DIPLOMARBEIT

Spectral Theory for Nonlinear Operators

ausgefuhrt am Institut fur

Analysis und Scientific Computing

der Technischen Universitat Wien

unter der Anleitung von

Ao.Univ.Prof. Dipl.-Ing. Dr.techn. Michael

Kaltenback

durch

Andreas Widder, B.Sc.

Matr. Nr. 0625002

Otto Glockel-Gasse 7

7210 Mattersburg

Wien, 6.05.2012

Contents

1 Introduction 1

1.1 Spectral theory for linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Spectral theory for nonlinear operators . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Preliminary considerations 4

2.1 Characteristics of nonlinear operators . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Dugundji Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Set-valued maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 The Antipodal Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 The FMV-spectrum 24

3.1 FMV-regular operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Stable solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 FMV-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Subdivision of the FMV-spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Special classes of operators . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Eigenvalues and approximate eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Numerical range of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Applications 48

4.1 Maps on spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 The nonlinear Fredholm alternative . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 1

Introduction

1.1 Spectral theory for linear operators

We first want to recapitulate some basic facts about spectral theory for a bounded linear

operator T that operates on a Banach space X over the field K = R or K = C. The set

of all such operators will be denoted as B(X) and will be equipped with the operator norm

‖T‖ = sup‖Tx‖ : ‖x‖ = 1.

We start with defining the resolvent set

ρ(T ) := λ ∈ K : λI − T is bijective,

where I is the identity operator on X. It can easily be shown that ρ(T ) coincides with the set

of points such that λI −T is invertible and its inverse is also a bounded operator. Therefore, it

also coincides with the set of points where λI − T is a homeomorphism. The inverse operator

R(λ, T ) := (λI − T )−1 is called the resolvent operator of T at λ. Since λ ∈ ρ(T ) and µ ∈ Kwith |µ− λ| < ‖R(λ, T )‖−1 implies µ ∈ ρ(T ), ρ(T ) is an open subset of K.

The spectrum of the operator T is definded as

σ(T ) := K\ρ(T ),

which is in turn closed. In the case of K = C the set ρ(T ) is also always nonempty and the

spectral radius r(T ) := sup|λ| : λ ∈ ρ(T ) can be calculated by the Gelfand formula

r(T ) = limn→∞

n√‖Tn‖.

The estimate r(T ) ≤ ‖T‖ is true, even in the case K = R. Consequently σ(T ) is always bounded

and, therefore, compact.

A very important property of the spectrum is the spectral mapping theorem. That is, for any

1

polynomial p(λ) = anλn + · · ·+ a1λ+ a0 we get the identity

σ(p(T )) = p(σ(T ))

Here p(T ) denotes the operator anTn + · · ·+ a1T + a0I and p(σ(T )) = p(λ) : λ ∈ σ(T ). This

theorem is one of the starting points for the spectral theorem, which allows the representation

of certain linear operators as integrals over the spectrum.

Another important property of the spectrum is its upper semi-continuity. Since this fact often

remains unmentioned, we will give a short proof.

Proposition 1.1 For T ∈ B(X) let G ⊆ K be an open set with σ(T ) ⊆ G. Then there exists a

δ > 0 such that σ(S) ⊆ G for every S ∈ B(X) with ‖S − T‖ < δ.

Proof: We define the closed set F = K\G. Since every λ in F belongs to ρ(T ), R(λ, T ) lies

in B(X). We are going to show that also λ ∈ ρ(S) for all S ∈ B(X) satisfying

‖S − T‖ < 1

‖R(λ, T )‖.

To do so, we use the well known fact from the theory of Neumann series that I−R is invertible

for any R with ‖R‖ < 1. Therefore, the formula

∥∥I − (λI − T )−1(λI − S)∥∥ = ‖R(λ, T ) ((λI − T )− (λI − S))‖ ≤

≤ ‖R(λ, T )‖ ‖(λI − T )− (λI − S)‖ = ‖R(λ, T )‖ ‖S − T‖ < 1

implies that (λI − T )−1(λI − S) is invertible. Consequently, λI − S is invertible, i.e. λ ∈ ρ(S).

The fact that ‖R(λ, T )‖ → 0 as |λ| → ∞ implies that actually

δ := infλ∈F

1

‖R(λ, T )‖> 0.

2

Heuristically, upper semi-continuity assures that the spectrum does not ’expand suddenly’.

However it is not lower semi-continuous, which means that even under small perturbations it

can ’collapse’.

1.2 Spectral theory for nonlinear operators

Since spectral theory is so fruitful in the case of linear operators, it is natural to try to extend

its principles to nonlinear operators. In order to justify the label ’spectral theory for nonlinear

operators’, we would like the spectrum of nonlinear operators to have similar properties as

in the linear case. It should even be identical to the classical spectrum when applied to a

linear operator. Since this task is complicated enough, we will restrict ourselves to continuous

operators.

2

The first question when talking about a spectral theory for nonlinear operators is then, of

course, how to define the spectrum. A first straightforward attempt could simply be, as in the

linear case, to define the spectrum of an operator T as the set of all points λ such that λI − Tis not bijective. But simple examples show that in that case the spectrum would fail to have

even any basic properties like being closed, bounded or nonempty. For linear operators, the

linearity guarantees the linearity of its inverse and the open mapping theorem its continuity.

Thus, bijectivity is equivalent to being a homeomorphism. Since this tool is not available for

nonlinear operators, these two properties are not equivalent anymore. Defining the spectrum

as all λ such that λI − T is not a homeomorphism will consequently lead to a different kind of

spectrum.

Although this approach turns out to be just as dissapointing as the previous one, it points out

an important difference between the spectral theory for linear and nonlinear operators. The

linearity of an operator is responsible for the fact that many different properties are equivalent to

bijectivity. So in the linear theory the spectrum contains information about all these properties,

while at the same time one has to check only bijectivity, which is comparatively easy to handle.

If we want to deal with nonlinear operators, we have to look at all these different properties

seperately. So for any property A that makes sense for a nonlinear operator, we can define the

A-resolvent

ρA(T ) := λ ∈ K : λI − T has property A

and the A-spectrum

σA(T ) = K\ρA.

The only restriction for this property A is that it should be equivalent to bijectivity in case

of linear operators, so that the spectrum coincides with the classical one in the linear case.

Therefore, there are many different spectra to be considered.

None of them has yet lead to results of the same extent as the linear spectrum. By the above

considerations this is not completely unexpected. It is possible, and not very unlikely, that

there is no particular nonlinear spectrum that is as all-encompassing. However, the study of

these many different spectra is not in vain, and leads to interesting results.

In this thesis we will present one of these spectra, the Furi-Martelli-Vignoli-spectrum or FMV-

spectrum for short. This example will show how a spectral theory for nonlinear operators can

be developed and we will provide some results from this new theory.

3

Chapter 2

Preliminary considerations

In this chapter we discuss some results which will be used throughout this paper. We will

restrict ourselves to results pertaining to the nonlinearity of operators. Well known results that

strictly deal with linear operators (e.g. the open mapping theorem) will be cited without proofs

throughout this thesis. We will also assume knowledge of basic theorems about topological

properties of Banach spaces, like the Baire category theorem or the fact that closed balls are

compact only in finite dimensional spaces.

2.1 Characteristics of nonlinear operators

Let X and Y be two Banach spaces and T : X → Y a continuous operator, which in general

will be nonlinear. By C(X,Y ) we denote the set of all continuous operators from X into Y .

Of course, this set forms a linear space. C(X) := C(X,X) is an algebra with respect to the

composition. However, in the case of linear operators the space B(X,Y ) is normed by the

operator norm. As indicated in the introduction, the significance of the operator norm here

is due to the linearity of the operator. In the case of nonlinear operators one has to consider

multiple seminorms or other characteristics. We only present the four characteristics that we

will use in the following.

Definition 2.1 For T ∈ C(X,Y ) we define

[T ]Q := lim sup‖x‖→∞

‖T (x)‖‖x‖

and

[T ]q := lim inf‖x‖→∞

‖T (x)‖‖x‖

,

as elements of [0,∞]. If [T ]Q < ∞, we call T quasibounded. By Q(X,Y ) we denote the set of

all quasibounded continuous maps of X into Y .

4

In particular, the fact that [T ]Q = λ or [T ]q = λ implies that there exists an unbounded sequence

(xn)n∈N in X such that limn→∞ ‖T (xn)‖/‖xn‖ = λ. Furthermore, the inequality [T ]q ≤ [T ]Q

is obviously true. Consequently, [T ]Q = 0 actually implies that limn→∞ ‖T (xn)‖/‖xn‖ = 0 for

every sequence (xn)n∈N with ‖xn‖ → ∞.

We now gather some more properties of these characteristics.

Proposition 2.2 Let T, S ∈ C(X,Y ) and R ∈ C(Y,Z). Then the following holds:

(i) [T ]q > 0 implies that T is coercive, i.e. lim‖x‖→∞

‖T (x)‖ =∞.

(ii) One of the quantities on the left being finite, [T ]q − [S]Q ≤ [T + S]q ≤ [T ]q + [S]Q.

(iii) One of the quantities on the left being finite, |[T ]q − [S]q| ≤ [T − S]Q. In particular,

[T − S]Q = 0 implies [T ]q = [S]q.

(iv) [T−1]Q = [T ]−1q if T is a homeomorphism and either T is linear or X and Y are finite

dimensional.

(v) [R T ]q ≥ [R]q[T ]q.

Proof: If [T ]q > 0, then for sufficiently large ‖x‖ there exists a k > 0 such that ‖T (x)‖ ≥ k ‖x‖.This verifies (i).

For (ii) consider

[T + S]q = lim inf‖x‖→∞

‖T (x) + S(x)‖‖x‖

≤ lim inf‖x‖→∞

‖T (x)‖+ ‖S(x)‖‖x‖

≤ lim inf‖x‖→∞

‖T (x)‖‖x‖

+ lim sup‖x‖→∞

‖S(x)‖‖x‖

= [T ]q + [S]Q.

For the second inequality replace T by T + S and S by −S.

Similarly, consider

[T − S]Q = lim sup‖x‖→∞

‖T (x)− S(x)‖‖x‖

≥ lim sup‖x‖→∞

∣∣∣∣‖T (x)‖ − ‖S(x)‖‖x‖

∣∣∣∣ ≥ lim sup‖x‖→∞

(‖T (x)‖‖x‖

− ‖S(x)‖‖x‖

)≥ lim sup‖x‖→∞

‖T (x)‖‖x‖

− lim inf‖x‖→∞

‖S(x)‖‖x‖

≥ lim inf‖x‖→∞

‖T (x)‖‖x‖

− lim inf‖x‖→∞

‖S(x)‖‖x‖

= [T ]q − [S]q,

which for symmetry reasons proves (iii).

In (iv), the two assumptions that T is linear and that X and Y are finite dimensional both

assure that ‖x‖ → ∞⇔ ‖T (x)‖ → ∞. We therefore can consider the chain of equalities

[T−1]Q = lim sup‖y‖→∞

∥∥T−1(y)∥∥

‖y‖= lim sup‖x‖→∞

‖x‖‖T (x)‖

=

(lim inf‖x‖→∞

‖T (x)‖‖x‖

)−1=

1

[T ]q.

5

Finally, to see (v), consider

[R T ]q = lim inf‖x‖→∞

‖R(T (x))‖‖x‖

= lim inf‖x‖→∞

‖R(T (x))‖‖T (x)‖

‖T (x)‖‖x‖

≥ lim inf‖x‖→∞

‖R(T (x))‖‖T (x)‖

lim inf‖x‖→∞

‖T (x)‖‖x‖

≥ lim inf‖y‖→∞

‖R(y)‖‖y‖

lim inf‖x‖→∞

‖T (x)‖‖x‖

= [R]q[T ]q.

The last inequality holds true, because if ‖T (x)‖9∞, then [T ]q = 0 and the inequality holds,

and if ‖T (x)‖ → ∞ we can set T (x) = y to see that the inequality holds. 2

Next, we are going to introduce a tool that is often used in nonlinear functional analysis. In

the following, Bε(z) will always denote the open ball with radius ε and center z.

Definition 2.3 Let X be a Banach space and M ⊆ X. The Hausdorff measure of noncompact-

ness of M is defined as

α(M) := infε > 0| ∃m ∈ N, z1, . . . , zm ⊆ X : M ⊆ Bε(z1) ∪ · · · ∪Bε(zm) ∈ [0,∞].

A finite set z1, . . . , zm ⊆ X with M ⊆ Bε(z1) ∪ · · · ∪Bε(zm) is called a finite ε-net for M .

So if M has a measure of noncompactness ε, it can be covered by finitely many open balls of

any radius greater than ε.

Proposition 2.4 Let X be a Banach space, M,N ⊆ X, λ ∈ K, and z ∈ X. Then the measure

of noncompactness has the following properties:

(i) α(M) = α(M).

(ii) α(M) = 0 if and only if M is precompact, i.e. has a compact closure.

(iii) α(M) <∞ if and only if M is bounded.

(iv) M ⊆ N ⇒ α(M) ≤ α(N).

(v) |α(M)− α(N)| ≤ α(M +N) ≤ α(M) + α(N), where for the first inequality to hold either

α(M) or α(N) needs to be finite.

(vi) α(λM) = |λ|α(M).

(vii) α(M + z) = α(M).

(viii) α(M ∪N) = maxα(M), α(N).

(ix) α(co(M)) = α(M), where co(M) denotes the convex hull of M .

(x) α(Br(z)) = 0 if dimX <∞, and α(Br(z)) = r if dimX =∞.

6

Proof: The first four assertion are straightforward to verify. To see (v), observe, that if

z1, . . . , zm is a finite ε-net for M , and w1, . . . , wn is a finite η-net for N , then zi +wj |i =

1, . . . ,m; j = 1, . . . , n is a finite (ε+η)-net for M+N , which shows α(M+N) ≤ α(M)+α(N).

The first inequality in (v) immediately follows from this one. Moreover, if z1, . . . , zm is a

finite ε-net for M then λz1, . . . , λzm is a finite |λ|ε-net for λM , so (vi) follows. Similarly,

(vii) follows from the observation that z1, . . . , zm is a finite ε-net for M if and only if z1 +

z, . . . , zm + z is a finite ε-net for M + z. For (viii) we use the fact that if z1, . . . , zm is a

finite ε-net for M and w1, . . . , wn is a finite η-net for N , then z1, . . . , zm ∪ w1, . . . , wnis a finite δ-net for M ∪N , where δ = maxε, η.To see that (ix) holds true, by (iv) it suffices to show that α(co(M)) ≤ α(M). So for η > α(M)

choose a finite η-net z1, . . . zm and define N = co(z1 . . . zm). Any x ∈ co(M) can be

written as a convex combination x =∑aixi with xi ∈ M and

∑ai = 1. For every xi there is

a zj(i) with |xi − zj(i)| < η. Setting z =∑aizj(i) we get

‖x− z‖ =∥∥∥∑ aixi −

∑aizj(i)

∥∥∥ =∥∥∥∑ ai(xi − zj(i))

∥∥∥ ≤∑ |ai|∥∥xi − zj(i)∥∥ ≤∑ |ai|η = η.

Because z ∈ N , we have dist(x,N) ≤ η. Since N is compact, we can find a finite ε-net

w1 . . . wn for N and arbitrary ε > 0. This is then a finite (η + ε)-net for co(M).

Since every bounded set in a finite dimensional space is precompact, the first assertion of (x)

is trivial. To see the second one, by (vi) and (vii) it suffices to show that α(B1(0)) = 1. Since

B1(0) can be covered by itself, we have α(B1(0)) ≤ 1. Assume α(B1(0)) < 1. Then B1(0) can

be covered by finitely many open balls with radius η ∈ (0, 1). Again by (vi) and (vii) we can

in turn cover each of those balls with finitely many open balls with radius η2, which gives us a

finite cover of B1(0) by such sets. Since ηn → 0, by iterating this process we get a finite cover

of B1(0) with open balls with a radius smaller than ε for every ε > 0. This shows, that B1(0)

is precompact, i.e. B1(0) is compact. Thus, we get a contradiction to the well known fact that

this is only the case in finite dimensional spaces. 2

Unlike the characteristics we introduced before, the following characteristics can also be de-

fined for an arbitrary subset of a Banach space.

Definition 2.5 Let Z ⊆ X. For T ∈ C(Z, Y ) we define

[T ]A := infk : k > 0, α(T (M)) ≤ kα(M) for all bounded M ∈ Z

and

[T ]a := supk : k > 0, α(T (M)) ≥ kα(M) for all bounded M ∈ Z

as elements of [0,∞].

We call [T ]A the measure of noncompactness of T and denote by A(Z, Y ) the set of all continuous

maps T from Z into Y with [T ]A <∞.

7

Note that in finite dimensional spaces we always have [T ]A = 0 and [T ]a = ∞. In infinite

dimensional spaces, where this characteristic is of more use, we get the equivalent representations

[T ]A = sup∞>α(M)>0

α(T (M))

α(M)

and

[T ]a = inf∞>α(M)>0

α(T (M))

α(M).

Sets with α(M) = 0 can be left out here, since the continuity of T assures that also α(T (M)) = 0.

This can be seen by considering α(T (M)) ≤ α(T (M)) = 0. From this representation it is also

clear that an operator T with [T ]A <∞ maps bounded sets into bounded sets.

These two characteristics are also closely related to two important properties of operators.

Definition 2.6 Let T ∈ C(X,Y ).

• The operator T is called compact, if T (M) is precompact for every bounded set M ⊆ X.

• The operator T is called proper, if the preimage T−1(N) is compact for every compact set

N ⊆ Y .

Proposition 2.7 Let X, Y , and Z be Banach spaces. For T, S ∈ C(X,Y ) and R ∈ C(Y,Z) the

following assertions hold true:

(i) T is compact if and only if [T ]A = 0.

(ii) If [T ]a > 0 and [T ]q > 0, then T is proper.

(iii) [T ]a > 0 implies that T is proper on closed bounded sets.

(iv) One of the quantities on the left being finite, [T ]a − [S]A ≤ [T + S]a ≤ [T ]a + [S]A.

(v) One of the quantities on the left being finite, |[T ]a − [S]a| ≤ [T − S]A. In particular,

[T − S]A = 0 implies [T ]a = [S]a.

(vi) [T−1]A = [T ]−1a if T is a homeomorphism and either T is linear or X and Y are finite

dimensional.

(vii) [R]a[T ]a ≤ [R T ]a ≤ [R]A[T ]a, where the second inequality holds if [R]A <∞.

Proof: The first assertion follows immediately from the definition of a compact operator and

the definition of the measure of noncompactness of an operator.

In (ii), because [T ]a > 0, we may find a k > 0 such that α(T (M)) ≥ kα(M) for each bounded

M ∈ X. As [T ]q > 0, Lemma 2.2,(i) shows that T is coercive. Therefore, for any compact set

N ∈ Y , T−1(N) is bounded and

α(T−1(N)) ≤ 1

kα(T (T−1(N))) ≤ 1

kα(N) = 0.

8

Thus T−1(N) is precompact. Since T is continuous, T−1(N) is also closed and therefore com-

pact. The same reasoning shows that T is proper on closed bounded sets if only [T ]a > 0.

(iv) and (v) are proven similarly as in Lemma 2.2.

(vi) is trivial if X and Y are finite dimensional. If one is infinite dimensional, the assumption

that T is linear assures that T maps bounded sets into bounded sets. Also, due to T being a

homeomorphism, α(T (M)) = 0 if and only if α(M) = 0. We therefore get the chain of equalities

[T−1]A = sup∞>α(N)>0

α(T−1(N))

α(N)= sup∞>α(M)>0

α(M)

α(T (M))=

(inf

∞>α(M)>0

α(T (M))

α(M)

)−1=

1

[T ]a.

Finally, let kT > 0 such that α(T (M)) ≥ kTα(M) for all bounded M ∈ X. Further, let kR > 0

such that α(R(M)) ≥ kRα(M) for all bounded M ∈ Y . Then α(R T (M)) ≥ kRα(T (M)) ≥kRkTα(M) for all bounded M ⊆ X. This shows the first inequality in (vii). For the second

inequality, consider

[R T ]a = inf∞>α(M)>0

α(R T (M))

α(M)= inf

∞>α(M)>0∞>α(T (M)) 6=0

α(R T (M))

α(T (M))

α(T (M))

α(M)

≤ sup∞>α(N)>0

α(R(N))

α(N)inf

∞>α(M)>0

α(T (M))

α(M)= [R]A[T ]a.

2

For linear operators the characteristic [T ]q and [T ]a are linked to the injectivity and bi-

jectivity of the operator T . [T ]A and [T ]Q on the other hand can be linked to the operator

norm.

Lemma 2.8 Let T : X → Y be linear and bounded. Then the following holds

(i) [T ]Q = ‖T‖.

(ii) [T ]A ≤ ‖T‖.

(iii) If [T ]q > 0, then T is injective.

(iv) If T is bijective, then [T ]q > 0 and [T ]a > 0.

Proof: The first assertion follows immediately from the linearity of T . Also, if z1, . . . , zmis a finite ε-net for a bounded set M , then Tz1, . . . , T zm is obviously a finite ‖T‖ ε-net for

T (M), i.e. α(T (M)) ≤ ‖T‖α(M). Hence, [T ]A ≤ ‖T‖.For (iii), assume T is not injective. Then there exist x 6= y ∈ X with Tx = Ty. Let (xn)n∈N

and (yn)n∈N be sequences with xn → x and yn → y. Then ‖n(xn − yn)‖ → ∞ for n → ∞.

Using the linearity of T we get the contradiction

[T ]q = lim inf‖z‖→∞

‖Tz‖‖z‖

≤ limn→∞

‖T (n(xn − yn))‖‖n(xn − yn)‖

= limn→∞

n ‖T (xn − yn)‖n ‖xn − yn‖

=‖Tx− Ty‖‖x− y‖

= 0.

9

To see (iv), note that because T is linear, bijective, and bounded, it is invertible and its inverse

is also a linear and bounded operator. Using Proposition 2.2,(iv) we get

[T ]q = [T−1]−1Q =∥∥T−1∥∥−1 =

1

‖T−1‖> 0,

and using Proposition 2.7,(vi) we get

[T ]a =1

[T−1]A≥ 1

‖T−1‖> 0.

2

2.2 The Dugundji Extension Theorem

Dugundjis extension theorem assures that for any continuous map T : A → Y that is defined

on a closed set A ⊆ X, there exists a continuous extension T of T to X, i.e. T : X → Y and

T (x) = T (x) for x ∈ A, with the additional restriction that the range of T is in some sense

constrained by the original range of T .

In its original version X is allowed to be an arbitrary metric space and Y a locally convex linear

space. We will give a simpler version of this theorem for mappings between two Banach spaces

X and Y . First we need some definitions.

Definition 2.9 (i) Let A = (Ai)i∈I and B = (Bj)j∈J be systems of subsets of a set X. B is

called a refinement of A, if for every j ∈ J there is an i ∈ I with Bj ⊆ Ai.

(ii) Let X be a Hausdorff space and U = (Ui)i∈I be a cover of X. U is called locally finite,

if every point x ∈ X has a neighborhood O(x) such that O(x) intersects at most finitely

many elements of U .

(iii) A Hausdorff space X is called paracompact if for every open cover U of X there exists an

open cover V of X, such that V is a locally finite refinement of U .

The following definition will be needed in the next section, but is closely related to the definitions

above.

Definition 2.10 Let U = (Ui)i∈I be an open cover of X. A system of real valued continuous

functions (fi)i∈I is called a partition of unity subordinate to U if

(a) fi(x) ≥ 0 for all x ∈ X and all i ∈ I,

(b) (supp(fi))i∈I is a locally finite system, where supp(f) := x ∈ X : f(x) 6= 0,

(c) supp(fi) ⊆ Ui for all i ∈ I,

10

(d)∑

i∈I fi(x) = 1 for all x ∈ X.

The following theorem contains two well known topological results.

Theorem 2.11 (i) Every metric space is paracompact.

(ii) To every open cover U of a paracompact space there exists a a partition of unity subordinate

to U .

Proof: For example [Q]. 2

We are now able to formulate our version of Dugundjis theorem.

Theorem 2.12 (Dugundji Extension Theorem) Let X and Y be Banach spaces and let

T : C → K be a continuous map, where C ⊆ X is closed and K ⊆ Y is convex.

Then there exists a continuous mapping T : X → K such that T (x) = T (x) for x ∈ C.

Proof: For each x in X\C set rx = 13dist(x,C). Then diamBrx(x) ≤dist(Brx , C).

The collection (Brx(x))x∈X\C is an open cover of X\C. By Theorem 2.11,(i) it has a locally

finite refinement (Oi)i∈I which is an open cover of X\C.

Define q : X\C → (0,∞) by

q(x) =∑i∈I

dist(x,X\Oi).

Since (Oi)i∈I is locally finite, the sum contains only finitely many not vanishing terms and q is

a continuous function. Further, since the (Oi)i∈I form an open cover of X\C, q(x) > 0.

Next we define for i ∈ I and x ∈ X\C

ρi(x) =dist(x,X\Oi)

q(x).

We then have 0 ≤ ρi(x) ≤ 1 and∑

i∈I ρi(x) = 1.

Note that since (Oi)i∈I is a refinement of (Brx(x))x∈X\C , for any Oi there exists x0 ∈ X\C with

dist(C,Oi) ≥ dist(C,Brx0 (x0)) ≥ diamBrx0 (x0) > 0.

For each i ∈ I, we can therefore choose an xi ∈ C such that dist(xi, Oi) ≤ 2dist(C,Oi), and

define

T (x) =

T (x) for x ∈ C,∑

i∈I ρi(x)T (xi) for x /∈ C.

Obviously T : X → K and T is an extension of T . Further, T is continuous on the interior

of C as well as on X\C. In order to show that T is continuous, it suffices to prove that T is

continuous on ∂C.

11

Let x ∈ ∂C. Since T is continuous, for given ε > 0 we find a δ > 0 such that ‖T (x)− T (y)‖ ≤ εfor y ∈ C with ‖x− y‖ ≤ δ. For y ∈ X\C we have

∥∥∥T (x)− T (y)∥∥∥ =

∥∥∥∥∥T (x)−∑i∈I

ρi(y)T (xi)

∥∥∥∥∥ ≤∑i∈I

ρi(y) ‖T (x)− T (xi)‖ .

If ρi(y) 6= 0 then dist(y,X\Oi) > 0, i.e. y ∈ Oi. Taking the infimum over all w ∈ Oi in

‖y − xi‖ ≤ ‖y − w‖+ ‖w − xi‖ we obtain

‖y − xi‖ ≤ diamOi + dist(xi, Oi).

Now, Oi ⊆ Brx0 (x0) for some x0 ∈ X\C. Since

diamOi ≤ diamBrx0 (x0) ≤ dist(Brx0 , C) ≤ dist(C,Oi),

we get

‖y − xi‖ ≤ 3dist(C,Oi) ≤ 3 ‖y − x‖ .

Thus, for i such that ρi(y) 6= 0 we get ‖x− xi‖ ≤ ‖y − x‖ + ‖y − xi‖ ≤ 4 ‖x− y‖ . Hence,

‖x− y‖ ≤ δ/4 implies ‖x− xi‖ ≤ δ, and, therefore, ‖T (x)− T (xi)‖ ≤ ε. Finally,∥∥∥T (x)− T (y)∥∥∥ ≤∑

i∈Iρi(y) ‖T (x)− T (xi)‖ ≤ ε

∑i∈I

ρi(y) = ε.

2

This theorem has two very useful corollaries.

Corollary 2.13 Let X and Y be Banach spaces and let T : C → Y be continuous, where

C ⊆ X is closed.

Then there exists a continuous extension T of T to X with T (X) ⊆ co(T (C)).

Corollary 2.14 Let X and Y be Banach spaces and let T : C → Y be compact, where C ⊆ Xis closed and bounded.

Then there exists a compact extension T of T to X with T (X) ⊆ co(T (C)).

Proof: Since T is compact and C bounded, T (C) is precompact. By Lemma 2.4(ix), so is

co(T (C)). Therefore, an extension T of T as in Theorem 2.12 is also compact. 2

12

2.3 Set-valued maps

For a set X we will denote the power set of X by P(X). By a multivalued (or set-valued) map

between two sets X and Y we mean a map T : X → P(Y ). It assigns to each point of X a

subset of Y . Every map T : X → Y can be identified with a set-valued map T ′ : X → P(Y )

by setting T ′(x) = T (x). Such maps are then referred to as single-valued maps.

For a set-valued map we define the image of a set M as

T (M) :=⋃x∈M

T (x)

and the preimage of a set A as

T−1(A) := x ∈ X : T (x) ∩A 6= ∅.

Note that unlike for single-valued maps the inclusion T (T−1(A)) ⊆ A need not hold. However,

unless T−1(A) = ∅, we have T (T−1(A))∩A 6= ∅. For single-valued maps this definition coincides

with the classical preimage of a set. Furthermore, we call the set

G(T ) := (x, y) : x ∈ X, y ∈ T (x)

the graph of T .

For set-valued maps we have the following notions of continuity:

Definition 2.15 Let X and Y be topological spaces and T : X →P(Y ) a set-valued map.

(i) T is called upper semi-continuous if T−1(A) is closed for all closed sets A ⊆ Y .

(ii) T is called lower semi-continuous if T−1(A) is open for all open sets A ⊆ Y .

For single-valued maps both upper and lower semi-continuity are obviously equivalent to conti-

nuity. In the following we will also need a different characterization of semi-continuity.

Proposition 2.16 Let X and Y be topological spaces and T : X →P(Y ) a set-valued map.

(i) T is upper semi-continuous if and only if for every x ∈ X and every open set V in Y with

T (x) ⊆ V there exists a neighbourhood U(x) such that T (U(x)) ⊆ V .

(ii) T is lower semi-continuous if and only if for every x ∈ X, y ∈ T (x) and every neighbour-

hood V (y) of y there exists a neighbourhood U(x)of x such that

T (u) ∩ V (y) 6= ∅, for all u ∈ U(x).

Proof:

(i) Suppose T fulfills the assumptions and A ⊆ Y is closed. Choose x ∈ (T−1(A))C , then

T (x) ⊆ AC . Since AC is open, there exists a neighbourhood U(x) of x, such that

13

T (U(x)) ⊆ AC . Therefore U(x) ⊆ (T−1(A))C and it follows that T−1(A) is closed.

Conversely, suppose T−1(A) is closed for all closed A ⊆ Y . Let x ∈ X and V ⊆ Y be

open with T (x) ⊆ V . Then V C is closed and by assumption so is T−1(V C). Moreover,

x /∈ T−1(V C). Hence, there exists a neighbourhood of x with U(x) ⊆ (T−1(V C))C . This

neighbourhood apparently satisfies T (U(x)) ⊆ V .

(ii) Suppose T fulfills the assumptions and A ⊆ Y is open. Assume that x ∈ T−1(A) and

choose y ∈ T (x) ∩ A. Since A is open, there exists a neighbourhood of y with V (y) ⊆ A.

Because of our assumption, there exists a neighbourhood U(x) of x with T (u)∩ V (y) 6= ∅for all u ∈ U(x). This means U(x) ⊆ T−1(V (y)) ⊆ T−1(A). It follows that T−1(A) is

open.

Conversely, suppose T−1(A) is open for every open set A ⊆ Y . Assume that x ∈ X,

y ∈ T (x), and V (y) is an open neighbourhood of y. Then T−1(V (y)) is open and x ∈T−1(V (y)). Therefore, U(x) = T−1(V (y)) is a neighbourhood of x. It follows that T (u)∩V (y) 6= ∅ for all u ∈ U(x).

2

With this characterization of upper semi-continuity it is easy to see that Proposition 1.1 indeed

shows the upper semi-continuity of the classical spectrum. It also clarifies what we meant with

the heuristical explanation that the values of a lower semi-continuous map cannot ’collapse’ and

the values of an upper semi-continuous map cannot ’suddenly expand’.

We now give a condition for a set-valued map to be upper semi-continuous, which will be needed

later on. For this, let X be linear space and p be a seminorm on X. For x ∈ X we denote by

Uδ(x) the p-neighbourhood Uδ(x) := y ∈ X : p(x− y) < δ.

Definition 2.17 A set-valued map T : X → P(K) is called closed if the graph of T is closed

in X ×K, i.e. yn ∈ T (xn), yn → y and p(xn − x)→ 0 imply that y ∈ T (x).

Lemma 2.18 Let T : X → P(K) be closed. Then for every x ∈ X and y ∈ K\T (x) there

exists δ > 0 and an open set Vy ⊆ K such that y ∈ Vy and T (Uδ(x)) ∩ Vy = ∅.

Proof: Assume the assertion does not hold true for x ∈ X and y ∈ K\T (x). Let (δn)n∈N be

a null sequence. Since T (Uδn(x)) ∩ Bδn(y) 6= ∅, we can find xn ∈ Uδn(x) and yn ∈ Bδn(y) such

that yn ∈ T (xn). Further, we get that yn → y and p(xn − x)→ 0. Since T is closed, this shows

that y ∈ T (x), which contradicts our choice of y. 2

Lemma 2.19 Let T : X →P(K) be a closed map. If

supy∈T (x)

|y| ≤ p(x) for all x ∈ X,

then T is upper semi-continuous.

14

Proof: Let x ∈ X and let V ⊆ K be open with T (x) ⊆ V . By Proposition 2.16,(i), we have

to show that there exists a δ > 0 with T (Uδ(x)) ⊆ V .

Choose η > 0 with T (Uη(x))\V 6= ∅ (if this is not possible, there is nothing to prove). For

z ∈ Uη(x) we have

supλ∈T (z)

|λ| ≤ p(z) ≤ p(x) + η.

Consequently, T (Uη(x)) is bounded, and so the set C := T (Uη(x))\V is compact.

Let y ∈ C be arbitrary. Since y /∈ T (x) and T is closed, by Lemma 2.18 we find δ(y) > 0 and

an open Vy ⊆ K with y ∈ Vλ and T (Uδ(y)(x)) ∩ Vy = ∅. Obviously, Vy : y ∈ C is an open

cover of C. Since C is compact, we get C ⊆ Vy1 ∪ · · · ∪Vym for suitable y1, . . . , ym ∈ C. Putting

δ = minη, δ(y1), . . . , δ(ym) we see that T (Uδ(x))∩C = ∅. Since T (Uδ(x)) ⊆ T (Uη(x)), we get

T (Uδ(x)) ⊆ V . 2

For the rest of this section we will only deal with lower semi-continuity.

Lemma 2.20 Let T : X → P(Y ) be lower semi-continuous, where X and Y are topological

spaces. If S : X → P(Y ) is such that T (x) = S(x) for all x ∈ X, then S is lower semi-

continuous.

Proof: Assume S is not lower semi-continuous. Then there exists an x ∈ X, y ∈ S(x),

and an open set V (y) ⊆ Y with y ∈ V (y) such that for all neighbourhoods U(x) of x there

exists an u ∈ U(x) with S(u) ∩ V (y) = ∅. Since V (y) is open, this also means T (u) ∩ V (y) =

S(u)∩ V (y) = ∅, which consequently implies T (u)∩ V (y) = ∅. If y ∈ T (x), this contradicts the

lower semicontinuity of T .

If y /∈ T (x), we have y ∈ T (x). Hence, V (y)∩T (x) 6= ∅. Further, V (y) is an open neighbourhood

V (z) of every z ∈ V (y) ∩ T (x). As above we get T (u) ∩ V (z) = T (u) ∩ V (y) = ∅. 2

Lemma 2.21 Let T : X →P(Y ), where Y is a Banach space. Let T be lower semi-continuous,

O ⊆ Y open, f : X → Y a continuous map, and suppose that T (x) ∩ (f(x) + O) 6= ∅ for all

x ∈ X. Then S : X →P(Y ), defined by S(x) = T (x) ∩ (f(x) +O), is lower semi-continuous.

Proof: Let x ∈ X, y ∈ S(x), and V (y) be an open set with y ∈ V (y). Then y ∈ T (x)

and (f(x) + O) ∩ V (y) is an open neighbourhood of y. Therefore, there exists an o ∈ O with

y = f(x) + o and an ε > 0 such that f(x) + Bε(o) ⊆ (f(x) + O) ∩ V (y). Since T is lower

semi-continuous, there exists a neighbourhood U(x) of x such that for all u ∈ U(x) we have

T (u) ∩ (f(x) +Bε/2(o)) 6= ∅.Since f is continuous, there exists a neighbourhood U(x) of x, such that ‖f(x)− f(u)‖ < ε/2

for all u ∈ U(x). Let a ∈ f(x) + Bε/2(o). Then for all u ∈ U(x) we get ‖a− (f(u) + o)‖ ≤‖a− f(x)− o‖+ ‖f(x) + o− f(u)− o‖ < ε, i.e. a ∈ f(u) +Bε(o) ⊆ (f(u) +O).

Set U(x) = U(x) ∩ U(x). Then for any u ∈ U(x) we have T (u) ∩ (f(x) + Bε/2(o)) 6= ∅, since

15

u ∈ U(x). But for any a ∈ T (u)∩(f(x)+Bε/2(o)) we have a ∈ (f(x)+Bε/2(o)), hence a ∈ V (y).

Further, u ∈ U(x) yields a ∈ (f(u) +O). Hence, a ∈ T (u) ∩ (f(u) +O) ∩ V (y) = S(u) ∩ V (y).

Therefore, S(u) ∩ V (y) 6= ∅ for all u ∈ U(x). 2

Lemma 2.22 Let T = X → P(Y ) be lower semi-comtinuous and Y be a Banach space.

Suppose f : X → R is continuous and f(x) ≥ 0 for all x ∈ X. Define S : X →P(Y ) by

S(x) =

T (x) ∩Bf(x)(0) if f(x) > 0,

0 if f(x) = 0.

If S(x) 6= ∅ for all x ∈ X, then S is lower semi-continuous.

Proof: Let x ∈ X, y ∈ S(x), and V (y) be open with y ∈ V (y). First, suppose f(x) > 0. Then

f(x) > ‖y‖. Hence, there exists an ε > 0 with f(x) > ‖y‖ + ε. Since f is continuous, there

exists an open neighbourhood U(x) of x such that f(u) > ‖y‖+ ε for all u ∈ U(x). Moreover,

there exists an open neighbourhood U(x) of x such that T (u) ∩ B‖y‖+ε(0) ∩ V (y) 6= ∅ for all

u ∈ U(x), because T is lower semi-continuous and B‖y‖+ε(0) ∩ V (y) is an open neighbourhood

of y. So for u in the open set U(x) = U(x) ∩ U(x) we get

S(u) ∩ V (y) = T (u) ∩Bf(u) ∩ V (y) ⊇ T (u) ∩B‖y‖+ε ∩ V (y) 6= ∅.

Now suppose f(x) = 0. Then y = 0 and V (y) is a neighbourhood of 0, i.e. there is an ε > 0 such

that Bε(0) ⊆ V (y). Since f is continuous, there is a open neighbourhood U(x) of x such that

f(u) < ε for u ∈ U(x). For f(u) > 0 we have S(u) = T (u)∩Bf(u)(0) ⊆ Bf(u)(0) ⊆ Bε(0) ⊆ V (y).

For f(u) = 0 we have S(u) = 0 ⊆ V (y). In any case, S(u) ∩ V (y) 6= ∅. 2

Finally, we define the objects that will be studied for the rest of this section.

Definition 2.23 Let T : X → P(Y ) be a set-valued map. A single-valued map t : X → Y is

called a selection of T , if

t(x) ∈ T (x) for all x ∈ X.

The existence of a selection is obviously equivalent to the fact, that T (x) 6= ∅ for all x ∈ X.

Selections are an important tool when dealing with set-valued maps. Therefore, it is of interest

to show the existence of selections with additional properties. The following important theorem

by Michael provides conditions for the existence of a continuous selection.

Theorem 2.24 (Michael’s Selection Theorem) Let X be a paracompact space, Y a Ba-

nach spaces, and T : X →P(Y ) a lower semi-continuous set-valued map. If T (x) is nonempty,

closed, and convex for all x ∈ X, then there exists a continuous selection t : X → Y of T.

16

Proof: For y ∈ Y and A ⊆ Y set d(y,A) = infa∈A ‖y − a‖. As a first step, we show that for

each ε > 0 there exists a continuous map f : X → Y such that

d(f(x), T (x)) < ε for all x ∈ X. (2.1)

Fix ε > 0 and choose any selection m : X → Y . Since T is lower semi-continuous, for each

x ∈ X there exists an open neighbourhood U(x) of x such that

T (u) ∩Bε(m(x)) 6= ∅ for allu ∈ U(x). (2.2)

Let (fi)i∈I be a partition of unity subordinate to the open cover (U(x))x∈X of X. For every

i ∈ I choose an xi ∈ X such that supp(fi) ⊆ U(xi) if supp(fi) 6= ∅ and set

f(x) =∑i∈I

fi(x)m(xi).

Then f is a continuous function of X into Y . If fi(x) > 0 for some i, then x ∈ U(xi) and by

(2.2)

m(xi) ∈ T (x) +Bε(0).

Since T (x) + Bε(0) is convex, we get f(x) ∈ T (x) + Bε(0). Hence, d(f(x), T (x)) < ε, i.e. f

satisfies (2.1).

In the second step we construct the requested selection. Set εn = 2−n. We will inductively

define a sequence (fn)n∈N of continuous mappings fn : X → Y with

d(fn(x), T (x)) < εn, x ∈ X, n = 1, 2, . . . (2.3)

d(fn(x), fn−1(x)) < εn−1, x ∈ X, n = 2, 3, . . . (2.4)

As we showed in the first step, there exists f1 with d(f1(x), T (x)) < 1/2, x ∈ X.

Assume that n ≥ 2 and we already have constructed f1, . . . , fn−1. For each x ∈ X we define

G(x) = (fn−1(x) +Bεn−1(0)) ∩ T (x)

By the induction hypothesis, G(x) is not empty. Since T (x) convex, so is G(x). Furthermore,

G : X →P(Y ) is lower semi-continuous by Lemma 2.21. So we can apply the first part of our

proof also to G, since the only additional property of T is that T (x) is closed, which was not

used in the first part. Therefore, there exists a continuous map fn : X → Y such that (2.3)

holds. By construction, fn also satisfies (2.4).

Since∑∞

n=1 εn converges, (fn)n∈N is a uniform Cauchy sequence and, therefore, converges to a

continuous map t : X → Y . Since T (x) is closed, d(t(x), T (x)) = 0 implies that t is a selection

of T . 2

17

Corollary 2.25 Let X be paracompact, Y a Banach space, and T : X → P(Y ) a lower

semi-continuous map such that T (x) is nonempty, closed, and convex for every x ∈ X. Set

m(x) = inf‖y‖ : y ∈ T (x) and suppose f : X → R is continuous weith f(x) ≥ 0 for all x,

and f(x) > m(x) whenever m(x) > 0. Then there exists a continuous selection t of T such that

‖t(x)‖ ≤ f(x) for all x ∈ X.

Proof: The map S : X →P(Y ) defined by

S(x) =

T (x) ∩Bf(x)(0) if f(x) > 0,

0 if f(x) = 0,

is lower semi-continuous by Lemma 2.22. Hence, R : X → P(Y ), defined by R(x) = S(x), is

lower semi-continuous by Lemma 2.20. Furthermore, R(x) is nonempty, closed, and convex for

all x ∈ X. By Theorem 2.24, there exists a selection t of R. By construction t is also a selection

of T and fulfils ‖t(x)‖ ≤ f(x). 2

Theorem 2.26 Let T : X → Y be a continuous linear surjection from a Banach space X onto

a Banach space Y . Then there exists a continous function s : Y → X and a constant M > 0

such that for every y ∈ Y

• s(y) ∈ T−1(y),

• ‖s(y)‖ ≤M ‖y‖.

Proof: First, we show that there exists an M > 0, such that m(y) = inf‖x‖ : x ∈ T−1(y) ≤M ‖y‖ and m(y) < M ‖y‖ whenever m(y) > 0. Since T is surjective, by the open mapping

theorem the image of B1(0) ⊆ X under T is an open neighbourhood of 0 ∈ Y . In particular,

there exists a δ > 0 such that Sδ = y ∈ Y : ‖y‖ = δ ⊆ T (B1(0)). For an arbitrary 0 6= y ∈ Ywe have y0 = δy ‖y‖−1 ∈ Sδ. Hence, there exists a x0 ∈ B1(0) with T (x0) = y0 and, further,

T (x0 ‖y‖ δ−1) = y. Therefore, m(y) ≤ ‖x0 ‖y‖ /δ‖ ≤ δ−1 ‖y‖. Now set M = λδ−1 with a λ > 1.

Define S : S1 →P(X) by S(x) = T−1(x). Then for any open U ⊆ X the set S−1(U) = y ∈S1 : T−1(y) ∩ U 6= ∅ = T (U) ∩ S1 is open in S1 by the open mapping theorem. Hence, S is

lower semi-continuous. Moreover, S(y) is nonempty, closed, and convex for all y ∈ S1. So we

can apply Corollary 2.25 to S with f(x) = M ‖x‖ and get a continuous function s : S1 → X

with s(y) ∈ T−1(y) and ‖s(y)‖ ≤M ‖y‖. Setting

s(y) =

‖y‖ s( y

‖y‖) if y 6= 0,

0 if y = 0,

we get a map from the whole space Y into X with the mentioned properties. 2

18

2.4 The Antipodal Theorem

To formulate and prove the Antipodal Theorem we first need to establish the fixed point index

and other preliminary results.

Definition 2.27

• Let G be a nonempty and open bounded set in a Banach space X. Then V (G,X) denotes

the set of all compact maps T : G → X such that T has no fixed points on the boundary

∂G of G, i.e. @x ∈ ∂G : T (x) = x.

• Two maps T, S ∈ V (G,X) are called compactly homotopic on ∂G if there exists a contin-

uous map H with the following properties

(i) H : ∂G× [0, 1]→ X is compact,

(ii) H(x, 0) = T (x) and H(x, 1) = S(x) for x ∈ ∂G,

(iii) H(x, t) 6= x for all (x, t) ∈ ∂G× [0, 1].

We write T ∼= S. The map H is called a compact homotopy.

Proposition 2.28 Let G be a nonempty open bounded subset of a Banach space X and let

T, S ∈ V (G,X). Then the following holds true:

(i) The relation ∼= is an equivalence relation.

(ii) infx∈∂G ‖x− T (x)‖ > 0.

(iii) If supx∈∂G

‖T (x)− S(x)‖ < infx∈∂G

‖x− T (x)‖, then T ∼= S.

Proof: The relation ∼= is reflexive, since T ∼= T by H(x, t) = T (x). If T ∼= S by H, then

S ∼= T by H1 = H(x, 1 − t), so ∼= is symmetric. To see that ∼= is also transitive, let T ∼= S by

H1 and S ∼= R by H2, then T ∼= R by

H(x, t) =

H1(x, 2t) for 0 ≤ t ≤ 1

2 ,

H2(x, 2t− 1) for 12 < t ≤ 1.

To prove (ii), we show that (I − T )(∂G) is closed. Becaus of 0 /∈ (I − T )(∂G), this gives

the assertion. So let (yn)n∈N be a sequence in (I − T )(∂G) that converges to a y ∈ X. By

Corollary 2.14 we can extend T to a compact map T on X. Since ∂G is closed and bounded,

and [I− T ]a ≥ [I]a− [T ]A = 1 > 0 by Lemma 2.7,(iii), point (ii) of the same lemma tells us that

(I − T ) is proper on ∂G, so I − T is also proper there. Because (yn)n∈N ∪ y is compact, also

(I − T )−1((yn)n∈N ∪ y) is compact. If we choose xn ∈ (I − T )−1(yn) ∩ ∂G, then (xn)n∈N has

a convergent subsequence xnk→ x ∈ ∂G. Because I − T is continuous, we get (I − T )(x) = y.

Thus, y ∈ (I − T )(∂G).

19

For (iii) let H(x, t) = (1− t)T (x) + tS(x). Then H is continuous and compact on ∂G× [0, 1],

H(x, 0) = T (x) and H(x, 1) = S(x) for x ∈ ∂G, and for all (x, t) ∈ ∂G× [0, 1] we have

‖H(x, t)− x‖ = ‖T (x)− x− t(T (x)− S(x))‖ ≥ ‖T (x)− x‖+ t ‖T (x)− S(x)‖ > 0.

2

The next lemma is especially interesting in connection with point (iii) of the previous proposi-

tion.

Lemma 2.29 Let X and Y be Banach spaces and M ⊆ X be nonempty and bounded. Let

T : M → Y be a compact operator. Then to each ε > 0, there exists a compact operator

P : M → Y such that supx∈M ‖T (x)− P (x)‖ ≤ ε and P (M) is contained in finite dimensional

subspace of Y .

Proof: Fix ε > 0. Since M is bounded, T (M) is precompact. Hence, there exist yi ∈ T (M),

i = 1, . . . , N , such that mini=1,...,N ‖T (x)− yi‖ < ε for all x ∈ M . Define continuous functions

pi : M → R by pi(x) = max(ε− ‖T (x)− yi‖ , 0). Then for each x ∈M there exists at least one

i such that pi(x) 6= 0. Therefore, we can define P : M → Y by

P (x) =

∑Ni=1 pi(x)yi∑Ni=1 pi(x)

.

For all x ∈M we then have

‖P (x)− T (x)‖ =

∥∥∥∥∥∑N

i=1 pi(x)(yi − T (x))∑Ni=1 pi(x)

∥∥∥∥∥ ≤∑N

i=1 pi(x)ε∑Ni=1 pi(x)

≤ ε.

By construction, P (M) lies in the finite dimensional subspace spanned by yi : i = 1, . . . , N.Finally, since T (M) is bounded, so is P (M). Since a bounded set in a finite dimensional space

is precompact, P is compact. 2

Definition 2.30 Let X be a Banach space. An integer valued function i(T,G), where G is a

nonempty open bounded subset of X and T ∈ V (G,X), is called the fixed point index of T on

G if the following axioms are satisfied.

(A1) If T (x) = x0 for all x ∈ G and some fixed x0 ∈ G, then i(T,G) = 1.

(A2) If i(T,G) 6= 0, then there exists an x ∈ G such that T (x) = x.

(A3) If there are open Gi ⊆ X, i = 1, . . . , n, such that G =⋃ni=1Gi and Gi ∩Gj = ∅ for i 6= j,

then

i(T,G) =n∑i=1

i(T,Gi)

20

whenever T ∈ V (G,X) and T |Gi∈ V (Gi, X) for all i.

(A4) If T ∼= S, then i(T,G) = i(S,G).

Despite their importance, we will not give a proof of the following three results, since they

are beyond the scope of the present thesis. The proof of the first of these theorems includes a

lengthy construction of the fixed point index, while the other two proofs necessitate a detailed

knowledge of this construction. For a proof of these theorems see for example Zeidler [Z].

Theorem 2.31 For every Banach space X there exists a unique fixed point index.

Theorem 2.32 Let G be a nonempty open bounded region and T ∈ V (G,X). Let Ω ⊆ X be a

bounded region with G ⊆ Ω and h : Ω→ X be a linear map such that I−h is compact. Suppose

h : Ω→ h(Ω) is a homeomorphism, T (G) ⊆ Ω, and 0 /∈ h (I − T )(∂G). Then

i(h T h−1, h(G)) = i(T,G).

Theorem 2.33 If T ∈ V (G,X) and if T (G) lies entirely in a closed linear subspace Y of X,

then i(T,G) = i(T,G ∩ Y ), where the right hand side is the fixed point index in Y .

The last two lemmas that we need before we are able to proof the Antipodal Theorem deal

with fixed point free extensions of fixed point free maps.

Lemma 2.34 Let A and B be closed and bounded subsets of a Banach space X with A ⊆ B.

Let H : A× [0, 1]→ X be compact and H(x, t) 6= x for all (x, t) ∈ A× [0, 1].

If H(., 1) has a fixed point free compact extension H1 : B → X, then H has a compact extension

H : B × [0, 1]→ X with H(x, t) 6= x for all (x, t) ∈ B × [0, 1]

Proof: Set

H0(x, t) =

H1(x) for x ∈ B, t = 0

H(x, t) for x ∈ A, t ∈ [0, 1],

and extend H0 by Corollary 2.14 to a compact map H0 : B × [0, 1] → X. Let B0 = x ∈ B :

∃t ∈ [0, 1] with H0(x, t) = x. Then B0 is closed and A∩B0 = ∅. Therefore, by Corollary 2.13,

we can extend the map

a(x) =

0 for x ∈ B0

1 for x ∈ A

to a continuous map a : B → [0, 1]. Finally, we set H(x, t) = H0(x, a(x)t) for (x, t) ∈ B × [0, 1].

Then H is the desired extension of H. In fact, if H(x, t) = x, then H0(x, τ) = x for some τ , i.e.

x ∈ B0. Therefore, a(x) = 0 and further H0(x, 0) = x, i.e. H1(x) = x, which is impossible by

hypothesis. 2

21

Lemma 2.35 Let A and B be compact subsets of a proper linear subspace of RN with N ≥ 2

and A ⊆ B. Then every continuous fixed point free map f : A → RN has a continuous fixed

point free extension f to B.

Proof: We may assume without loss of generality that A and B lie in the subspace with

coordinates (ζ1, . . . , ζM , 0, . . . , 0). We also set F (x) = f(x)− x.

Since F has no zeroes on A, we get α = infx∈A |F (x)| > 0. We extend F continuously to

F : B → RN by Corollary 2.13. By the Weierstrass approximation theorem, there exists a map

G : B → RN with supx∈B |F −G| < α/3, and such that the components of G are polynomials.

Since ∂Gi(x)/∂ζj = 0 for j = M + 1, . . . , N , we have det(G′(x)) = 0 on B. By Sard’s Theorem,

G(B) has no interior points. Hence, there exists an x0 such that x0 /∈ G(B) and |x0| < α/3.

For (x, t) ∈ (A× [0, 1]) ∪ (B × 1) define h(x, t) = (1− t)F (x) + t(G(x)− x0). Then h(x, t) 6= 0

since |h(x, t)| ≥ |F (x)|− |F (x)−G(x)|− |x0| > 0. This also implies h(x, 1) 6= 0 for x ∈ B. Thus

we can apply Lemma 2.34 to H(x, t) = x + h(x, t). In particular, we get that the extention

H(x, 0) : B → RN is fixed point free and H(x, 0) = f(x) for x ∈ A. 2

Theorem 2.36 (Antipodal Theorem) Let X be a Banach space and let T : BR(0) → X

be compact. If T has no fixed points on ∂BR(0) and if T satisfies the antipodal condition

T (−x) = −T (x) for x ∈ ∂BR(0), then the fixed point index i(T,BR(0)) is odd.

Proof: First, we proof this theorem for dim(X) <∞. For this we can assume that X = RN

for some N ≥ 1.

Q is called an orthant of RN if

Q = x ∈ RN : 0 ≤ ejxj for j = 1, . . . , N,

where (e1, . . . eN ) ∈ −1, 1N is fixed. Per definition, there are 2N orthants in RN . Let

Q1, . . . , Q2N−1 be the 2N−1 orthants of RN with xN ≥ 0 for all x ∈ Qj . Then RN =

Q1 ∪ · · · ∪Q2N−1 ∪ −Q1 ∪ · · · ∪ −Q2N−1 .1

Now fix r ∈ (0, R) and define Bj as the interior of Qj ∩BR(0)\Br(0). Then

BR(0) = Br(0) ∪B1 ∪ · · · ∪B2N−1 ∪ −B1 ∪ · · · ∪ −B2N−1 .

We show that there exists a continuous function S : BR(0)→ RN with

S(x) = T (x) for all x ∈ ∂BR(0), (2.5)

S(x) = 0 for all x ∈ Br(0), (2.6)

1This is the multidimensional equivalent of the following easy construction in R2: Divide the plane into its fourquadrants. Choose Q1 and Q2 as the first and second quadrant, respectively. Then −Q1 is the third quadrantand −Q2 the fourth. So R2 = Q1 ∪Q2 ∪ −Q1 ∪ −Q2.

22

S(x) 6= x for all x ∈ ∂(±Bj), j = 1, . . . , 2N−1, (2.7)

S(−x) = −S(x) for all x ∈ BR(0). (2.8)

Because of (2.6) we only have to define S on ±Bj . There S is already defined on ∂Bj ∩ ∂BR(0)

and ∂Bj ∩ ∂Br(0) by (2.5) and (2.6), and S is fixed point free there. The remaining border of

Bj can be divided into N parts, each of which lies in a N − 1 dimensional hyperplane of RN .

We start by applying Lemma 2.35 to each of these parts in order to extend S continuously

to ∂B1 such that (2.7) is fulfilled. We now can extend S continuously from ∂B1 to B1 using

Theorem 2.12. Further, we define S on −B1 by S(x) = −S(−x) for x ∈ −B1, such that (2.8) is

fulfilled. We repeat this procedure consecutively with B2 to B2N−1 . Note that when considering

Bj , S is already defined on Bj ∩ Bi for i < j, so we can skip defining S on that part of the

border of Bj .

Now (2.6) implies that i(S,Br(0)) = 1 by axiom (A1) of the fixed point index. Using the home-

omorphism h : x 7→ −x we can use Theorem 2.32 (with Ω = BR(0)) to see that i(S(x), Bj) =

i(−S(−x),−Bj) = i(S(x),−Bj). Axiom (A3) of the fixed point index then gives

i(S,BR(0)) = i(S,Br(0)) +2N−1∑j=1

(i(S,Bj) + i(S,−Bj)) = i(S,Br(0)) + 22N−1∑j=1

i(S,Bj),

i.e. i(S,BR(0)) is odd. Since S(x) = T (x) for x ∈ ∂BR(0), Proposition 2.28 together with

axiom (A4) of the fixed point index show that i(T,BR(0)) = i(S,BR(0)).

Now let dim(X) =∞. First, we observe that (T (x)− T (−x))/2 is compact and coincides with

T (x) for x ∈ ∂BR(0). By Proposition 2.28 and (A4), this yields i(T (x), BR(0)) = i((T (x) −T (−x))/2, BR(0)). By Lemma 2.29, there is a compact operator S : BR(0)→ Y , where Y is a

finite dimensional subspace of X, such that supx∈BR(0) ‖T (x)− S(x)‖ < infx∈∂BR(0) ‖T (x)− x‖.As above, for P (x) = (S(x) − S(−x))/2 we have i(P,BR(0)) = i((T (x) − T (−x))/2, BR(0)).

But since P maps BR(0) into a finite dimensional subspace Y , Theorem 2.33 shows that

i(P,BR(0)) = i(P,BR(0) ∩ Y ), the latter of which is odd by the first part of the proof, since

P (−x) = −P (x). 2

23

Chapter 3

The FMV-spectrum

3.1 FMV-regular operators

FMV-regularity, named after its inventors M. Furi, M. Martelli, and A. Vignoli, will play the

essential role in the definition of the FMV-spectrum as explained in the introduction.

3.1.1 Stable solvability

Stable solvability is a property of operators that is important in the definition of FMV-regularity

and of the FMV-spectrum. It assures solvability of certain types of equations.

In the following, X and Y will always denote Banach-spaces.

Definition 3.1 A continuous operator T : X → Y is called stably solvable, if, given any

compact operator S : X → Y with [S]Q = 0, the equation T (x) = S(x) has a solution x ∈ X.

Lemma 3.2 Let T ∈ C(X,Y ) be stably solvable. Then T is surjective.

Proof: For y ∈ Y the operator S(x) ≡ y fulfils [S]A = [S]Q = 0. By the stable solvability of

T , there is an x ∈ X with T (x) = S(x) = y. 2

In general, the converse of this lemma does not hold true. We will however show later that in

the case of linear operators stable solvability reduces to surjectivity.

The notion of stable solvability can be extended further. This more general definition will be

usefull in the study of FMV-regular operators.

Definition 3.3 For k ≥ 0, we call an operator T ∈ C(X,Y ) k-stably solvable, if, given any

operator S ∈ C(X,Y ) with [S]A ≤ k and [S]Q ≤ k, the equation T (x) = S(x) has a solution

x ∈ X.

Obviously, 0-stably solvable operaters are exactly the stably solvable operators. Moreover,

every k-stably solvable operator is certainly k′-stably solvable for 0 ≤ k′ < k. This motivates

the following definition.

24

Definition 3.4 For T ∈ C(X,Y ) we call

µ(T ) := infk : k ≥ 0, T is not k-stably solvable

the measure of stable solvability of T .

We call an operator T ∈ C(X,Y ) strictly stably solvable if µ(T ) > 0, i.e. T is k-stably solvably

for some k > 0.

For the rest of this section we return to stably solvable operators and show some of their

properties.

Lemma 3.5 Let T ∈ C(X,Y ) with [T ]q > 0. Then T is stably solvable if and only if the

equation T (x) = S(x) has a solution x ∈ X for every compact operator S : X → Y for which

the set x ∈ X : S(x) 6= 0 is bounded.

Proof: Since every operator S with S(x) = 0 outside a bounded set fulfils [S]Q = 0, this

direction of the proof is trivial.

So let S be a compact operator from X into Y with [S]Q = 0. For n ∈ N define the operator

Sn(x) = dn(‖x‖)S(x), where

dn(t) =

1 if 0 ≤ t ≤ n,2− 1

n t if n ≤ t ≤ 2n,

0 if t ≥ 2n.

Then x ∈ X : Sn(x) 6= 0 is bounded and Sn compact. Hence, by assumption, there exists an

xn ∈ X with Sn(xn) = T (xn). If ‖xn‖ ≤ n for some n ∈ N, then T (xn) = Sn(xn) = S(xn) and

we are finished.

So assume ‖xn‖ > n for all n ∈ N. Then ‖xn‖ → ∞ as n→∞, and we get the contradiction

[T ]q = lim inf‖x‖→∞

‖T (x)‖‖x‖

≤ limn→∞

‖T (xn)‖‖xn‖

= limn→∞

dn(‖xn‖)‖S(xn)‖‖xn‖

≤ limn→∞

‖S(xn)‖‖xn‖

≤ lim sup‖x‖→∞

‖S(x)‖‖x‖

= 0.

2

Corollary 3.6 The identity operator is stably solvable.

Proof: Since [I]q = 1 we can restrict ourselves to compact maps S with x ∈ X : S(x) 6= 0is bounded to prove stable solvability. For such S, there obviously exists an r > 0 such that

S(Br(0)) ⊆ Br(0). But then Schauders fixed point theorem tells us that S has a fixed point,

i.e. S(x) = I(x). 2

25

Theorem 3.7 Let T ∈ C(X) be stably solvable, B ⊆ X a nonempty closed subset, and let

H : B → X be a continuous operator. Assume that H(B) is bounded and

T−1(co(H(B))) ⊆ B, i.e. T (y) ∈ co(H(B))⇒ y ∈ B.

Moreover, suppose that the equality

α(T (M)) = α(H(M))

implies the precompactness of M for any M ⊆ B.

Then the equation T (x) = H(x) has a solution x ∈ X.

Proof: We construct a sequence (xn)n∈N as follows. Let x0 ∈ B be arbitrary. Due to

Lemma 3.2 we can choose inductively xn ∈ T−1(H(xn−1)) for n ≥ 1, so T (xn) = H(xn−1).

By construction, the set A = x0, x1, . . . satisfies T (A) = T (xo) ∪H(A). Therefore, we get

α(T (A)) = α(T (xo) ∪H(A)) = α(H(A)). Hence, A is precompact by assumption. Moreover,

by construction A ⊆ B and consequently H(A) ⊆ H(B) is bounded.

Next, we show that A′ ⊆ T−1(H(A′)), where A′ denotes the set of all accumulation points of

A. Note that A′ 6= ∅ since A is precompact. So for x ∈ A′ we find a subsequence (xnk)k∈N of

(xn)n∈N such that xnk→ x, hence, also H(xnk−1) = T (xnk

) → T (x) as k → ∞. Therefore, if

we choose y ∈ A′ to be a limit point of (xnk−1)k∈N, we get H(y) = T (x).

Next, we consider the family

M = M : M ⊇ A′, M = M, T−1(co(H(M))) ⊆M.

Since B ∈M, this family is nonempty. Denote by M0 the intersection of all M ∈M. Note that

M0 is nonempty and closed, since M ⊇ A′ 6= ∅ for each M ∈M and each M is closed.

Set M1 = T−1(co(H(M0))). Then for all M ∈M we get

M1 = T−1(co(H(M0))) ⊆ T−1(co(H(M))) ⊆M,

hence M1 ⊆M0.

Further, M1 is closed and satisfies both

M1 = T−1(co(H(M0))) ⊇ T−1(co(H(A′))) ⊇ A′

and

T−1(co(H(M1))) ⊆ T−1(co(H(M0))) = M1.

Therefore, M1 ∈ M and M0 ⊆ M1. Consequently, T−1(co(H(M0))) = M1 = M0. Since T is

stably solvable and, therefore, surjective by Lemma 3.2, this in turn implies T (M0) = co(H(M0))

26

and, hence,

α(T (M0)) = α(co(H(M0))) = α(H(M0)).

Thus, by assumption, M0 is compact, and by the continuity of H also H(M0) is compact.

By Corollary 2.13, we find a continuous operator G : X → X with G(x) = H(x) for x ∈ M0

and G(X) ⊆ co(H(M0)). This shows that G(X) is precompact, and in turn that G is a

compact operator. Furthermore, [G]Q = 0 since G(X) is bounded. Since T is stably solv-

able, there exists an x ∈ X with T (x) = G(x). But from T (x) ∈ co(H(M0)) it follows that

x ∈ T−1(co(H(M0))) = M0. Thus, T (x) = G(x) = H(x). 2

Corollary 3.8 (Darbo fixed point theorem) Let B ⊆ X be nonempty, bounded, closed,

and convex. Let S ∈ C(B,B) and suppose [S]A < 1. Then S has a fixed point.

Proof: Choose H = S and T = I in Theorem 3.7, the latter of which is possible because of

Corollary 3.6 2

Corollary 3.9 Let T ∈ C(X) with [T ]A < 1 and [T ]Q < 1. Then T has a fixed point.

Proof: Let b ∈ ([T ]Q, 1). Then ‖T (x)‖ ≤ b‖x‖ for all x ∈ X with ‖x‖ ≥ r for sufficiently large

r > 0. Because of [T ]A <∞, T (Br(0)) is bounded. Hence, there exists a d > 0 with ‖T (x)‖ < d

for x ∈ Br(0). Altogether we have ‖T (x)‖ ≤ b‖x‖+ d for all x ∈ X.

Setting R = d/(1− b), for x ∈ BR(0) we get

‖T (x)‖ ≤ b‖x‖+ d ≤ bR+ d =bd+ d(1− b)

1− b=

d

1− b= R.

This shows, that T maps BR(0) into itself and the assertion follows from Corollary 3.8. 2

Theorem 3.10 Let T ∈ B(X,Y ). Then T is stably solvable if and only if T is surjective.

Proof: By Lemma 3.2, we only have to prove that a surjective operator is stably solvable. So

let T ∈ B(X,Y ) be surjective. By Theorem 2.26 we may find a continuous function s : Y → X

such that s(y) ∈ T−1(y) for all y ∈ Y and ‖s(y)‖ ≤M‖y‖ for some M > 0.

Now let G : X → Y be compact with [G]Q = 0. Since s maps bounded sets onto bounded sets,

the map G s : Y → Y is compact.

Since [G]A = [G]Q = 0, we can use the same technique as at the beginning of the proof of

Corollary 3.9 to show that for every ε > 0 we can find a d > 0 such that ‖G(x)‖ ≤ ε‖x‖+ d for

27

all x ∈ X. For any sequence (yn)n∈N with ‖yn‖ → ∞ we then get

limn→∞

‖G s(yn)‖‖yn‖

≤ limn→∞

ε‖s(yn)‖+ d

‖yn‖≤ lim

n→∞

εM‖yn‖+ d

‖yn‖= lim

n→∞

(εM +

d

‖yn‖

)= εM.

Since ε > 0 was arbitrary, we get limn→∞‖Gs(yn)‖‖yn‖ = 0. Hence, [G s]Q = 0.

Thus, we can use Corollary 3.9 to see that Gs has a fixed point y ∈ Y . Consequently, x = s(y)

satisfies

Tx = Ts(y) = y = G(s(y)) = G(x),

i.e. T is stably solvable. 2

3.1.2 FMV-regularity

Definition 3.11 An operator T ∈ C(X,Y ) is called FMV-regular, if T is stably solvable and

fulfils [T ]q > 0 and [T ]a > 0.

By what we have shown in chapter 2, amongst the properties of FMV-regular operators are,

that they are proper and coercive. Both of these properties are due to the fact, that [T ]q > 0,

and [T ]a > 0. But the positivity of these two characteristics also has an important influence on

the stable solvability of the operator.

Lemma 3.12 Every FMV-regular operator is strictly stably solvable. More precisely, the esti-

mate

µ(T ) ≥ min [T ]q, [T ]a

holds true.

Proof: Fix k with k < [T ]q and k < [T ]a. We have to show that the equation T (x) = S(x)

has a solution x ∈ X for any S ∈ C(X,Y ) with [S]Q ≤ k and [S]A ≤ k.

Choose two numbers b and c such that [S]Q < b < c < [T ]q. Then ‖S(x)‖ ≤ b‖x‖ and

‖T (x)‖ ≥ c‖x‖ for ‖x‖ ≥ r with sufficiently large r > 0. Since [S]A <∞, S(Br(0)) is bounded

by some constant R > 0. Hence, ‖S(x)‖ ≤ R+ b‖x‖ for all x ∈ X. Put

ρ =R

c− b.

By choosing R sufficiently large, we may assume without loss of generality that ρ ≥ r. We

now want to apply Theorem 3.7 with B = Bρ(0). It is clear that S(Bρ(0)) is bounded, because

S(Bρ(0)) ⊆ BR+bρ(0).

Fix x ∈ X with T (x) ∈ BR+bρ(0). If ‖x‖ > ρ were true (which would also imply ‖x‖ > r), we

28

would get

R+ bρ ≥ ‖T (x)‖ ≥ c‖x‖ > cρ,

i.e.

R

c− b> ρ,

contraditing our choice of ρ. Therefore,

T−1(co(S(Bρ(0)))) ⊆ T−1(co(BR+bρ(0))) = T−1(BR+bρ(0)) ⊆ Bρ(0).

Furthermore, for all M ⊆ Bρ(0) with α(M) > 0, by assumption we have

α(T (M)) ≥ [T ]aα(M) > [S]Aα(M) ≥ α(S(M)).

Thus, all hypotheses of Theorem 3.7 are satisfied. Hence the equation T (x) = S(x) has a solu-

tion x ∈ X. 2

Lemma 3.13 Let T, S ∈ C(X,Y ), let T be k-stably solvable. Further, assume that k ≥ [S]A

and k ≥ [S]Q. Then T +S is k′-stably solvable for 0 ≤ k′ ≤ k−max[S]A, [S]Q. In particular,

µ(T + S) ≥ µ(T )−max[S]A, [S]Q.

Proof: Let H ∈ C(X,Y ) with [H]A ≤ k −max[S]A, [S]Q and [H]Q ≤ k −max[S]A, [S]Q.We have to show that the equation T (x) +S(x) = H(x) has a solution x ∈ X. But [H −S]A ≤[H]A + [S]A ≤ k and [H − S]Q ≤ [H]Q + [S]Q ≤ k. Since T is k-stably solvable, the equation

T (x) = (H − S)(x) has a solution x ∈ X. 2

Theorem 3.14 Let T, S ∈ C(X,Y ). If T is FMV-regular with min[T ]q, [T ]a > [S]A and

min[T ]q, [T ]a > [S]Q, then T + S is also FMV-regular.

Proof: By Lemma 3.12, T is k-stably solvable for k < min[T ]q, [T ]a. By Lemma 3.13, T +S

is stably solvable. Moreover, [T + S]q ≥ [T ]q − [S]Q > 0 and [T + S]a ≥ [T ]a − [S]A > 0, by

Lemma 2.2,(ii) and Lemma 2.7,(iv) respectively. 2

Heuristically, this theorem shows that FMV-regularity is stable under perturbations that are

sufficiently compact and quasibounded.

FMV-regularity is also invariant under linear isomorphisms.

29

Theorem 3.15 Let T ∈ C(X,Y ) and let S ∈ B(Y,Z) be bijective. Then the operator S T ∈C(X,Z) is FMV-regular if and only if T is FMV-regular.

Proof: First, let T be FMV-regular. By definition [T ]a > 0 and [T ]q > 0. Moreover [S]a > 0

and [S]q > 0 by Lemma 2.8,(iv). So Proposition 2.2,(v) and Proposition 2.7,(vii) show that

[S T ]a ≥ [S]a[T ]a > 0

and

[S T ]q ≥ [S]q[T ]q > 0,

respectively. It remains to show that S T is stably solvable. So let G ∈ C(X,Z) satisfy

[G]A = [G]Q = 0. Then obviously S−1 G is still compact and [S−1 G]Q ≤ ‖S−1‖[G]Q = 0.

Since T is stably solvable, there exists an x ∈ X with T (x) = S−1G(x). Hence, ST (x) = G(x)

and S T is stably solvable.

Conversely, suppose S T is FMV-regular. Since S−1 is a linear bijection, we can use the first

part of the proof to see that T = S−1 (S T ) is FMV-regular. 2

Finally, we want to show, that for linear operators FMV-regularity reduces to a very simple

property.

Theorem 3.16 Let T ∈ B(X,Y ). Then T is FMV-regular if and only if T is bijective.

Proof: If T is FMV-regular, then [T ]q > 0. By Lemma 2.8,(iii), T is injective. Since T is

also stably solvable, T is surjective by Theorem 3.10.

Conversely, let T be a bijection. Then [T ]a > 0 and [T ]q > 0 by Lemma 2.8,(iv). Again by

Theorem 3.10, T is also stably solvable. 2

3.2 Topological properties

Definition 3.17 For T ∈ C(X) we call the set

ρFMV (T ) := λ ∈ K : λI − T is FMV-regular

the FMV-resolvent set and its complement

σFMV (T ) := K\ρFMV (T )

the FMV-spectrum of T .

Thanks to our extensive preparations, the following proofs concerning topological properties of

σFMV (T ) are very simple. One of the most important properties that we wanted a nonlinear

30

spectrum to have, has already been proven for the FMV-spectrum.

Proposition 3.18 σ(T ) = σFMV (T ) for T ∈ B(X).

Proof: This is a simple consequence of Theorem 3.16. 2

Proposition 3.19 The FMV-spectrum σFMV (T ) is closed for all T ∈ C(X).

Proof: Fix λ ∈ ρFMV (T ) and let 0 < δ < min[λI − T ]a, [λI − T ]q. Now let µ ∈ K with

|µ− λ| < δ. Because of

[(µ− λ)I]A = |µ− λ| < [λI − T ]a

and

[(µ− λ)I]Q = |µ− λ| < [λI − T ]q,

it follows from Theorem 3.14 that µI − T = (λI − T ) + (µ − λ)I is FMV-regular. This shows

that λ is an interior point of ρFMV (T ). Thus, ρFMV (T ) is open in K. 2

Proposition 3.20 Let T ∈ C(X) and suppose [T ]A < ∞ and [T ]Q < ∞. Then σFMV (T ) is

bounded. In fact, it is contained in Bmax[T ]A,[T ]Q(0).

Proof: Let λ ∈ K with |λ| > [T ]Q and |λ| > [T ]A. We will show, that λ ∈ ρFMV (T ).

First of all, we have [λI − T ]q ≥ |λ| − [T ]Q > 0 and [λI − T ]a ≥ |λ| − [T ]a > 0 by Lemma

2.2,(ii) and Lemma 2.7,(iv), respectively. It remains to show that λI − T is stably solvable. So

let S ∈ C(X) be compact with [S]Q = 0. Then the operator H := (T + S)/λ satisfies both

[H]A =

[T + S

λ

]A

=1

|λ|[T + S]A ≤

1

|λ|([T ]A + [S]A) =

[T ]A|λ|

< 1

and

[H]Q =

[T + S

λ

]Q

=1

|λ|[T + S]Q ≤

1

|λ|([T ]Q + [S]Q) =

[T ]Q|λ|

< 1.

By Corollary 3.9, H has a fixed point x ∈ X, from which the assertion follows. 2

The previous proposition motivates the following definitions.

Definition 3.21 We define the set

AQ(X) := A(X,X) ∩Q(X,X) = T ∈ C(X) : [T ]A <∞ and [T ]Q <∞.

31

For T ∈ AQ(X) define

pAQ(T ) := max[T ]A, [T ]Q.

Further, for T ∈ C(X) we call

rFMV (T ) := sup|λ| : λ ∈ σFMV (T )

the FMV spectral radius of T . If σFMV (T ) = ∅, we set rFMV (T ) = 0.

The following assertion has been proven in Proposition 3.20 and 3.19

Corollary 3.22 Let T ∈ AQ(X). Then σFMV (T ) is compact. Furthermore,

rFMV (T ) ≤ pAQ(T ).

It is easy to see, that pAQ is a seminorm on AQ. Also, note that B(X) ⊆ AQ(X) and pAQ(T ) =

‖T‖ for T ∈ B(X). This follows from Lemma 2.8,(i) and (ii). Hence, Corollary 3.22 includes

the well known result that for a linear operator the spectral radius of the classical spectrum is

bounded by its operator norm.

Further, AQ(X) can be equipped with a topology defined by the seminorm pAQ. The discussion

above then also shows that the following proposition includes the result of Proposition 1.1.

Proposition 3.23 The multivalued map σFMV : AQ(X) → P(K), which assignes to each

T ∈ AQ(X) its FMV-spectrum, is upper semi-continuous.

Proof: First, we want to show, that the map σFMV is closed. To do so, choose a sequence

(Tn)n∈N in AQ(X) and a sequence (λn)n∈N in ⊆ K such that

λn ∈ σFMV (Tn), λn → λ, pAQ(T − Tn)→ 0.

Since

pAQ((λnI − Tn)− (λI − T )) ≤ |λn − λ|+ pAQ(T − Tn)→ 0,

the sequence (λnI − Tn)n∈N converges to λI − T in AQ(X). If λI − T were FMV-regular, then

for sufficiently large n, λnI−Tn = (λI−T )+((λnI−Tn)−(λI−T )) would also be FMV-regular

by Theorem 3.14, contradicting our choice of λn. Therefore, λ ∈ σFMV (T ) and σFMV is closed.

The assertion now follows from Lemma 2.19, since by Corollary 3.22 for all T ∈ AQ(X) we have

supλ∈σFMV (T )

|λ| = rFMV (T ) ≤ pAQ(T ).

2

32

3.3 Subdivision of the FMV-spectrum

Definition 3.24 For T ∈ C(X) we set

σδ(T ) := λ ∈ K : λI − T is not stably solvable,

σa(T ) := λ ∈ K : [λI − T ]a = 0,

σq(T ) := λ ∈ K : [λI − T ]q = 0,

and

σπ(T ) := σa(T ) ∪ σq(T ).

By definition, we then have

σFMV = σδ ∪ σπ = σδ ∪ σa ∪ σq.

However, this decomposition need not be disjoint.

The seminorm pAQ can be usefull to compare the spectrum und subspectra of two operators.

Lemma 3.25 Let T, S ∈ C(X) such that pAQ(T−S) = 0. Then σq(T ) = σq(S), σa(T ) = σa(S),

and σδ(T ) = σδ(S), and hence σFMV (T ) = σFMV (S).

Proof: By Proposition 2.2,(iii), [T − S]Q = 0 implies σq(T ) = σq(S), while [T − S]A implies

σa(T ) = σa(S) by Proposition 2.7,(v).

Assume λI−S is stably solvable and let H ∈ C(x) with pAQ(H) = 0. Since pAQ(H+(T−S)) = 0,

there exists an x ∈ X with λx−S(x) = H(x) +T (x)−S(x), i.e. λx−T (x) = H(x). So λI −Tis also stably solvable. Hence, σδ(T ) = σδ(S). 2

We now want to show that these subdivisons have their counterparts in the linear theory.

Definition 3.26 For T ∈ B(X) we define the approximate point spectrum

σapp(T ) := λ ∈ K : ∃xn, n ∈ N, ‖xn‖ = 1, ‖(λI − T )(xn)‖ → 0

and the defect spectrum

σd(T ) := λ ∈ K : λI − T is not surjective.

Just as the FMV-spectrum is the union of σδ(T ) and σπ(T ), for linear operators we have

σ(T ) = σapp(T ) ∪ σd(T ). We show that σapp(T ) and σd(T ) correspond with σδ(T ) and σπ(T ),

respectively. In particular, they are the same in the case of linear operators.

Lemma 3.27 If T ∈ B(X), then σa(T ) ⊆ σq(T ).

Proof: If dim(X) < ∞, then σa(T ) = ∅. So we can restrict ourselves to infinite dimensional

spaces. Let λ ∈ σa(T ). Then for every n ∈ N there exists a bounded set Mn with α(Mn) > 0

33

and α((λI − T )(Mn)) < α(Mn)/(2n).

If (λI − T ) is not injective on Mn, choose x1, 6= x2 ∈Mn with (λI − T )(x1) = (λI − T )(x2) and

set yn = x1 − x2.If (λI − T ) is injective on Mn, define

K = infx 6=y

x,y∈Mn

‖(λI − T )(x)− (λI − T )(y)‖‖x− y‖

.

Assume K > 0 and fix k ∈ (0,K). Then for x, y ∈Mn we have

‖(λI − T )(x)− (λI − T )(y)‖ ≥ k‖x− y‖.

If z1, . . . , zn is a finite ε-net for (λI − T )(Mn), then (λI − T )−1(z1), . . . , (λI − T )−1(zn) is

a finite ε/k−net for Mn, i.e. α((λI − T )(Mn)) ≥ kα(Mn). This in turn means 1/(2n) > k.

Hence, 1/n > K.

We therefore can choose x1, x2 ∈ Mn with ‖(λI − T )(x1) − (λI − T )(x2)‖ < ‖x1 − x2‖/n. Set

yn = x1 − x2. This is of course also possible in the case of K = 0.

We need to show that

lim inf‖y‖→∞

‖(λI − T )(y)‖‖y‖

= 0.

For now we have

limn→∞

‖(λI − T )(yn)‖‖yn‖

= 0.

If (yn)n∈N has a subsequence (ynk)k∈N with ‖ynk

‖ → ∞, then λ ∈ σq(T ) and we are done. If it

has subsequence (ynk)n∈N that is both bounded and bounded away from zero, set xk = kynk

to

see that λ ∈ σq(T ). If ‖ynk‖ → 0, set xk = ynk

/‖ynk‖2. We then get ‖xk‖ → ∞ and

limk→∞

‖(λI − T )(xk)‖‖xk‖

= limk→∞

‖ynk‖−2‖(λI − T )(ynk

)‖‖ynk‖−1

= limk→∞

‖(λI − T )(ynk)‖

‖ynk‖

= 0,

hence, λ ∈ σq(T ). 2

Theorem 3.28 If T ∈ B(X), then σδ(T ) = σd(T ) and σπ(T ) = σapp(T ).

Proof: That σδ(T ) = σd(T ) is a simple consequence of Theorem 3.10. To see that σq(T ) =

σapp(T ) consider the equation

lim inf‖x‖→∞

‖λx− T (x)‖‖x‖

= lim inf‖x‖→∞

∥∥∥∥λ x

‖x‖− T

(x

‖x‖

)∥∥∥∥ = lim inf‖y‖=1

‖λy − T (y)‖.

34

Since σq(T ) = σπ(T ) by Lemma 3.27, we are done. 2

One of the important properties of the approximate point spectrum is that it is closed and

contains the boundary of the classical spectrum. The same is true for its nonlinear counterpart.

Lemma 3.29 Let T ∈ C(X), then σa(T ), σq(T ), and σπ(T ) are closed.

Proof: Let (λn)n∈N ⊆ σq(T ) converge to a λ. By Proposition 2.2,(iii), we get

[λI − T ]q = |[λI − T ]q − [λnI − T ]q| ≤ [λI − T − λnI + T ]Q = |λn − λ| → 0.

Hence, σq(T ) is closed. Similarly, σa(T ) is closed and, therefore, also σπ(T ) = σa(T )∪σq(T ). 2

Theorem 3.30 Let T ∈ C(X). Then

∂σFMV (T ) ⊆ σπ(T ).

Proof: First, we want to show, that σFMV (T )\σπ(T ) is an open subset of K. So fix a

λ ∈ σFMV (T )\σπ(T ). It suffices to show that there exists a δ > 0, such that µI − T is not

stably solvable for |µ− λ| < δ.

If this is not true, we find a sequence (λn)n∈N with λn → λ such that λnI −T is stably solvable

for all n. Since λI − T is not stably solvable, there exists a compact operator S with [S]Q = 0

such that λx − T (x) 6= S(x) for all x ∈ X. On the other hand, since all λnI − T are stably

solvable, we get a sequence (xn)n∈N with λnxn − T (xn) = S(xn).

In case ‖xn‖ → ∞ we get

‖λxn − T (xn)‖‖xn‖

≤ ‖λxn − λnxn‖‖xn‖

+‖λnxn − T (xn)‖

‖xn‖= |λ− λn|+

‖S(xn)‖‖xn‖

→ 0.

But this means [λI−T ]q = 0, contradicting λ /∈ σπ. Hence, there exists a bounded subsequence

(xnk)k∈N. This implies

‖(λ− T − S)(xnk)‖ = ‖λxnk

− (T − S)(xnk)‖ = ‖λxnk

− λnkxnk‖ ≤ |λ− λnk

| ‖xnk‖ → 0.

Furthermore, by Proposition 2.4,(v), [λ−T −S]a = [λ−T ]a > 0. Thus, by Proposition 2.7,(ii),

the preimage of the compact set (λ−T −S)(xnk)∪0, which in particular includes (xnk

)k∈N,

is compact. We can therefore assume without loss of generality, that xnk→ x for some x ∈ X.

By continuity, λx− T (x)− S(x) = 0. This contradiction shows that our assertion was true.

Now let λ ∈ ∂σFMV (T ), and assume λ /∈ σπ(T ). Then λ ∈ σFMV (T )\σπ(T ) which is an in Kopen subset of σFMV (T ), contradicting λ ∈ ∂σFMV (T ). 2

35

Corollary 3.31 Let T ∈ C(X) and let C ⊆ K\σπ(T ) be a connected set. Then either C ⊆σFMV (T ) or C ⊆ ρFMV (T ).

Proof: Let K be the connected component of K\σπ(T ) which contains C. Set

K0 = ν ∈ K : νI − T is FMV-regular = K ∩ ρFMV (T )

and assume K0 is not empty. By Theorem 3.30, the boundary of K0 relative to K is empty.

Therefore, K0 is open and closed in K. Since K is connected, it follows, that K0 = K. 2

3.3.1 Special classes of operators

We now look at two special classes of operators for which we can give more detailed information

about their FMV-spectrum. First, we look at compact operators in infinite dimensional Banach

spaces. The restriction to infinite dimensional spaces is to be expected, since in finite dimen-

sional spaces every continuous operator is compact. Hence, compactness only yields additional

properties in infinite dimensions. In the following proposition, for a closed set Σ ⊆ K with

0 /∈ Σ, we denote by c0[Σ] the connected component of K\Σ containing 0.

Proposition 3.32 Let X be an infinite dimensional Banach space and suppose T ∈ C(X) is

compact. Then the following is true:

(i) σa(T ) = 0, hence σπ = 0 ∪ σq(T ).

(ii) T is not surjective. In particular, 0 ∈ σδ(T )

(iii) Either 0 ∈ σq(T ), or c0[σq(T )] ⊆ σδ(T ).

(iv) If 0 /∈ σq(T ) and σFMV (T ) is bounded, then σq(T ) contains a positive and a negative value.

(v) If σFMV (T ) 6= K, then σq(T ) 6= ∅.

Proof: Assertion (i) simply follows from the fact, that [λI−T ]a = |λ|, which is a consequence

of Proposition 2.4,(v). For (ii), assume that T is surjective. For each y ∈ X we get y ∈ T (Bn(0))

for some n ∈ N. We therefore have the representation

X =∞⋃n=1

T (Bn(0)) =∞⋃n=1

T (Bn(0)).

By Baire’s catgegory theorem, the countable union of closed sets without interior points has no

interior points. Since X does have an interior point, at least one of the compact sets T (Bn(0))

contains an interior point. In particular, T (Bn(0)) contains a closed ball, which in turn is

compact. Since closed balls in a space X are compact if and only if X is finite dimensional,

this contradicts our choice of X as an infinite dimensional Banach space. 0 ∈ δδ(T ) now follows

36

from Lemma 3.2.

To see that (iii) holds true, first observe that the condition 0 /∈ σq(T ) together with (i) implies

that 0 is an isolated point of σπ(T ) since σq(T ) is closed. Below we will show that λI−T is not

surjective for |λ| small enough. By Lemma 3.2, this implies that 0 is an interior point of σδ(T )

and thus also of the FMV-spectrum. Theorem 3.30 and (i) then tell us that ∂σFMV (T ) ⊆ σq(T ).

Hence, σδ(T ) has no boundary in c0[σq(T )]. σδ(T )∩ c0[σq(T )] is therefore both open and closed

in the connected set c0[σq(T )], i.e. σδ(T )∩ c0[σq(T )] is either empty or coincides with c0[σq(T )].

Since 0 ∈ σδ(T ) by (ii), the latter must be true, which proves (iii).

To show that λI − T is not surjective for |λ| small enough if 0 /∈ σq(T ), assume that this is

not the case. Then there exists a sequence (λn)n∈N in K converging to zero, such that λnI − Tis surjective for all n. Fix a ∈ (0, 12 [T ]q), and choose R > 0 such that ‖T (x)‖ ≥ 2a‖x‖ for

‖x‖ > R. Taking b = 2aR we then have ‖T (x)‖ ≥ 2a‖x‖ − b for all x ∈ X. Consequently, for

|λ| ≤ a we have ‖λx− T (x)‖ ≥ a‖x‖ − b.Now fix y ∈ B1(0). By assumption, we find a sequence (xn)n∈N in X such that λnxn−T (xn) = y

for all n. Without loss of generality we may assume that |λn| ≤ a for all n, and thus

1 > ‖y‖ = ‖λnxn − T (xn)‖ ≥ a‖xn‖ − b.

Hence, xn ∈ Br(0) with r = (1 + b)/a. Since λnxn → 0, we conclude that T (xn) → −y. But

since y ∈ B1(0) was arbitrary, we see that B1(0) ⊆ T (Br(0)), which is impossible as T (Br(0))

is precompact.

(iv) is an immediate consequence of (iii), since for instance R+∩σq(T ) = ∅ yields the unbound-

edness of σδ(T ) ⊆ σFMV (T ). Finally, if σq(T ) = ∅, then σδ(T ) = K by (iii). Thus, (v) holds

true. 2

Definition 3.33 An operator T ∈ C(X) is called asymptotically linear if there exists an operator

T ′ ∈ B(X) with [T − T ′]Q = 0. T ′ is called the asymptotic derivative of T .

If an operator is asymptotically linear, its asymptotic derivate is uniquely defined, since for two

asymptotic derivates T ′1 and T ′2 we get

‖T ′1 − T ′2‖ = [T ′1 − T ′2]Q ≤ [T ′1 − T ]Q + [T − T ′2]Q = 0.

For asymptotically linear operators the spectrum of its asymptotic derivate can be used to gain

information about its FMV-spectrum.

Proposition 3.34 Let T ∈ C(X) be an asymptotically linear operator with asymptotic derivate

T ′. Then the following is true:

(i) σq(T ) = σq(T′).

(ii) ∂σ(T ′) ⊆ σq(T ), in particular, σq(T ) is not empty for K = C.

37

(iii) If T − T ′ is compact, then σFMV (T ) = σ(T ′), σa(T ) = σa(T′), σq(T ) = σq(T

′), and

σδ(T ) = σδ(T′).

Proof: First, from the definition of the asymptotic derivative and Proposition 2.2,(iii) we get

∣∣[λI − T ]q − [λI − T ′]q∣∣ ≤ [λI − T − λI + T ′]Q = [T ′ − T ]Q = 0,

from which (i) follows immediately . To see that (ii) holds, use Proposition 3.18, Theorem 3.30,

Lemma 3.27, and (i) to get

∂σ(T ′) = ∂σFMV (T ′) ⊆ σπ(T ′) = σq(T′) = σq(T ).

Finally, if T−T ′ is compact, i.e. [T−T ′]A = 0, then pAQ(T−T ′) = max[T−T ′]A, [T−T ′]Q = 0.

Hence, (iii) holds by Lemma 3.25. 2

3.4 Eigenvalues and approximate eigenvalues

Definition 3.35 For T ∈ C(X) define the point spectrum

σp(T ) := λ ∈ K : ∃x ∈ X, x 6= 0 with λx− T (x) = 0.

Every λ ∈ σp(T ) is called an eigenvalue of T .

Eigenvalues play an important role in classical spectral theory. Therefore, an important property

of the classical spectrum is the fact that σp(T ) ⊆ σ(T ) for T ∈ B(X). However, the FMV-

spectrum does not have this property. For example, for an operator T ∈ C(X) with pAQ(T ) = 0

we get σFMV (T ) ⊆ 0, since rFMV (T ) ≤ pAQ(T ). So if T (x) 6= 0 for all x ∈ X, we get

σp(T ) ⊆ K\0, i.e. σp(T ) ∩ σFMV (T ) = ∅.At first, this seems to be a rather big flaw in the theory of the FMV-spectrum, given the

importance of the point spectrum in linear theory. However, the question arises, whether the

role of eigenvalues is as important in the nonlinear theory as it is in the linear theory, or whether

their importance is due to the linearity of the operators. In fact, for nonlinear operators many

different notions of eigenvalues have been studied, and many of them seem to be more useful

for nonlinear operators than the point spectrum. The type of eigenvalues that fits particularly

well with the FMV-spectrum are approximate eigenvalues.

Definition 3.36 For T ∈ C(X), σq(T ) is called the approximate point spectrum of T . Every

λ ∈ σq(T ) is called an approximate eigenvalue.

We have already defined an approximate point spectrum for linear operators. However, we

have also shown that these two definitions coincide for linear operators, see Lemma 3.27 and

38

Theorem 3.28. The name approximate eigenvalue stems from the fact that λ ∈ σq(T ) if and

only if there exists an unbouded sequence (xn)n∈N in X with

limn→∞

‖λxn − T (xn)‖‖xn‖

= 0.

For a linear operator T ∈ B(X) we obviously have σp(T ) ⊆ σq(T ). However, this inclusion may

be strict. For compact linear operators the relation between these two sets is very close.

Lemma 3.37 Let T ∈ B(X) be compact. Then σp\0 = σq\0.

Proof: Let λ ∈ σq\0. Let (xn)n∈N be a sequence in X with

limn→∞

‖λxn − T (xn)‖‖xn‖

= 0,

and define yn = xn‖xn‖ . Then (yn)n∈N is a bounded sequence, and, since T is compact, there

exists a subsequence (ynk)k∈N such that T (ynk

)→ x. Because of

‖λynk− T (ynk

)‖ = ‖λ xnk

‖xnk‖− T

(xnk

‖xnk‖

)‖ =‖λxnk

− T (xnk)‖

‖xnk‖

→ 0,

we get ynk→ x/λ. If we define y = x/λ, then we get

T (y) = x = λy

by continuity reasons. Since ‖y‖ = 1, we have λ ∈ σp(T ). 2

Furthermore, there is also a very good topological reason why the approximate point spec-

trum is a reasonable choice to substitute the point spectrum. To show this, we first need yet

another notion of eigenvalue.

Definition 3.38 For T ∈ C(X), we call λ ∈ K an unbounded eigenvalue of T , if there exists

an unbounded sequence (xn)n∈N in X such that λxn − T (xn) = 0 for all n. The set

σ0p(T ) := λ ∈ K : λ unbounded eigenvalue of T

is called the unbounded point spectrum of T .

Obviously, we have the two inclusions σ0p(T ) ⊆ σp(T ) and σ0p(T ) ⊆ σq(T ) for all T ∈ C(X). For

T ∈ B(X) however, we even get σ0p(T ) = σp(T ). So the unbounded point spectrum is always

included in the FMV-spectrum and for linear operators it coincides with the point spectrum.

However, the following proposition will show that the approximate point spectrum has a topo-

logical property the unbounded point spectrum lacks. It also shows that the unbounded point

spectrum is never ’too far away’ from the approximate point spectrum.

Proposition 3.39 The multivalued map σq : AQ(X) → K, T 7→ σq(T ) is the closure of the

multivalued map σ0p : AQ(X)→ K, T 7→ σ0p(T ).

39

In particular, for T ∈ AQ(X) we have

σq(T )=λ ∈ K| ∃(Tn)n∈N ⊆ AQ(X), (λn)n∈N ⊆ K : λn ∈ σ0p(Tn), pAQ(Tn − T )→ 0, λn → λ

=λ ∈ K| ∃S ∈ AQ(X) : pAQ(T − S) = 0, λ ∈ σ0p(S).

Proof: Fix λ ∈ σq(T ) and let (xn)n∈N be a sequence satisfying the conditions ‖xn‖ → ∞ and

‖λxn − T (xn)‖/‖xn‖ → 0. Without loss of generality we may assume, that ‖xm − xn‖ > 2 for

m 6= n. Put

ηn(x) = max1− ‖x− xn‖, 0,

so that ηn(xm) = δmn. Now define S : X → X by

S(x) = T (x) +

∞∑n=1

ηn(x)(λxn − T (xn)).

By definition, ηn(x) 6= 0 for only one n, because if ηn(x) 6= 0, then ‖x − xn‖ < 1. So ‖x −xm‖ ≥ |‖x− xn‖ − ‖xn − xm‖| > 1 and, hence, ηm(x) = 0 for m 6= n, i.e. S is well defined.

Furthermore, [S]Q <∞. In fact, because of [T ]Q <∞ and our choice of (xn)n∈N, we get

[S]Q = lim sup‖y‖→∞

‖S(y)‖‖y‖

≤ lim sup‖y‖→∞

‖T (y)‖‖y‖

+ lim sup‖y‖→∞

‖∑∞

n=1 ηn(y)(λxn − T (xn))‖‖y‖

= [T ]Q + lim sup‖y‖→∞

ηn(y)(y)‖λxn(y) − T (xn(y))‖‖y‖

≤ [T ]Q + limn→∞

‖λxn − T (xn)‖‖xn‖ − 1

= [T ]Q,

where n(y) is the one n where ηn(y) 6= 0, if such an n exists. Next, let (yk)k∈N be a bounded

sequence. Then ‖xn − yk‖ < 1 can only be true for finitely many n. Hence, (S − T )(yk) ∈spanλxn1 − T (xn1), · · · , λxnm − T (xnm) for some m ∈ N. It therefore has a convergent

subsequence, i.e. S − T is compact. Thus, we get

[S]A = sup∞>α(M)>0

α(S(M))

α(M)≤ sup∞>α(M)>0

(α(T (M))

α(M)+α((S − T )(M))

α(M)

)= [T ]A.

In summary, S ∈ AQ(X).

By definition, S(xn) = λxn for all n, i.e. λ ∈ σ0p(S). Further, pAQ(S − T ) = 0. In fact,

[T − S]Q ≤ lim sup‖y‖→∞

‖∑∞

n=1 ηn(y)(λxn − T (xn))‖‖y‖

= 0,

as has been shown above, and [T − S]A = 0 since S − T is compact. Thus, we have proven the

inclusion

σq(T ) ⊆ λ ∈ K| ∃S ∈ AQ(X) : pAQ(T − S) = 0, λ ∈ σ0p(S).

40

The inclusion

λ ∈ K| ∃S ∈ AQ(X) : pAQ(T − S) = 0, λ ∈ σ0p(S) ⊆

λ ∈ K| ∃(Tn)n∈N ⊆ AQ(X), (λn)n∈N ⊆ K : λn ∈ σ0p(Tn) for all n, pAQ(Tn − T )→ 0, λn → λ

is trivial.

Now we show that the map T 7→ σq(T ) is closed. Let (λn)n∈N and (Tn)n∈N be sequences, such

that λn → λ in K, pAQ(T − Tn)→ 0, and λn ∈ σq(Tn) for all n. Since [λnI − Tn]q = 0 for all n,

we can find a sequence (yn)n∈N with ‖yn‖ > n and

‖λnyn − Tn(yn)‖‖yn‖

<1

n.

Since, without loss of generality, [T−Tn]Q < 1/n, we can choose yn so that they fulfil additionally

‖(T − Tn)(yn)‖‖yn‖

<1

n.

We conclude

[λI − T ]q ≤ limn→∞

‖λyn − T (yn)‖‖yn‖

≤ limn→∞

(‖(λ− λn)(yn)‖

‖yn‖+‖λnyn − Tn(yn)‖

‖yn‖+‖(T − Tn)(yn)‖

‖yn‖

)≤ lim

n→∞

(|λ− λn|+

1

n+

1

n

)= 0.

Hence, λ ∈ σq(T ). Since σ0p(T ) ⊆ σq(T ), the graph of σq thus contains the closure of the graph

of σ0p(T ). This yields the remaining inclusion. 2

Finally, we bring another theorem that shows that the approximate point spectrum is a good

substitute of the point spectrum.

Definition 3.40 An operator T ∈ C(X) is called odd if T (−x) = −T (x) for all x ∈ X.

An operator T ∈ C(X) is called asymptotically odd if there exists an odd operator T ∈ C(X)

with [T − T ]Q = 0.

Obviously, every T ∈ B(X) is odd.

Theorem 3.41 Let T ∈ C(X) be asymptotically odd and compact. Then every λ ∈ σFMV (T )

is an asymptotic eigenvalue of T .

Proof: First, fix λ ∈ σFMV (T ), λ 6= 0. From [λI − T ]a ≥ |λ| − [T ]A = |λ| > 0 we get that

λ ∈ σδ(T ) or λ ∈ σq(T ). We have to exclude the case that solely the first possibilty holds true,

i.e. we have to show that λI − T is stably solvable if λ /∈ σq(T ). Assume further that T itself

is odd.

Since [λI − T ]q > 0, by Lemma 3.5 we can restrict ourselves to compact operators S ∈ C(X)

41

where S(x) = 0 for ‖x‖ ≥ R with suitable R > 0 in order to prove stable solvability. We may

further assume that λx−T (x) 6= 0 for ‖x‖ > R, because otherwise we would have [λI−T ]q = 0.

Define H : BR(0)→ X by H(x) = (T (x)+S(x))/λ. Then H coincides with T/λ on ∂BR(0) and

is odd there. By Theorem 2.36, H has a fixed point x ∈ X. So x = H(x) = (T (x) + S(x))/λ,

i.e. λx− T (x) = S(x), and λI − T is stably solvable.

Now assume that T is asymptotically odd. Let G be an odd operator with [T −G]Q = 0. Define

T ∈ C(X) by T (x) = (T (x)− T (−x))/2. Then T is odd, compact, and

‖T (x)− T (x)‖‖x‖

=‖T (x)− T (x)−T (−x)

2 ‖‖x‖

=12‖T (x) + T (−x)‖

‖x‖

≤ 1

2

‖T (x)−G(x) +G(x) + T (−x)‖‖x‖

≤ 1

2

‖T (x)−G(x)‖‖x‖

+1

2

‖T (−x)−G(−x)‖‖ − x‖

.

Hence, [T−T ]Q = 0. Since T−T is compact, we have pAQ(T−T ), and so σFMV (T ) = σFMV (T )

and σq(T ) = σq(T ) by Lemma 3.25. By what we have proven so far we may conclude that

σFMV (T )\0 = σFMV (T )\0 ⊆ σq(T ) = σq(T ).

It remains to deal with the case λ = 0. In finite dimensional spaces, we always have σa = ∅, so

the above argument also works for λ = 0. In infinite dimensional spaces, we have 0 ∈ σFMV (T )

by Proposition 3.32. But 0 is also in σq(T ), because if this were not the case, then by Proposition

3.32,(iii) we get

c0[σq(T )] ⊆ σδ(T ) ⊆ σFMV (T ) ⊆ σq(T ) ∪ 0,

i.e. c0[σq(T )] ⊆ 0. But this is impossible since c0[σq(T )] is both open and nonempty. 2

Corollary 3.42 Let T ∈ B(X) be compact. Then every λ ∈ σ(T )\0 is an eigenvalue of T .

Proof: As T is linear, we can apply Theorem 3.41 and Lemma 3.37 to show that

σ(T )\0 = σFMV (T )\0 = σq(T )\0 = σp(T )\0.

2

Corollary 3.42 is an important result of linear spectral theory. However, when viewed from

the perspective of the aproximate point spectrum, we see that it is actually a result about a

larger class of operators, which are in general not linear. Only in the linear case this results

conveniently coincides with one about eigenvalues. Furthermore, even in the linear case, when

talking about eigenvalues we have to exclude the value 0, whereas if we talk about aproximate

eigenvalues it describes the complete spectrum of the operator.

42

3.5 Numerical range of operators

Definition 3.43 Let H be a Hilbert space with inner product 〈·, ·〉 and let T ∈ B(H). Denote

by S1(H) the unit sphere in H. Then the set

W (T ) := 〈Tx, x〉 : x ∈ S1(H) =

〈Tx, x〉‖x‖2

: x ∈ H,x 6= 0

is called the numerical range of T .

In linear theory, the numerical range has some very interesting properties. Unfortunately,

there is no definition of a numerical range for nonlinear operators that encompasses all these

properties. However, the property σ(T ) ⊆W (T ) can, in some sense, be transfered to nonlinear

operators. We already showed that for any operator the inclusion ∂σFMV (T ) ⊆ σπ(T ) holds

true. In the case of linear operators this reduces to ∂σ(T ) ⊆ σq(T ). Since W (T ) is convex, the

statements σ(T ) ⊆ W (T ) and σq(T ) ⊆ W (T ) are therefore equivalent. We want to define a

numerical range for nonlinear operators that retains the latter of these properties.

We need the following construction.

Definition 3.44 For T ∈ C(H) define the operator T⊥ : H\0 → H by

T⊥(x) := φT (x)x

where

φT (x) =〈T (x), x〉‖x‖2

.

T⊥ can be extended continuously to 0 if T (0) = 0. However, since we are only interested in

the behaviour of T⊥ outside of bounded sets, we will not restrict ourselves to such operators.

Instead we will treat T⊥ as a continuous operator on the whole space H. This can be realised

by defining T⊥ on B1(0) to be a continuous extension of T⊥ from S1(H) to B1(0). So if for two

operators T and S the operators T⊥ and S⊥ coincide on H\B1(0), we write T⊥ = S⊥ by using

the same extension to B1(0). We gather a few properties of the operator T⊥.

Proposition 3.45 Let T, S ∈ C(H) and λ ∈ K. Then the following is true

(i) T⊥⊥ = T⊥.

(ii) (T + S)⊥ = T⊥ + S⊥.

(iii) (λT )⊥ = λT⊥.

(iv) [T⊥]Q ≤ [T ]Q.

(v) [T⊥]q ≤ [T ]q.

43

Proof: The properties (i)-(iii) follow directly from the definition of T⊥. The remaining two

properties immediately follow from the estimate

‖T⊥(x)‖‖x‖

= |φT (x)| ≤ ‖T (x)‖‖x‖

.

2

Definition 3.46 For T ∈ C(H) we call

WFMV (T ) := σq(T⊥)

the numerical range of T .

We gather some properties of WFMV (T ).

Proposition 3.47 Let T, S ∈ C(H), z ∈ H, and µ ∈ K. Then WFMV (T ) has the following

properties.

(i) WFMV (T + S) ⊆WFMV (T ) +WFMV (S) if [T⊥]Q <∞.

(ii) WFMV (µT ) = µWFMV (T ).

(iii) WFMV (T + z) = WFMV (T ).

(iv) WFMV (µI − T ) = µ −WFMV (T ).

(v) WFMV (T ) = W (T ) if T ∈ B(H).

(vi) σq(T ) ⊆WFMV (T ).

Proof: If λ ∈WFMV (T +S) = σq(T⊥+S⊥), then there exists an unbouded sequence (xn)n∈N

in H such that φT (xn) + φS(xn) converges to λ. Since [T⊥]Q < ∞, the sequence φT (xn) is

bounded. So without loss of generality we may assume that φT (xn) converges to some µ ∈ K.

Consequently, φS(xn) converges to λ − µ. We can conclude µ ∈ σq(T⊥) = WFMV (T ) and

λ− µ ∈ σq(S⊥) = WFMV (S). Thus, (i) holds true.

The assertion (ii) follows from the obvious equality φµT = µφT . Similarly, (iii) follows from

the equality

φT+z(x) =〈T (x) + z, x〉‖x‖2

= φT (x) +〈z, x〉‖x‖2

,

where the last term tends to zero as ‖x‖ → ∞. Further, the equality φµI−T = µ − φT implies

(iv).

44

To prove (v), let T be linear. Then we get the chain of equalities

WFMV (T ) = σq(T⊥) =

λ ∈ K : lim inf

‖x‖→∞

‖λx− T⊥(x)‖‖x‖

= 0

=

λ ∈ K : inf

‖x‖=1‖λx− T⊥(x)‖ = 0

=

λ ∈ K : inf

‖x‖=1|λ− 〈Tx, x〉| = 0

= W (T ).

Finally, Proposition 3.45,(v) gives the implication λ ∈ σq(T )⇒ λ ∈ σq(T⊥). 2

The assertion (v) in the proposition above shows that the definition of the numerical range

for nonlinear operators yields nearly the same as the classical definition when applied to linear

operators. Note that even though we have proven that σq(T ) ⊆ WFMV (T ), this does not give

us any further information on σa(T ) or σδ(T ). However, having an additional tool to localize

σq(T ) can be usefull, as we shall see in Section 4.3.

Next, we want to give some topological information about the numerical range.

Theorem 3.48 Let H be a Hilbert space. WFMV (T ) allows the representation

WFMV (T ) =⋂n∈N

φT (H\Bn(0)). (3.1)

Suppose T ∈ C(H) satisfies [T⊥]Q <∞. Then the set WFMV (T ) is nonempty and compact. In

particular WFMV (T ) ⊆ B[T⊥]Q(0). If H 6= R then WFMV (T ) is also connected.

Proof: First, we show that (3.1) holds true. So let λ ∈ WFMV (T ). Then we may choose

a sequence (xn)n∈N in H with ‖xn‖ > n and φT (xn) → λ. If λ /∈ φT (H\Bm(0)) for some

m ∈ N, we can find a neighbourhood Uλ of λ such that Uλ ∩ φT (H\Bn(0)) = ∅ for n ≥ m.

However, φT (xn) ∈ φT (H\Bn(0)) since ‖xn‖ > n. Thus φT (xn) /∈ Uλ, contradicting the fact

that φT (xn)→ λ. Therefore, the inclusion ’⊆’ holds in (3.1).

Conversely, if λ belongs to the intersection in (3.1), we may find a sequence (xn)n∈N in H with

‖xn‖ > n and φT (xn) → λ. This implies that λ ∈ σq(T⊥) and the second inclusion in (3.1) is

proven.

The assumption [T⊥]Q <∞ implies that the set φT (H\Bn(0)) is bounded for sufficiently large

n. Moreover, H\Bn(0) is nonempty (and in the case of H 6= R also connected). Since φT is

continuous there, φT (H\Bn(0)) is also nonempty (and connected). Since the right hand side in

(3.1) is now the intersection of a decreasing sequence of nonempty, compact (and connected)

sets, WFMV (T ) is also nonempty, compact (and connected).

For the final assertion we again use the fact, that if λ ∈ WFMV (T ), there exists a sequence

(xn)n∈N in H such that ‖xn‖ > n and φT (xn)→ λ. Therefore,

[T⊥]Q = lim sup‖x‖→∞

|φT (x)| ≥ limn→∞

|φT (xn)| = |λ|.

2

45

Theorem 3.48 allows us to give a stricter estimate on the FMV spectral radius for operators

on Hilbert spaces.

Theorem 3.49 For T ∈ C(H) we have rFMV (T ) ≤ max[T ]A, [T⊥]Q.

Proof: Let |λ| > max[T ]A, [T⊥]Q. Then [λI − T ]a ≥ |λ| − [T ]A > 0 by Proposition

2.7,(iv). Moreover, λ /∈ σq(T ) since σq(T ) ⊆ WFMV (T ) ⊆ B[T⊥]Q(0) by Proposition 3.47,(vi)

and Theorem 3.48. So it remains to show that λI − T is stably solvable. Let S ∈ C(H)

with [S]A = [S]Q = 0. By Lemma 3.5 we can also assume that x ∈ H : S(x) 6= 0 is

bounded. The stable solvability is obviously equivalent to the existence of fixed point for the

map G(x) = λ−1(T (x) + S(x)). Define πn ∈ C(H) by

πn(x) = min1, n

‖x‖x.

Obviously, πn maps H into Bn(0). We now need an estimate for α(πn(M)) for a bounded set

M ⊆ X. If M ⊆ Bn(0), then πn(M) = M , hence, α(πn(M)) = α(M). Let M ⊆ X\Bn(0) and

let z1, · · · , zm be a finite ε-net for M with ε > α(M). Then for every x ∈ M there is a zi

with ‖zi − x‖ < ε. First, assume that zi /∈ Bn(0). Obviously, πn(tx) = πn(x) for every t > 1.

We conclude

‖πn(x)− πn(zi)‖ = ‖πn(x‖zi‖)− πn(zi‖x‖)‖ =

∥∥∥∥ n

‖x‖‖zi‖x− n

‖x‖‖zi‖zi

∥∥∥∥=

n

‖x‖‖zi‖‖x− zi‖ <

1

nε ≤ ε.

Also, if zi ∈ Bn(0), then

‖πn(x)− πn(zi)‖ = ‖πn(x)− zi‖ ≤ ‖x− zi‖ < ε.

Hence, πn(zi) : i = 1, · · · ,m is a finite ε-net for πn(M) and α(πn(M)) ≤ α(M). For arbitrary

M ⊆ X we therefore get

α (πn(M)) = α(πn

(M ∩Bn(0)

)∪ πn

(M\Bn(0)

))= max

α(πn

(M ∩Bn(0)

)), α(πn

(M\Bn(0)

))≤ α(M)

Using this, we are able to see that

[πn G]A = sup∞>α(M)>0

α(πn G(M))

α(M)≤ sup∞>α(M)>0

α(G(M))

α(M)= [G]A

= [λ−1(T (x) + (S(x))]A ≤ |λ|−1[T (x)]A < 1.

By the Darbo fixed point theorem, Corollary 3.8, πn G has a fixed point xn ∈ Bn(0). If

‖xn‖ = ‖πn G(xn)‖ < n, then we have xn = πn G(xn) = G(xn) and we are done.

46

Assume that ‖πn G(xn)‖ = ‖xn‖ ≥ n for all n. Then for each n, there exists a cn ≤ 1 such

that xn = cnG(xn), i.e. λxn = cn(T (xn) + S(xn)). Since x ∈ H : S(x) 6= 0 is bounded, we

have [T⊥]Q = [T⊥+S⊥]Q, because this seminorm only takes the behaviour of operators outside

bounded sets into account. We end up with the contradiction

|λ| > [T⊥]Q = [T⊥ + S⊥]Q = lim sup‖x‖→∞

‖(T⊥ + S⊥)(x)‖‖x‖

≥ lim supn→∞

|〈T (xn) + S(xn), xn〉|‖xn‖2

≥ lim supn→∞

|〈λxn, xn〉|‖xn‖2

= |λ|.

2

47

Chapter 4

Applications

4.1 Maps on spheres

As a first example, we want to show how the FMV-spectrum can be used to analyze maps on

spheres.

Definition 4.1 Let X be a Banach space. Then for r > 0

Sr(X) := x ∈ X : ‖x‖ = r

is the sphere with radius r in X. For a map T : Sr(X)→ X we define the map T : X → X by

T (x) :=

‖x‖T(rx

‖x‖

)if ‖x‖ 6= 0

0 if ‖x‖ = 0.

Finally, a map T : X → X is called positively homogenous if T (λx) = λT (x) for all λ ≥ 0.

It is obvious that T is continous if T is, and that T is positively homogenous. Clearly, any

positively homogenous map fulfils T (0) = 0. In some situations, this is the only root of T .

Lemma 4.2 Let T ∈ C(X) be positively homogenous and assume that [T ]a > 0. Then the

equation T (x) = 0 has only the trivial solution x = 0 if and only if [T ]q > 0.

Proof: First, let [T ]q > 0. Assume 0 6= x ∈ X with T (x) = 0 and take any sequence (xn)n∈N

in X with xn → x. Then we get the contradiction

[T ]q = lim inf‖y‖→∞

‖T (y)‖‖y‖

≤ limn→∞

‖T (nxn)‖‖nxn‖

= limn→∞

n‖T (xn)‖n‖xn‖

=‖T (x)‖‖x‖

= 0.

Therefore, T (x) = 0 if and only if x = 0.

Now let [T ]q = 0. Then there exists a sequence (xn)n∈N in X such that limn→∞ ‖T (xn)‖/‖xn‖ =

0. For yn = xn/‖xn‖ we have ‖yn‖ = 1 and

‖T (yn)‖ =

∥∥∥∥T ( xn‖xn‖

)∥∥∥∥ =

∥∥∥∥ 1

‖xn‖T (xn)

∥∥∥∥ =‖T (xn)‖‖xn‖

→ 0.

48

For A = yn : n ∈ N we have 0 = α(T (A)) ≥ [T ]aα(A). Hence, A is precompact and we

can assume without loss of generality that yn converges to some y ∈ X with ‖y‖ = 1. The

continuity of T implies that T (y) = 0. 2

The next lemma follows easily from Lemma 4.2.

Lemma 4.3 Let T : Sr(X)→ X be continuous. Then

[T ]q = inf‖T (x)‖ : x ∈ Sr(X),

and

[T ]Q = sup‖T (x)‖ : x ∈ Sr(X).

Furthermore, if [λI − T ]a > 0, then λ ∈ σq(T )⇔ λr ∈ σp(T ).

Proof: The representation of [T ]q and [T ]Q follow directly from the definition of T . The last

assertion follows from Lemma 4.2, which asserts that λ ∈ σq(T )⇔ λ ∈ σp(T )⇔ ∃x ∈ X\0 :

T (x) = λx⇔ ∃x ∈ X\0 : T (r x‖x‖) = λ

r rx‖x‖ . 2

First, we deal with spheres in a finite dimensional spaces. We give a fixed point theorem

and a condition for the surjectivity of an operator.

Theorem 4.4 Let T : Sr(Kn) → Sr(Kn) be continuous and assume that T is not surjective.

Then T has a fixed point and an antipodal point (i.e. a point x ∈ Kn with T (x) = −x).

Proof: Since pAQ(T ) = [T ]Q = r, we have that λ ∈ ρFMV (T ) for all λ > r by Corollary

3.22. Moreover, if λ ∈ σq(T ), then there is an x ∈ Sr(Rn) with λr x = T (x) by Lemma 4.3.

In particular, r = ‖T (x)‖ = ‖λr x‖ = |λ|r ‖x‖ = |λ|. Therefore, σq(T ) ⊆ Sr(K). But, because

T is not surjective, 0 ∈ σFMV (T ). Since the FMV-spectrum is closed and σa(T ) = ∅, we get

∂σFMV (T ) ⊆ σπ(T ) = σq(T ) ⊆ Sr(K) by Theorem 3.30. This yields σFMV (T ) = Br(0) and

hence, ∂σFMV (T ) = σq(T ) = Sr(K). In particular −r and r both lie in σq(T ) and, hence, −1

and 1 lie in σp(T ) by Lemma 4.3. 2

Theorem 4.5 If T : Sr(Kn)→ SR(Kn) is continuous and odd, then T is surjective.

Proof: Using the same technique as in the proof of the previous theorem, we achieve

σq(T ) ⊆ SRr

(K). Since T is odd, we have σFMV (T ) = σq(T ) by Theorem 3.41. Hence,

0 ∈ ρFMV (T ). In particular, T is surjective. Now let y ∈ SR(Kn). Then there is an x ∈ X with

T (x) = y. But this means T (r x‖x‖) = y

‖x‖ , i.e. R = ‖T (r x‖x‖)‖ = ‖y‖

‖x‖ = R‖x‖ . Hence, ‖x‖ = 1 and

we get T (rx) = y. So T is surjective. 2

49

Next, we want to consider infinite dimensional spaces. There, the above fixed point theorem

can be extended to the following result.

Theorem 4.6 (Birkoff-Kellog Theorem) Let X be an infinite dimensional Banach space

and let T : Sr(X)→ X be compact such that

infx∈Sr(x)

‖T (x)‖ > 0.

Then T has a positive and a negative eigenvalue.

Proof: Let (xn)n∈N be a bounded sequence in X. Because T is compact, there is a subse-

quence (xnk)k∈N such that T (r

xnk‖xnk

‖) converges. Since the sequence is bounded, we can assume

without loss of generality, that ‖xnk‖ also converges. Therefore T (xn) has a convergent sub-

sequence and T is compact. Since T (S1(X)) is bounded, we have [T ]Q < ∞ by Lemma 4.3,

and, therefore, T ∈ AQ(X). Hence, σFMV (T ) is bounded. But by assumption and by Lemma

4.3 [T ]q > 0, i.e. 0 /∈ σq(T ). So by Proposition 3.32,(iv), σq(T ) contains both a positive and a

negative value. But since [λI − T ]a = |λ| > 0 for all λ 6= 0, by Lemma 4.3 these values when

divided by r are in fact eigenvalues of T . 2

Corollary 4.7 Let X be an infinite dimensional Banach space and let T : Sr(X) → Sr(X) be

compact. Then T has a fixed point and an antipodal point.

Proof: Let λ± be the positive or negative eigenvalue of T which exists by Theorem 4.6, and

x± an eigenvector to λ±. Then r = ‖T (x±)‖ = ‖λ±x±‖ = |λ±| ‖x±‖ = |λ|r, i.e. |λ| = 1. Hence,

λ± = +1 or −1 respectively. 2

The next theorem shows that in contrast to finite dimensional spaces, for an operator between

spheres in an infinite dimensional space compactness and oddity are mutually exclusive.

Theorem 4.8 Let X be an infinite dimensional Banach space. Then T : Sr(X)→ SR(X) can

not be both compact and odd.

Proof: Assume, that T is both compact and odd. Then T is also compact and odd as seen in

the beginning of the proof Theorem 4.6. Since T is bounded away from 0, we get 0 /∈ σq(T ). But

because T is compact we have 0 ∈ σFMV (T ) by Prosposition 3.32. Thus, because of Theorem

3.41, we get the contradiction 0 ∈ σFMV (T ) = σq(T ). 2

The following theorem is one of the most useful results in topology. Here, our proof of the

theorem is of special interest. Because even though we state this theorem in its original form

for a sphere, the proof can be used verbatim for any point symmetric set, i.e. a set M where

x ∈ M implies −x ∈ M , for which c0[M ] or c0[K\M ] (depending on whether 0 ∈ M or

not) is bounded, and that allows a positively homogenous extension of the considered function

T : M ⊆ Kn+1 → Kn.

50

Theorem 4.9 (Borsuk-Ulam Theorem) Let T : Sr(Kn+1)→ Kn be continuous. Then there

exists an x ∈ Sr(Kn+1) with T (x) = T (−x).

Proof: Identify Kn with a subspace of Kn+1. Define S : Sr(Kn+1) → Kn+1 by S(x) =

T (x) − T (−x). Since we are in a finite dimensional space, we have σa(S) = ∅, in particular

0 /∈ σa(S). On the other hand, S is not surjective because its values lie in a proper subspace

of Kn+1. So 0 ∈ σδ(S) by Lemma 3.2, and by Theorem 3.41 0 ∈ σq(S). Hence, 0 ∈ σp(S) by

Lemma 4.3, i.e. there is an x ∈ Sr(Kn+1) with T (x)− T (−x) = 0. 2

4.2 Differential equations

We want to show through an example how some basic properties of the FMV-spectrum can be

used to deal with nonlinearities in differential equations. We consider the three point boundary

value problem

x(t) = µf(t, x(t), x(t)) + y(t),

x(0) = 0, x(1) = αx(η).

(4.1)

Here, f : [0, 1] × R × R → R is a continuous function, y ∈ L1[0, 1], η ∈ (0, 1), and α ≥ 0. This

problem is called a three point boundary problem, because the values at three different points

are used in the boundary conditions.

We are looking for solutions x of (4.1) in the Sobolev space W 2,1[0, 1], i.e. the set of all absolutely

contiuous functions on [0, 1], such that x is also absolutely continuous and x ∈ L1[0, 1].

In order to find this solution, consider the Banach space X = C1[0, 1] of all continuously

differentiable function on [0, 1] equipped with the norm ‖x‖X = max‖x‖∞, ‖x‖∞, where

‖ · ‖∞ denotes the supremum norm on C[0, 1]. Further, set Y = L1[0, 1] and let ‖ · ‖1 denote its

usual norm. Finally, define a linear operator L : D(L) ⊆ X → Y , where

D(L) = x ∈W 2,1[0, 1] : x(0) = 0, x(1) = αx(η),

by Lx = x and a real valued function c(α, η) by

c(α, η) = 1 +αη + 1

|1− αη|=

2

1−αη if αη < 1,2αηαη−1 if αη > 1.

Lemma 4.10 If αη 6= 1, the operator L : D(L) → Y is bijective. In this case its inverse

satisfies

‖L−1‖ ≤ c(α, η),

and is compact as an operator from Y into X.

51

Proof: Let x, y ∈ D(L) so that Lx = Ly. Then, L(x− y) = 0, i.e. x(t)− y(t) = ct+ d with

c, d ∈ R. Since x − y ∈ D(L), we get (x − y)(0) = 0, which yields d = 0 and (x − y)(1) =

α(x− y)(η), which yields c = αηc, and therefore c = 0 since αη 6= 1. Hence, L is injective.

Next, define a linear operator K : Y → D(L) by

Ky(t) =

∫ t

0(t− s)y(s)ds+

αt

1− αη

∫ η

0(η − s)y(s)ds− t

1− αη

∫ 1

0(1− s)y(s)ds.

Easy calculations show that LKy = y, Ky(0) = 0, and Ky(1) = αKy(η). Therefore, L is

bijective and K is its inverse. We also easily see that for x = Ky we get

‖x‖∞ ≤ 1‖y‖1 +αη

|1− αη|‖y‖1 +

1

|1− αη|‖y‖1 =

(1 +

αη + 1

|1− αη|

)‖y‖1,

and similarly

‖x‖∞ ≤(

1 +αη + 1

|1− αη|

)‖y‖1.

Hence,

‖L−1‖ = ‖K‖ ≤ c(α, η).

Finally, let (yn)n∈N be a bounded sequence in Y , i.e. ‖yn‖1 < M for an M > 0. Since K is a

bounded operator, (Kyn)n∈N is bounded in X. In particular, (Kyn)n∈N is uniformly bounded.

The above representation of K shows that

‖Kyn(t1) − Kyn(t2)‖∞ ≤∥∥∥∥∫ t1

0|(t1 − t2)yn(s)|ds

∥∥∥∥∞

+

∥∥∥∥∫ t2

t1

|(t2 − s)yn(s)|ds∥∥∥∥∞

+ |t1 − t2|∥∥∥∥ α

1− αη

∫ η

0|(η − s)yn(s)|ds

∥∥∥∥∞

+ |t1 − t2|∥∥∥∥ 1

1− αη

∫ 1

0|(1− s)yn(s)|ds

∥∥∥∥∞

≤ |t1 − t2|∥∥∥∥∫ t1

0|yn(s)|ds

∥∥∥∥∞

+ |t2 − t1|∥∥∥∥∫ t2

t1

|yn(s)|ds∥∥∥∥∞

+ |t1 − t2|∣∣∣∣ αη

1− αη

∣∣∣∣ ∥∥∥∥∫ η

0|yn(s)|ds

∥∥∥∥∞

+ |t1 − t2|∣∣∣∣ 1

1− αη

∣∣∣∣ ∥∥∥∥∫ 1

0|yn(s)|ds

∥∥∥∥∞

≤ |t1 − t2|(M +M +

∣∣∣∣ αη

1− αη

∣∣∣∣M +

∣∣∣∣ 1

1− αηM

∣∣∣∣)= |t1 − t2|M

(2 +

∣∣∣∣ αη

1− αη

∣∣∣∣+

∣∣∣∣ 1

1− αη

∣∣∣∣) .Thus, ‖Kyn(t1) − Kyn(t2)‖∞ < ε for |t1 − t2| < δ with a small enough δ > 0, i.e. (Kyn)n∈N

is equicontinuous. By the Arzela-Ascoli Theorem, (Kyn)n∈N has an in C[0, 1] convergent sub-

sequence. Using the same argument for the sequence of their derivatives, we conclude that

(Kyn)n∈N has an in X convergent subsequence. This shows that K is compact. 2

52

In particular, Lemma 4.10 shows that the linear version of our equation (4.1), where µ = 0, has

a unique solution for every y ∈ L1[0, 1], if only αη 6= 1. It should be noted that the biggest

difficulty in the proof of Lemma 4.10 lies in actually finding the explicit representation of L−1,

which we simply presented as given.

However, we want to focus on the question how one can deduce the existence of solutions to

the nonlinear problem (4.1) from this. Using only some basic properties of the FMV-spectrum,

this can be achieved by a simple growth condition on f .

Theorem 4.11 Let αη 6= 1. Suppose |f(t, u, v)| ≤ p(t)|u|+ q(t)|v|+ r(t) for suitable functions

p, q, r ∈ L1[0, 1]. Then the boundary value problem (4.1) has a solution for every function

y ∈ L1[0, 1], provided that

|µ|(‖p‖1 + ‖q‖1) <1

c(α, η).

Proof: Define an operator F : X → Y by

F (x)(t) = f(t, x(t), x(t)).

Clearly, F maps bounded sets into bounded sets. Hence, L−1F : X → X is compact, i.e.

[L−1F ]A = 0. For the seminorm pAQ(L−1F ) we get the estimate

pAQ(L−1F ) = [L−1F ]Q ≤ ‖L−1‖ [F ]Q

≤ c(α, η) lim sup‖x‖X→∞

‖p‖1‖x‖∞ + ‖q‖1‖x‖∞ + ‖r‖1‖x‖X

≤ c(α, η) lim sup‖x‖X→∞

(‖p‖1 + ‖q‖1)‖x‖X + ‖r‖1‖x‖X

≤ c(α, η) (‖p‖1 + ‖q‖1)

<1

|µ|.

By Corollary 3.22, 1/µ /∈ σFMV (L−1F ). Hence, 1/µ−L−1F is FMV-regular and, in particular,

surjective. Thus, there exists an x ∈ X such that

1

µx− L−1F (x) =

1

µL−1y,

which in particular implies x ∈ D(L).

Applying the operator µL we therefore get Lx− µF (x) = y, i.e.

x(t) = µf(t, x(t), x(t)) + y(t).

2

We now turn to ordinary differential equations in finite dimensional spaces. The following

53

theorem extends a well known result of the linear theory, as we will explain afterwards.

Theorem 4.12 Let T ∈ C(Kn). Assume that there exists a linear bijection A ∈ B(Kn) such

that [(A T A−1)⊥]Q < ∞ and WFMV (A T A−1) ⊆ K− = λ ∈ K : Re(λ) < 0. Then all

solutions of the autonomous differential equation

z(t) = T (z(t))

are bounded as t→∞

Proof: Put w = Az. Then the differential equation z = T (z) becomes w = A T A−1(w) =

S(w). So it is enough to show that all solutions of the differential equation w = S(w) are

bounded for t → ∞. Since [S⊥]Q < ∞, WFMV (S) is compact by Theorem 3.48. Because

of this and the fact that WFMV (S) ⊆ K−, there exists a δ > 0 so that Re(λ) < −δ for all

λ ∈WFMV (S). Now choose an ε > 0 with ε < −Re(λ)− δ for all λ ∈WFMV (S).

Using the representation (3.1) for the numerical range of S, we see that

WFMV (S) =

∞⋂n=1

φS(H\Bn(0)) = limN→∞

N⋂n=1

φS(H\Bn(0)).

This implies supλ∈WFMV (S)

(dist

(λ,⋂Nn=1 φS(H\Bn(0))

))< ε for N sufficiently large. In par-

ticular, supλ∈WFMV (S) |Re(φS(w)) − Re(λ)| < ε for ‖w‖ > N , which in turn implies that

Re(〈S(w), w〉) < −δ‖w‖2 whenever ‖w‖ > N .

Finally, let w(·) be a solution of w(t) = S(w(t)). Because of the above considerations we get

d

dt‖w(t)‖2 = 〈w(t), w(t)〉+ 〈w(t), w(t)〉 = 2 Re〈S(w(t)), w(t)〉 < −2δ‖w(t)‖2 < 0

if ‖w(t)‖ > N . This implies that w(t) is bounded. 2

Note that the condition [(A T A−1)⊥]Q < ∞ is in particular met if [T ]Q < 0. Because

in this case we can set y = A−1x and get

[A T A−1]Q‖A−1‖

=1

‖A−1‖lim sup‖x‖→∞

‖(A T A−1)(x)‖‖x‖

= lim sup‖y‖→∞

‖(A T )(y)‖‖A−1‖‖Ay‖

≤ lim sup‖y‖→∞

‖A‖ ‖T (y)‖‖y‖

= ‖A‖[T ]Q <∞,

since ‖x‖ → ∞ if and only if ‖A−1x‖ → ∞. Lemma 3.45,(iv) then gives [(A T A−1)⊥]Q ≤[A T A−1]Q ≤ ‖A−1‖‖A‖[T ]Q <∞.

If T is linear, it is a well known fact that the condition σ(T ) ⊆ K− implies that every solution

to the differential equation z = Tz is bounded. But σ(T ) ⊆ K− does not necessarily imply

that W (T ) ⊆ K−. However, it can be shown that there exists a linear ismorphism A, such that

WFMV (A T A−1) = W (A T A−1) ⊆ K−. Since also [T ]Q <∞, this shows that Theorem

4.12 is indeed an extension of this assertion about linear differential equations.

54

4.3 The nonlinear Fredholm alternative

The Fredholm alternative is an important and very useful theorem in the theory of linear

operators on a Banach space X. Its proof is in essence contained in Corollary 3.42.

Theorem 4.13 (Fredholm alternative) Let T ∈ B(X) be compact and 0 6= λ ∈ K. Then

either

• λ ∈ σ(T ), i.e. λ is an eigenvalue of T ,

or

• λI − T is bijective.

Proof: If λ is not an eigenvalue of T , then λ ∈ ρ(T ) by Corollary 3.42. Hence λI − T is

bijective. 2

The interesting implication here is, of course, that if λ is not an eigenvalue, then the equa-

tion λx− Tx = y has a unique solution for each y ∈ X.

In our theory the role of bijectivity has been taken over by FMV-regularity. The property of

FMV-regular operators, which can be used to guarantee solutions to certain equations, is stable

solvability. This motivates the following definition.

Definition 4.14 An operator T ∈ C(X) is called alternative if

σFMV (T ) = σπ(T ).

Again, the interesting implication here is that if λ /∈ σπ(T ), then λI − T is FMV-regular and,

in particular, stably solvable. In order to assert that λ /∈ σπ(T ), we have to check that both

λ /∈ σq(T ) and λ /∈ σa(T ).

The first of these conditions is the one that is easier to handle. By definition, λ ∈ σq(T ) if and

only if λ is an asymptotic eigenvalue. For λ ∈ σq(T ), we have to check whether there exists a

sequence (xn)n∈N in X with ‖xn‖ → ∞ such that

limn→∞

‖λxn − T (xn)‖‖xn‖

= 0.

Furthermore, according to Proposition 3.47,(vi), σq(T ) ⊆WFMV (T ). Thus, the implication

λ /∈WFMV (T )⇒ λ /∈ σq(T )

holds true and gives us a further tool to see whether λ ∈ σq(T ) or not.

The set σa(T ) is more complicated to handle. It is difficult to calculate [T ]a, as even the

calculation of the quotient α(T (M))/α(M) for fixed M can be very challenging. We give some

conditions which indicate that λ /∈ σa(T ).

Lemma 4.15 Let T ∈ C(X) fulfil [T ]a > 0. Then the following holds true

55

(i) There is a k > 0 such that α(T (M)) ≥ kα(M) for all bounded sets M ⊆ X.

(ii) T is proper on closed bounded sets.

(iii) If (xn)n∈N is bounded and (T (xn))n∈N converges, then (xn)n∈N has a convergent subse-

quence.

In particular, if λI − T does not meet one of these three conditions, then λ /∈ σa(T ).

Proof: (i) follows directly from the definition of [T ]a. (ii) has been proven in Proposition 2.7.

We show that (ii) ⇒ (iii). So let (xn)n∈N be bounded by R > 0 and assume that (T (xn))n∈N

converges to some y ∈ X. Then the set T (xn) : n ∈ N ∪ y is compact. Since T is proper

on BR(0) by assumption, the preimage of T (xn) : n ∈ N ∪ y is also compact. In particular,

(xn)n∈N lies in a compact set and, therefore, has a convergent subsequence. 2

Furthermore, we want to identify some classes of alternative operators for which there is no

need to be concerned with the subspectrum σa(T ) at all.

Proposition 4.16 Let T ∈ C(X) be alternative. Assume that one of the conditions

(i) X is finite dimensional,

(ii) T is compact,

(iii) T is asymptotically linear with asymptotic derivate T ′ such that T − T ′ is compact,

holds true. Then σFMV (T ) = σq(T ). In particular, for any λ ∈ K either

• λ ∈ σq(T ), i.e. λ is an asymptotic eigenvalue of T ,

or

• λI − T is FMV-regular.

Proof: Since T is alternative we have σFMV (T ) = σπ(T ) = σq(T ) ∪ σa(T ). Thus, we have to

show that σa(T ) ⊆ σq(T ).

If X is finite dimensional then σa(T ) = ∅ and we are done. If T is compact, then σa(T ) = 0by Proposition 3.32,(i), so we have to show that 0 ∈ σq(T ). But if this were not the case, then

Proposition 3.32,(iii) would give

c0[σq(T )] ⊆ σδ(T ) ⊆ σFMV (T ) ⊆ σq(T ) ∪ 0,

i.e. c0[σq(T )] ⊆ 0. But this is impossible since c0[σq(T )] is both open and nonempty.

Finally, if T fulfills (iii), then Proposition 3.34,(iii) and Lemma 3.27 show that

σa(T ) = σa(T′) ⊆ σq(T ′) = σq(T ).

2

56

Proposition 4.17 Let T ∈ B(X) be alternative. Then σ(T ) = σq(T ). In particular, either

• λ ∈ σq(T ), i.e. λ is an asymptotic eigenvalue of T ,

or

• λI − T is bijective.

Proof: By Lemma 3.27 we have σa(T ) ⊆ σq(T ). Since T is linear and alternative, this yields

σ(T ) = σFMV (T ) = σq(T ). 2

Now that we have explored the usefulness of the notion of alternative operators, we want to

identify some classes of operators as alternative. One such class has essentially already been

identified.

Theorem 4.18 Let T ∈ C(X) be compact and asymptotically odd. Then T is alternative.

Proof: Theorem 3.41 shows that σFMV (T ) = σq(T ). 2

We will identify some other classes of alternative operators. However, we will be able to do

so only in Hilbert spaces. So Theorem 4.18 is of special interest since it is also applicable in

Banach spaces.

For the rest of this section we will deal with operators on a complex Hilbert space H. In

Hilbert spaces, the previous theorem can be seen as a special case of a larger class of alternative

operators. In order to identify this class of operators, we will use some concepts of Fredholm

theory.

Definition 4.19 Let T ∈ B(H). We define the kernel of T by

ker(T ) := x ∈ H : Tx = 0

and the range of T by

ran(T ) := Tx : x ∈ H.

The operator T is called semi-Fredholm, if ran(T ) is closed and either dim ker(T ) < ∞ or

dim(ran(T ))⊥ <∞.

For a semi-Fredholm operator the number

ind(T ) := dim ker(T )− dim(ran(T ))⊥ ∈ Z ∪ ±∞

is called the Fredholm index of T . If |ind(T )| <∞, T is called Fredholm.

We denote by F(H) the set of all semi-Fredholm operators on H.

Out of the theory of Fredholm operators, we want to import the following two results. A proof

of these theorems can for example be found in [C].

57

Theorem 4.20 Let T ∈ B(H) be Fredholm. Then the following is equivalent.

(i) ind(T ) = 0.

(ii) There exists a compact operator K ∈ B(H) such that T +K is invertible.

Theorem 4.21 Let F(H) be equipped with the subspace topology of B(H). Then the map

ind : F(H) → Z ∪ ±∞ that assigns to each semi-Fredholm operator its Fredholm index is

continuous.

We are now able to identify another class of alternative operators.

Definition 4.22 For T ∈ B(H) we define the essential spectrum by

σes(T ) := λ ∈ K : λI − T is not semi-Fredholm or ind(λI − T ) 6= 0.

T is called balanced if σa(T ) = σes(T ).

Theorem 4.23 Let T ∈ B(H) be balanced and S ∈ C(H) be compact and asymptotically odd.

Then the operator T + S is alternative.

Proof: Choose λ /∈ σπ(T + S). We have to show that λI − (T + S) is FMV-regular.

Since S is compact we get [λI − T ]a = [λI − (T + S)]a > 0 because λ /∈ σa(T + S). Hence,

λ /∈ σes(T ), since T is balanced. This means that λI − T is Fredholm of index 0. By Theorem

4.20, there exists a compact operator K ∈ B(H) such that λI−(T +K) =: L is an isomorphism.

By Theorem 3.15 it is enough to show that the operator L−1(λI−(T +S)) = 1−L−1(S+K)

is FMV-regular.

Since S and K are both compact, L−1 (S +K) is compact and, therefore,

[1− L−1 (S +K)]a = 1.

Furthermore, we can use Proposition 2.2,(v) to show

[1− L−1 (S +K)]q = [L−1 (λI − (T + S))]q ≥ [L−1]q[λI − (T + S)]q > 0.

The last inequality holds true since λ /∈ σq(T + S) and [L−1]q > 0 by Lemma 2.8,(iv). Now,

L−1 (S + K) is not only compact, but also asymptotically odd. Thus, it is alternative by

Theorem 4.18. Hence, 1− L−1 (S +K) is FMV-regular. 2

We are now interested in identifying classes of balanced operators

Every linear operator T on a finite dimensional Hilbert space is balanced, since it can be easily

seen that σa(T ) = σes(T ) = ∅. In infinite dimensional Hilbert spaces the operators λI for λ ∈ Kare simple examples of balanced operators, since σa(λI) = σes(λI) = λ.To identify more interesting examples of balanced operators the following result will be very

useful.

58

Lemma 4.24 Let T ∈ B(H). Then [T ]a > 0 if and only if ran(T ) is closed and fulfils

dim ker(T ) <∞.

Proof: Suppose [T ]a > 0 and let (xn)n∈N be an arbitrary sequence in B1(0) ∩ ker(T ). Then

T (xn) = 0 for all n. In particular (T (xn))n∈N converges and by Lemma 4.15,(iii), the sequence

(xn)n∈N has a convergent subsequence. Consequently, the closed unit ball in the subspace ker(T )

is compact. Thus, ker(T ) is finite dimensional.

This in turn also implies that there exists a closed subspace H1 of H such that H = H1⊕ker(T ).

As T (H) = T (H1) we only need to show that T (H1) is closed. Note that the restriction T |H1 :

H1 → T (H1) is bijective. It is obviously surjective, but also injective, since T |H1(x) = T |H1(y)

for x 6= y would imply T |H1(x− y) = 0, which leads to the contradiction x− y ∈ ker(T ). Thus,

T |H1 has an inverse.

If we can show that this inverse is bounded, i.e. T |H1 fulfils ‖T |H1x‖ ≥ m‖x‖ for all x ∈ H1

with ‖x‖ = 1 and some m > 0, then T (H1) = T (H) is closed as the preimage of a closed set

under a continuous map. Assume the contrary. Then there exists a sequence (xn)n∈N in H1

with ‖xn‖ = 1 such that T |H1xn → 0. Using Lemma 4.15,(iii), we can assume without loss

of generality that (xn)n∈N converges to some x ∈ H1 with ‖x‖ = 1. Obviously Tx = 0, which

contradicts H1 ∩ ker(T ) = 0.Conversely, we have to show that if ran(T ) is closed and dim ker(T ) <∞, then [T ]a > 0. Since

dim ker(T ) < ∞, there exists a closed subspace H1 of H such that H = H1 ⊕ ker(T ). Let

P1 : H → H1 be the projection along ker(T ) onto H1 and set P2 = I − P1. Let T |H1 be as

above.

Since ran(T ) is closed, T |H1 is an isomorphism of H1 onto ran(T ) = ran(T |H1) by the open map-

ping theorem. Since T = T |H1P1 we have [T ]a ≥ [T |H1 ]a[P1]a by Proposition 2.7,(vii). Since P2

maps H into the finite dimensional space ker(T ), it is compact. Thus, [P1]a = [I−P2]a = [I]a =

1. Furthermore, [T |H1 ]a > 0 by Lemma 2.8,(iv). Hence, [T ]a ≥ [T |H1 ]a[P1]a = [T |H1 ]a > 0. 2

Corollary 4.25 For T ∈ B(H) we have σa(T ) ⊆ σes(T ) ⊆ σ(T ).

Proof: If λ /∈ σ(T ) then λI −T is bijective. So ran(λI −T ) = H is closed, dim ker(λI −T ) =

0 < ∞, and ind(λI − T ) = dim ker(λI − T )− dim(ran(λI − T ))⊥ = 0, i.e. λ /∈ σes(T ). Hence,

σes(T ) ⊆ σ(T ).

If λ /∈ σes(T ) then in particular ran(λI − T ) is closed and dim ker(λI − T ) < ∞. By Lemma

4.24 this implies λ /∈ σa(T ). Hence, σa(T ) ⊆ σes(T ). 2

In order to exploit the properties of Hilbert spaces to a fuller extent, we introduce the no-

tion of adjoint operators.

Definition 4.26 Let T ∈ B(H). By T ∗ we denote the bounded linear operator that is uniquely

defined by the equation

〈Tx, y〉 = 〈x, T ∗y〉 for all x, y ∈ H.

59

T ∗ is called the adjoint operator of T .

T is called normal, if TT ∗ = T ∗T , and selfadjoint, if T = T ∗.

We will use the following well known results about adjoint operators.

Theorem 4.27 Let T ∈ B(H).

(i) T is compact if and only if T ∗ is compact.

(ii) If T is normal, then ker(T ∗) = (ran(T ))⊥.

(iii) If T is normal, then H = ker(T )⊕ ran(T ).

(iv) T is selfadjoint if an only if 〈Tx, x〉 ∈ R for all x ∈ H.

Proof: See [R2] and [W]. 2

Using Theorem 4.27, we are able to identify some classes of balanced operators.

Proposition 4.28 Let T ∈ B(H) fulfil one of the following conditions.

(i) r(T ) = 0.

(ii) There is an n ∈ N such that Tn is compact.

(iii) T is normal.

Then T is balanced.

Proof: Since every linear operator on a finite dimensional Hilbert space is balanced, we may

assume that H is infinite dimensional.

(i): Using Proposition 2.7,(vii) and Lemma 2.8,(ii) we get

[T ]a = n√

[T ]na ≤n√

[Tn]a ≤ n√

[Tn]A ≤ n√‖Tn‖ → r(T ) = 0.

Thus, 0 ∈ σa(T ). Corollary 4.25 shows that 0 ⊆ σa(T ) ⊆ σes(T ) ⊆ σ(T ) = 0. Hence,

σa(T ) = σes(T ).

(ii): Let n be the smallest integer such that Tn is compact. As above, we get [T ]na ≤ [Tn]a ≤[Tn]A = 0 and thus 0 ∈ σa(T ). Since σa(T ) ⊆ σes(T ) by Corollary 4.25, we are finished once

we show that λ 6= 0 implies λ /∈ σes(T ).

If n > 1, we can write λnI − Tn = A (λI − T ), where

A = λn−1I + λn−2T + · · ·+ Tn−1.

So if λ 6= 0, we get 0 < |λ|n = [λnI − Tn]a ≤ [A]A[λ− T ]a by Proposition 2.7,(vii). This yields

[λI − T ]a > 0, which is also true if n = 1. Since T ∗n is also compact, we get [λI − T ∗]a > 0.

60

By Lemma 4.24 and Theorem 4.27,(ii), ran(λI − T ) is closed and both dim ker(λI − T ) and

dim(ran(λI − T ))⊥ = dim ker(λI − T ∗) are finite. Thus, λI − T is Fredholm.

Since (cT )n is compact for any c ∈ R, we get that λI− cT is Fredholm for any c ∈ R and λ 6= 0.

Theorem 4.21 tells us that the Fredholm index is continuous. Hence, ind(λI−T ) = ind(λI) = 0

and λ /∈ σes(T ).

(iii): Let λ /∈ σa(T ). By Lemma 4.24, dim ker(λI − T ) < ∞ and ran(λI − T ) is closed. As

λI − T is normal by assumption, Theorem 4.27,(iii) gives H = ker(λI − T )⊕ ran(λI − T ). For

the Fredholm index we get

ind(λI − T ) = dim ker(λI − T )− dim(ran(λI − T ))⊥ = dim ker(λI − T )− dim ker(λI − T ) = 0.

So we have σes(T ) ⊆ σa(T ) and by Corollary 4.25, σes(T ) = σa(T ). 2

Next, we want to show a third class of alternative operators. Since every linear selfadjoint

operator is obviously normal, the previous theorem shows that they are alternative. We can, in

a certain manner, extend the notion of selfadjoint operators to nonlinear operators.

Definition 4.29 An operator T ∈ C(H) is called selfadjoint, if 〈T (x), x〉 ∈ R for all x ∈ H.

By SA(H) we denote the set of all selfadjoint operators on H.

By Theorem 4.27,(iv), this coincides with the usual definition of selfadjoint linear operators.

Note that the equation 〈T (x), y〉 = 〈x, T (y)〉 can not be used for nonlinear operators, as its

validity for all x, y ∈ H already implies the linearity of T .

We will now give a condition under which selfadjoint operators are alternative.

Lemma 4.30 If T ∈ SA(H), then WFMV (T ) ⊆ R

Proof: For λ ∈WFMV (T ) = σq(T⊥) we have

0 = lim inf‖x‖→∞

∥∥∥λx− 〈T (x),x〉‖x‖2 x∥∥∥

‖x‖= lim inf‖x‖→∞

|λ− 〈T (x), x〉‖x‖2

|.

Since 〈T (x),x〉‖x‖2 ⊆ R, λ must also lie in R. 2

Theorem 4.31 Let T ∈ SA(H). Suppose T fulfils the following two conditions.

(i) σa(T ) ⊆ R.

(ii) There exists a λ1 with Imλ1 < 0 and a λ2 with Imλ2 > 0 such that λ1I − T and λ2I − Tare stably solvable.

Then T is alternative.

61

Proof: Because T is selfadjoint, we have WFMV (T ) ⊆ R by Lemma 4.30. Therefore, σπ(T ) =

σq(T )∪σa(T ) ⊆WFMV (T )∪σa(T ) ⊆ R by Proposition 3.47,(vi). Since ∂σFMV (T ) ⊆ σπ(T ) ⊆ Rby Theorem 3.30, this leaves only four possibilites: σFMV (T ) = C, σFMV (T ) = λ ∈ C : Imλ ≥0, σFMV (T ) = λ ∈ C : Imλ ≤ 0, and σFMV (T ) ⊆ R. The first three cases are impossible by

assumption (ii). Thus, σFMV (T ) ⊆ R. Since R has no interior points in C, every point of the

FMV-spectrum must lie in its boundary. Hence, σFMV (T ) ⊆ ∂σFMV (T ) ⊆ σπ(T ). 2

The final two results give special cases in which the conditions of the previous theorem are

met.

Proposition 4.32 Let T ∈ SA(H). Suppose T fulfils one of the following two conditions.

(i) T is assymptotically linear with asymptotic derivate T ′ and T − T ′ is compact.

(ii) T = R+ S where R ∈ B(H) is selfadjoint and S ∈ C(H) is compact.

Then T fulfils condition (i) in Theorem 4.31.

Proof: In the first case, we can use Proposition 3.34,(iii), Lemma 3.27, Proposition 3.47,(vi),

and Lemma 4.30 to see that

σa(T ) = σa(T′) ⊆ σq(T ′) = σq(T ) ⊆WFMV (T ) ⊆ R.

In the second case, we get σa(T ) = σa(R+ S) = σa(R) since S is compact. Using the linearity

of R we get

σa(T ) = σa(R) ⊆ σ(R) ⊆W (R) ⊆ R.

2

Proposition 4.33 Let T ∈ SA(H). Suppose T fulfils one of the following two conditions.

(i) There exists a λ ∈ R such that λI − T is FMV-regular.

(ii) max[T ]A, [T⊥]Q <∞.

Then T fulfils condition (ii) in Theorem 4.31.

Proof: In the first case, we can use the fact that ρFMV (T ) is open to see that there exists an

ε > 0 such that B2ε(λ) ⊆ ρFMV (T ). Hence, λ± iε are FMV-regular.

In the second case, Theorem 3.49 shows that σFMV (T ) is bounded. This obviously implies the

assumption. 2

62

Bibliography

[APV] J.Appell, E. De Pascale, A.Vignoli: Nonlinear Spectral Theory, Walter de Gruyter,

2004

[C] J.B.Conway: A Course in Functional Analysis, Springer, 1985

[D] J.Dugundji: An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 333-367

[FW] W.Feng, J.R.L.Webb: Surjectivity results for nonlinear mappings without oddness con-

ditions, Comment. Math. Univ. Carolinae 38,1 (1997), 15-28

[FMV] M.Furi, M.Martelli, A.Vignoli: Contributions to spectral theory for nonlinear op-

erators in Banach spaces, Ann. Mat. Pura Appl. 118 (1978), 229-294

[MA] J.Matousek: Using the Borsuk-Ulam theorem, Springer, 2003

[MI] E.Michael: Continuous selections I, Ann. of Math. 63 (1956), 361-382

[Q] B.v.Querenburg: Mengentheoretische Topologie, Springer, 1973

[R1] W.Rudin: Real and Complex Analysis, Third Edition, McGraw-Hill, Inc., 1987

[R2] W.Rudin: Functional Analysis, Second Edition, McGraw-Hill, Inc., 1991

[R3] W.Rudin: Principles of Mathematical Analysis, Third Edition, McGraw-Hill, Inc., 1976

[ST] K.Schmitt, R.C.Thompson: Nonlinear Analysis and Differential Equations - An Intro-

duction, http://www.math.utah.edu/∼ schmitt/ode1.pdf

[W] D.Werner: Funktionalanalysis, 6., korrigierte Auflage, Springer, 2007

[Z] E.Zeidler: Nonlinear Functional Analysis and its Applications I (Fixed-Point Theorems),

Springer, 1986

63