherestrictionT−λ I : W → W
H m D H
is an invertible operator.
For M-admissible operators T : X → X, X = Y is a real Banach space. Let
V be the span of φ , and set W = Ran(T − λ I), so that again there are
M X M
decompositions X = Y = W ⊕V, a projection Π : X → W and an invertible
X X
restriction T −λ I : W → W .
M X X
To unify notation, we write the functions of interest as
F = T −P : W ⊕V → W ⊕V
X Y
where the spaces are defined in terms of a privileged eigenpair (λ ,φ ) (p is m
p p
or M, depending on the type of amenability). We also write Π : Y → W .
Y
Define the translations P = P −γI,T = T −γI : X → Y and the restriction
γ γ
T = T −γI : W → W
γ,W W X Y
Proposition 1. Let F = T −P : X → Y for an amenable operator T : X → Y
and assume
(HLS) There exists γ ∈ R for which the restriction T : W → W is invertible
γ,W X Y
and the map ΠP : X → W is a Lipschitz function with a constant L
γ Y
satisfying LkT−1 k < 1.
γ,W
Then hypothesis (LS) holds for the map F : X → Y.
The projection Π in (HLS) does not appear in arguments when P is a Ne-
mitskii operator. Indeed, a function f whose derivatives takes values close to
λ gives rise to a Nemitskii map P(u) which looks pretty much like a multiple
p
of the identity. Thus, f acts rather homogeneously in all directions of space.
The situation for more general maps P is different: in adapted coordinates, as
presented in the Introduction, large distortions along vertical lines in the image
do not change the global nature of the fold. Such distortions correspond to large
values of P(u) along φ and are trivialized by the action of the projection Π.
m
Proof: The argument follows [5]. Write u = Πu+tφ = w +tφ , for w ∈ W .
p p X
For t ∈ R, define the projected restrictions F : W → W ,
t X Y
F (w) = ΠF(w+tφ ) = Π(T −P)(w+tφ ) = T w −ΠP (w +tφ ) .
t p p γ,W γ p
Set T w = y to obtain
γ,W
F ◦T−1 : W → W , F (y) = y −ΠP (T−1 y +tφ ) = y −K (y).
t γ,W Y Y t γ γ,W p t
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We bound variations of K : W → W . For z,z˜∈ W , s,s˜∈ R,
t Y Y Y
kK (z)−K (z˜)k = kΠP (T−1 z +sφ )−ΠP (T−1 z˜+s˜φ )k
s s˜ γ γ,W p γ γ,W p
≤ LkT−1 (z −z˜)k+C|s−s˜| ≤ ckz −z˜k+C|s−s˜|
γ,W
for c < 1 by (HLS) and C possibly large.
From the Banach contraction theorem, I − K : W → W are (uniform)
t Y Y
Lipschitz bijections. We show that the maps (Id−K )−1 : W → W are also
t Y Y
uniformly Lipschitz: set z = (Id−K )−1(y ), i = 1,2 and then
i t i
||z −z || ≤ ||y −y ||+||K (z )−K (z )|| ≤ ||y −y ||+c||z −z ||
1 2 1 2 t 1 t 2 1 2 1 2
so that
1
kz −z k ≤ ky −y k .
1 2 1 2
1−c
Thus F = (I −K )◦T : W → W are also uniformly bilipschitz homeo-
t t γ,W X Y
morphisms. We now show that
Φ−1 = Ψ = (F , Id) : Y → X
t
is a bilipschitz homeomorphism. Clearly, Ψ is a Lipschitz bijection. To handle
Ψ−1 = Φ, take v = y + sφ and v˜ = y˜ + s˜φ (the letter C represents different
constants along the computations):
||Φ(v)−Φ(v˜)|| ≤ C ||(F )−1(y)−(F )−1(y˜)|| +||sφ−s˜φ||
X s s˜ X X
(cid:0) (cid:1)
≤ C ||(F )−1(y)−(F )−1(y)|| +||(F )−1(y)−(F )−1(y˜)|| +|s−s˜|
s s˜ X s˜ s˜ X X
(cid:0) (cid:1)
For the second term use the Lipschitz bound for F−1. For the first, we prove
s˜
||F−1(y)−F−1(y)|| ≤ C|s−s˜|. As before, set w = F−1(y),w˜ = F−1(y) and
s s˜ s s˜
z = T ◦F−1(y) = (1−K )−1(y) , z˜= T ◦F−1(y) = (1−K )−1(y) .
γ,W s s γ,W s˜ s˜
The iterations yielding z and z˜,
z = 0, z = K (z )+y and z˜ = 0, z˜ = K (z˜ )+y
0 j+1 s j 0 j+1 s˜ j
imply the estimates
||z −z˜ || = ||K (z )−K (z˜ ))|| ≤ c ||z −z˜ ||+C |s−s˜| ,
j+1 j+1 s j s˜ j j j
and for j → +∞,
||T−1 ||
||F−1(y)−F−1(y)|| = ||w−w˜|| ≤ ||T−1 ||||z −z˜|| ≤ C γ,W |s−s˜| .
s s˜ γ,W 1−c
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Adding up,
||T−1 ||
γ,W
||Φ(v)−Φ(v˜)|| ≤ C |s−s˜|+Cky −y˜k +||sφ−s˜φ|| ,
X X X
1−c
completing the proof that Ψ : X → Y is a bilipschitz homeomorphism. Hypoth-
esis (LS) is clear from the following diagram, for Fa = F ◦Φ:
F
D = W ⊕V −→ H = W ⊕V
X Y
Ψ=(Ft,Id) ց ր Fa=F◦Φ=(Id,ha)
W ⊕V
Y
For a fixed z ∈ W , the inverse of a vertical line {(z,s),s ∈ R} under Ψ is a
Y
fiber {u(z,t) = w(z,t)+tφ , t ∈ R}. The height function is
p
h : D = W ⊕V → R, h(u) = hφ ,F(u)i = hφ , Tu−P(u)i .
X p p
In the M-amenable case, the dual operator T∗ : X∗ → X∗ also has a simple
eigenvalue λ , associated with the eigenvector φ∗ given by the functional which
M M
is zero on W and normalized so as hφ∗ ,φ i = 1 (notice the coupling between
X M M
X and X∗). To unify notation, set λ∗ = φ or φ∗ .
p m M
Proposition 2. For T : X → Y amenable and F = T −P : X → Y, assume
(HPR) There exist λ ,λ ,c ,c ∈ R with λ < λ < λ for which
− + − + − p +
∀ u ∈ X , hφ∗,P(u)i ≥ λ hφ∗,ui +c , λ hφ∗,ui +c .
p − p − + p +
and (HLS). Then F : X → Y satisfies hypothesis (PR).
Proof: From (HPR), height functions go to −∞ along fibers. Indeed,
h(u(z,t)) = hφ ,Tw(z,t)+tTφ i−hφ ,P(u(z,t))i
p p p
= tλ −hφ ,P(u(z,t))i ≤ (λ −λ ) t−c , (λ −λ ) t−c ,
p p p − − p + +
and the limits follow (t → −∞ from the first bound, t → ∞ from the second),
together with their uniformity in z. Since the homeomorphism Ψ : D → H
preserves the horizontal component, lim ha(z,t) = −∞.
|t|→∞
The properness ofF : D → H is equivalent tothat ofFa : W ⊕V → W ⊕V,
Y Y
which we prove. Take (z ,y ) = (z ,ha(z ,t )) → (z ,y ). If |t | → ∞, then
n n n n n ∞ ∞ n
ha(z ,t ) → −∞, contradicting the uniform convergence at z .
∞ n ∞
A sketch of proof of Theorem A
By hypothesis (LS), Theorem Areduces to thestatement thatthe proper map
Fa : W ⊕R → W ⊕R, (z,t) 7→ (z,ha(z,t))
Y Y
is either a homeomorphism or a global fold. But, by hypothesis (NT), this is
exactly Proposition 9 in [23].
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3 m-amenable operators
More general self-adjoint operators giving rise to global folds were presented be-
fore ([3]). Here we consider m-amenable operators T : D → H: more concretely,
the Banach spaces X and Y from the previous sections are Y = H = L2(M,dµ)
for a σ-finite measure space (M,µ) and X = D ⊂ H is the domain of self-
adjointness of T, equipped with the norm kuk = kuk + kTuk . Euclidean
D H H
space D = H = Rn is the case when µ is a finite collection of deltas.
The spectrum of T : D ⊂ H → H will be denoted by σ(T). Following [21],
we say that a function of u ∈ H is positive if u ≥ 0 a.e. and u 6= 0. A bounded
operator A : H → H is positivity preserving if Au is positive for all positive u.
A C1 function P : H → H is m-compatible with an m-amenable operator
T : D → H if it satisfies the following properties. Let µ = minσ(T)\{λ }.
m
(m-LS) There are a,b ∈ R, a < λ < b < µ, such that, for u ∈ D, DP(u) : H → H
m
is a bounded, symmetric operator with σ(DP(u)) ⊂ [a,b].
(m-PR) There exist λ ,λ ,c ,c ∈ R with λ < λ < λ for which
− + − + − m +
∀ u ∈ D , hφ ,P(u)i ≥ λ hφ ,ui +c , λ hφ ,ui +c .
m − m − + m +
(m-NT1) For every t > 0,u ∈ D, etDP(u) : H → H is positive preserving.
(m-NT2) For u,v ∈ D, if v −u > 0 a.e. then infσ(DF(v)) < infσ(DF(u)).
As we shall see in Proposition4, the m-amenability of T together with (m-LS)
above implies the m-amenability of any Jacobian DF(u) : D → H. In particular,
there is a continuous function λ : D → R which associates to u ∈ D thesmallest
m
(elementary) eigenvalue of DF(u) (and then λ (DF(u)) = infσ(DF(u))).
m
3.1 Proof of Theorem 1
We must verify hypotheses (LS), (PR) and (NT) of Theorem A for m-amenable
operators and m-compatible C1 perturbations.
Proposition 3. Let T : H → D be m-amenable, and P : H → H be a C1 map
for which (m-LS) and (m-PR) hold. Then F satisfies (HLS) and (HPR).
Proof: We show that (m-LS) implies (HLS) for γ = (a+b)/2. For P = P −γI
γ
and u(τ) = τu+(1−τ)u˜,
1
P (u)−P (u˜) = DP (u(τ)) u−u˜ dτ
γ γ Z γ
0 (cid:0) (cid:1)
8
and then, since kΠk = 1,
1
kΠP (u)−ΠP (u˜)k = k ΠDP (u(τ)) u−u˜ dτk ≤ (b−γ)ku−u˜k .
γ γ Z γ
0 (cid:0) (cid:1)
Thus L = b−γ and since kT−1 k = k(T −γI)−1k ≥ (µ−γ)−1, (HLS) follows.
γ,W W
Clearly (m-PR) implies (HPR).
In [23], (NT) was verified for weaker differentiability conditions, but less gen-
eral perturbations P: the argument was given for Nemitskii operators which are
only Lipschitz, and is presented in slightly modified form in Section 5.3. We need
an extension of the argument.
By (m-LS), for all u ∈ D, the operators DP(u) : H → H are bounded, and
by the Kato-Rellich theorem, the Jacobians DF(u) = T −DP(u) : D → H are
self-adjoint. For u,v ∈ D, define the mean Jacobian
1 1
MF(u,v) = DF(u+t(v −u))dt = T − DP(u+t(v −u))dt ,
Z Z
0 0
so that MF = T −MP(u,v), for a symmetric, bounded operator MP(u,v). No-
tice that MF(u,u) = DF(u) and MF(u,v) = MF(v,u): to simplify statements,
we treat Jacobians as special cases of mean Jacobians.
Lemma A below is exercise 91 of Chapter XIII from [21], a result by Faris
[13]. Lemma B is Theorem XIII.44 of [21]. A bounded self-adjoint operator
A : H → H is positivity improving if for any nonzero f ≥ 0, Af > 0 a.e.
Lemma A. Let T : D → H be a self-adjoint operator for which e−tT, t > 0,
is positivity improving. Let −V : H → H be a positivity preserving bounded
operator. Then, for t > 0, e−t(T+V) is positivity improving.
Lemma B. Let T : D → H be a self-adjoint operator that is bounded from below.
Suppose that e−tT is positivity preserving for all t > 0 and that E = minσ(T) is
an eigenvalue. Then the following are equivalent.
(a) E is a simple eigenvalue with a strictly positive eigenvector.
(b) For all t > 0, e−tT is positivity improving.
Proposition 4. For u,v ∈ D, if 0 ∈ σ(MF(u,v)) then 0 = minσ(MF(u,v)).
The operators MF(u,v) : D → H and MF(u,v) : D → H are m-amenable.
Proof: Clearly (m-LS) holds for DP(u) and MP(u,v). By the Weyl inequali-
ties, (m-LS) implies that the eigenvalues in σ(T)\{λ } are not perturbed enough
m
to reach 0, and thus only the smallest eigenvalue of MF(u,v) can be zero. We
now prove m-amenability of S = MF(u,v) = T −MP(u,v). For t > 0, e−tT is
positivity improving by (m-a) and etMP(u,v) is positivity preserving by (m-NT1).
By Lemma A, e−tS is positivity improving, so that (m-b) holds for S. By Lemma
B, condition (m-a) for S also holds.
9
We are ready for the proof of (NT). Suppose by contradiction that there is
g ∈ H with three preimages u ,u ,u ∈ D, F(u ) = F(u ) = F(u ) = g, so that
1 2 3 1 2 3
F(u )−F(u ) = 0 = F(u )−F(u ). (1)
2 1 3 2
The u ’s belong to the same fiber (the inverse under F of the vertical line through
i
g), so write u = t φ +w ,i = 1,2,3 for t < t < t and w ∈ W = hφ i⊥.
i i m i 1 2 3 i D m
From the fundamental theorem of calculus,
1
F(u )−F(u ) = DF(tu +(1−t)u ) dt (u −u )
i+1 i Z i+1 i i+1 i
0
= MF(u ,u )(u −u ) = 0 , i = 1,2 .
i+1 i i+1 i
Thus u −u ∈ kerMF(u ,u ) and form the proposition above, the functions
i+1 i i+1 i
u −u have a well defined sign. More,
i+1 i
hu −u , φ i = h(t φ +w )−(t φ +w ) , φ i
i+1 i m i+1 m i+1 i m i m
= h(t −t ) φ , φ i = t −t > 0
i+1 i m m i+1 i
and since φ > 0 we must have u > u > u a.e.
m 3 2 1
The operators MF(u ,u ),i = 1,2 both have a zero eigenvalue, at the bot-
i+1 i
tom of their spectra, by Proposition 4. As u + t(u − u ) < u + t(u − u )
1 2 1 2 3 2
for each t ∈ [0,1],thus the integrand defining MF(u ,u ) is smaller than that of
2 1
MF(u ,u ). Since the smallest eigenvalue λ (MF(u,v)) is the minimal value
3 2 m
of the associated quadratic form (and it is achieved!), zero cannot be a common
eigenvalue. The proof of (NT) is complete.
Finally, F : D → H cannot be a homeomorphism, a possibility left by Theo-
rem 1: at each fiber, Fa behaves like x → −x2, by the proof of (HPR).
4 M-amenable operators
LetX bearealBanachspace, B(X)bethespaceofboundedlinearoperatorsfrom
X to X equipped with the usual sup norm. Among the many almost equivalent
definitions of the essential spectrum σ (T) in [12], we use the fourth one,
e
σ (T) = {λ ∈ C : T −λ I : X → X is not a Fredholm operator of index zero} .
e
Denotethespectral radiusofT byr(T)andtheessential spectral radiusbyr (T).
e
The points of σ(T) outside of the circle {|z| ≤ r (T)} are isolated eigenvalues of
e
finite (algebraic) multiplicity. For the dual operator T∗ : X∗ → X∗, r(T∗) = r(T)
and r (T∗) = r (T), from the characterization in terms of Fredholm theory.
e e
10