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Spectral Analysis of Linear Operators
SMA 5878 Functional Analysis II
Alexandre Nolasco de Carvalho
Departamento de MatematicaInstituto de Ciencias Matematicas and de Computacao
Universidade de Sao Paulo
March 27, 2019
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Example
Let X = L2(0, π) and D(A0) = C 20 (0, π) the set of functions which
are twice continuously differentiable functions and have compactsupport in (0, π). Define A0 : D(A0) ⊂ X → X by
(A0φ)(x) = −φ′′(x), x ∈ (0, π).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It is easy to see that A0 is symmetric and that 〈A0φ, φ〉 ≥2π2 ‖φ‖
2X
for all φ ∈ D(A0).
From Friedrichs Theorem, A0 has a self-adjoint extension A thatsatisfies 〈Aφ, φ〉 ≥ 2
π2 ‖φ‖2X for all φ ∈ D(A).
Note that, the space X12 from Friedrichs theorem is, in this
example the closure of D(A) in the norm H1(0, π) and therefore
X12 = H1
0 (0, π).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
On the other hand D(A∗) is characterised by
D(A∗0) = {φ ∈ X : ∃φ∗ ∈ X such that 〈−u′′, φ〉 = 〈u, φ∗〉, ∀u ∈ D(A0)}
and A∗0φ=−φ′′ for all φ∈D(A∗
0).
Hence, D(A)=H2(0, π)∩H10 (0, π) and Au=−u′′ for all u∈D(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Also from Friedrichs Theorem we know that (−∞, 1π ) ⊂ ρ(A). In
particular 0 ∈ ρ(A) and if φ ∈ D(A), we have that
|φ(x)− φ(y)| ≤ |x − y |12‖φ′‖L2(0,π) = |x − y |
12 〈Aφ, φ〉
12 .
Hence, if B is a bounded subset of D(A) with the graph norm,then supφ∈B ‖φ′‖L2(0,π) < ∞ and the family B of functions isequicontinuous and bounded in C ([0, π],R) with the uniformconvergence topology.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It follows from the Arzela-Ascoli Theorem that B is relativelycompact in C ([0, π],R) and consequently B is relatively compactin L2(0, π).
From a previous exercise it follows that A−1 is a compact operator.
It follows that σ(A) = {λ1, λ2, λ3, · · · } where λn = n2 ∈ σp(A)
with eigenfunctions φn(x) =(2π
) 12 sen(nx), n ∈ N.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Min-Max Characterisation of Eigenvalues
In this section we introduce min-max characterisations ofeigenvalues of compact and self-adjoint operators. To presentthese characterisations we will employ the following result
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
LemmaLet H be a Hilbert space over K and A ∈ L(H) be a self-adjointoperator, then
‖A‖L(H) = sup‖u‖=1‖v‖=1
|〈Au, v〉| = sup‖u‖=1
|〈Au, u〉|.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: It is enough to prove that
‖A‖L(H) = sup‖u‖=1‖v‖=1
|〈Au, v〉| ≤ sup‖u‖=1
|〈Au, u〉| := a.
If u, v ′ ∈ H, ‖u‖ = ‖v ′‖ = 1, |〈Au, v ′〉| e iα = 〈Au, v ′〉 andv = e−iαv ′, we have that
|〈Au, v ′〉| = 〈Au, v〉 =1
4[〈A(u + v), u + v〉 − 〈A(u − v), u − v〉]
≤a
4[‖u + v‖2 + ‖u − v‖2] ≤ a.
This completes the proof.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
ExerciseShow that, if 0 6= A ∈ L(H) is self-adjoint, then A is notquasinilpotent.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
TheoremLet H be a Banach space over K and A ∈ K(H) be a self-adjointoperator such that 〈Au, u〉 ≥ 0 for all u ∈ H. Then,
1. λ1 :=sup{〈Au, u〉 :‖u‖=1} is an eigenvalue and exists v1∈H,‖v1‖=1 such that λ1=〈Av1, v1〉. Besides that Av1=λ1v1.
2. Inductively,
λn :=sup{〈Au, u〉 :‖u‖=1 and u⊥vj , 1≤ j≤n−1} ∈σp(A) (1)
and exists vn ∈ H, ‖vn‖ = 1, vn ⊥ vj , 1 ≤ j ≤ n − 1, suchthat λn = 〈Avn, vn〉. Besides that Avn = λnvn.
3. If Vn = {F ⊂ H : F is a vec. subspace of dimension n of H},
λn = infF∈Vn−1
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F}, n ≥ 1 and (2)
λn = supF∈Vn
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F}, n ≥ 1. (3)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: We consider only the case K = C and λ1 > 0 leaving theremaining cases as exercises for the reader.
1.Let {un} be a sequence in H with ‖un‖=1 and 〈Aun, un〉n→∞−→ λ1.
Taking subsequences if necessary, {un} converges weakly to v1 ∈ Hand {Aun} converges strongly to Av1.
Hence 〈Av1, v1〉 = λ1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Let us show that the sequence {un} converges strongly.
From the previous lemma we know that 0 < λ1 = ‖A‖L(H) andfrom the fact that {un} converges weakly to v1 we have that0 < ‖v1‖ ≤ 1. Hence,
limn→∞
‖Aun − λ1un‖2 = lim
n→∞‖Aun‖
2 − 2λ1 limn→∞
〈Aun, un〉+ λ21
= ‖Av1‖2 − λ2
1 ≤ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Since {Aun} converges strongly to Av1, {Aun − λ1un} convergesstrongly to zero and λ1 > 0, it follows that {un} converges stronglyto v1, ‖v1‖ = 1 and Av1 = λ1v1. This concludes the proof of 1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
2. The proof of this item follows from 1. simply noting that theorthogonal of Hn−1 = span{v1, · · · , vn−1} is invariant by A andrepeating the procedure for the restriction of A to H⊥
n−1, n ≥ 2.This concludes the proof of 2.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
3. We first prove expression (2). If G = span{v1, · · · , vn−1} wehave, from (1), that
λn = sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ G}
≥ infF∈Vn−1
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F}.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
On the other hand, let F ∈ Vn−1 and w1, · · · ,wn−1 anorthornormal set of F . Choose u =
∑ni=1 αivi such that ‖u‖ = 1
and u ⊥ wj , 1 ≤ j ≤ n − 1. Hence∑n
i=1 |αi |2 = 1 and
〈Au, u〉 =
n∑
i=1
|αi |2λi ≥ λn.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
This implies
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F} ≥ λn, for all F ∈ Vn−1
and completes the proof of (2).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
We now prove (3). If G = span{v1, · · · , vn} and u ∈ G , ‖u‖ = 1,we have that u =
∑ni=1 αivi with
∑ni=1 |αi |
2 = 1 e
〈Au, u〉 =n∑
i=1
|αi |2λi ≥ λn.
This implies that
supF∈Vn
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F} ≥ λn.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Conversely, given F ∈ Vn choose u ∈ F , ‖u‖ = 1, such thatu ⊥ vj , 1 ≤ j ≤ n − 1. It follows, from 2., that 〈Au, u〉 ≤ λn andconsequently
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F} ≤ λn, for all F ∈ Vn.
This completes the proof of (3).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
ExerciseIf A : D(A) ⊂ H → H is positive, self-adjoint and (〈Au, u〉 > 0 forall u ∈ D(A)) and has compact resolvent, find the min-maxcharacterisation for the eigenvalues of A.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Dissipative operators and numerical range
DefinitionLet X be a Banach space over K. The duality map J : X → 2X
∗is
a multivalued function defined by
J(x) = {x∗ ∈ X ∗ : Re〈x , x∗〉 = ‖x‖2, ‖x∗‖ = ‖x‖}.
From the Hanh-Banach Theorem we have that J(x) 6= ∅.
A linear operator A : D(A) ⊂ X → X is dissipative if for eachx ∈ D(A) there exists x∗ ∈ J(x) such that Re 〈Ax , x∗〉 ≤ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
ExerciseShow that, if X ∗ is uniformly convex and x ∈ X, then J(x) is aunitary subset of X ∗.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
LemmaThe linear operator A is dissipative if and only if
‖(λ− A)x‖ ≥ λ‖x‖ (1)
for all x ∈ D(A) and λ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: If A is dissipative, λ > 0, x ∈ D(A), x∗ ∈ J(x) andRe〈Ax , x∗〉 ≤ 0,
‖λx − Ax‖‖x‖ ≥ |〈λx − Ax , x∗〉| ≥ Re〈λx − Ax , x∗〉 ≥ λ‖x‖2
and (1) follows. Conversely, given x ∈ D(A) suppose that (1)holds for all λ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
If y∗λ ∈ J((λ− A)x) and g∗λ = y∗λ/‖y
∗λ‖ we have that
λ‖x‖≤‖λx − Ax‖=〈λx−Ax , g∗λ〉=λRe〈x , g∗
λ〉−Re〈Ax , g∗λ〉
≤ λ‖x‖ − Re〈Ax , g∗λ〉
(2)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Since the unit ball of X ∗ is compact in the weak∗-topology wehave that there exists g∗ ∈ X ∗ with ‖g∗‖ ≤ 1 such that g∗ is alimit point of the sequence {g∗
n} [there is a sub-net (see Apendix)of {g∗
n} that converges to g∗].
From (2) it follows that Re〈Ax , g∗〉 ≤ 0 and Re〈x , g∗〉 ≥ ‖x‖. ButRe〈x , g∗〉 ≤ |〈x , g∗〉| ≤ ‖x‖ and therefore Re〈x , g∗〉 = ‖x‖.
Taking x∗ = ‖x‖g∗ we have that x∗ ∈ J(x) and Re〈Ax , x∗〉 ≤ 0.Thus, for all x ∈ D(A) there exists x∗ ∈ J(x) such thatRe〈Ax , x∗〉 ≤ 0 and A e dissipative.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Theorem (G. Lumer)
Suppose that A is a linear operator in a Banach space X . If A isdissipative and R(λ0 − A) = X for some λ0 > 0, then A is closed,ρ(A) ⊃ (0,∞) and
‖λ(λ− A)−1‖L(X ) ≤ 1,∀λ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: If λ > 0 and x ∈ D(A), do Lemma 2 temos que
‖(λ− A)x‖ ≥ λ‖x‖.
Now R(λ0 − A) = X , ‖(λ0 − A)x‖ ≥ λ0‖x‖ for x ∈ D(A), so λ0 isin ρ(A) and A is closed. Let Λ = ρ(A) ∩ (0,∞).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Λ is an open subset of (0,∞) for ρ(A) is open, let us prove that Λis a closed subset of (0,∞) to conclude that Λ = (0,∞).
Suppose that {λn}∞n=1 ⊂ Λ, λn → λ > 0, if n is sufficiently large
we have that |λn − λ| ≤ λ/3 then, for all n sufficiently large,‖(λ−λn)(λn−A)−1‖≤|λn−λ|λ−1
n ≤1/2 and I+(λ−λn)(λn−A)−1
is in isomorphism of X .
Thenλ− A =
{I + (λ− λn)(λn − A)−1
}(λn − A) (3)
takes D(A) over X and λ ∈ ρ(A), as desired.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Corollary
Let A be a closed and densely defined linear operator. If both Aand A∗ are dissipative, then ρ(A) ⊃ (0,∞) and
‖λ(λ− A)−1‖ ≤ 1,∀λ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: From Theorem (G. Lummer) it is enough to prove thatR(I − A) = X .
Since A is dissipative and closed, R(I − A) is a closed subspace ofX .
Let x∗ ∈ X ∗ be such that 〈(I − A)x , x∗〉 = 0 for all x ∈ D(A).This implies that x∗ ∈ D(A∗) and (I ∗ − A∗)x∗ = 0.
Since A∗ is also dissipative it follows from the previous lemma thatx∗ = 0. Consequently R(I −A) is dense in X and since R(I −A) isclosed, R(I − A) = X .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
In several examples, the technique used to obtain estimates for theresolvent of a given operator and the localisation of its spectrum isthe localisation of the numerical range (defined next).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
If A is a linear operator in a complex Banach space X its numericalrange W (A) is the set
W (A) :={〈Ax , x∗〉 :x ∈D(A), x∗∈X ∗, ‖x‖=‖x∗‖= 〈x , x∗〉=1}. (4)
When X is a Hilbert space
W (A) = {〈Ax , x〉 : x ∈ D(A), ‖x‖ = 1}.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Theorem (Numerical Range)
Let A : D(A) ⊂ X → X be a closed densely defined operator andW (A) be the numerical range of A.
1. If λ /∈ W (A) then λ− A is injective, has closed image andsatisfies
‖(λ− A)x‖ ≥ d(λ,W (A))‖x‖. (5)
where d(λ,W (A)) is the distance of λ to W (A). Besidesthat, if λ ∈ ρ(A),
‖(λ− A)−1‖L(X ) ≤1
d(λ,W (A)). (6)
2. If Σ is open and connected in C\W (A) and ρ(A) ∩Σ 6= ∅,then ρ(A) ⊃ Σ and (6) is satisfied for all λ ∈ Σ.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: Let λ /∈ W (A). If x ∈ D(A), ‖x‖ = 1, x∗ ∈ X ∗, ‖x∗‖ = 1and 〈x , x∗〉 = 1 then,
0<d(λ,W (A))≤|λ−〈Ax , x∗〉|= |〈λx −Ax , x∗〉|≤‖λx −Ax‖ (7)
and therefore λ− A is one-to-one, has closed image and satisfies(5). If, besides that, λ ∈ ρ(A) then, (7) implies (6).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It remains to show that, if Σ intersects ρ(A) then, ρ(A) ⊃ Σ. Tothat end consider the nonempty set ρ(A) ∩Σ.
This set is clearly open in Σ.
But it is also closed since, if λn ∈ ρ(A) ∩ Σ and λn → λ ∈ Σ then,for sufficiently large n, |λ− λn| < d(λn,W (A)).
From this and (6) it follows that |λ− λn| ‖(λn − A)−1‖ < 1, for nsufficiently large. Consequently,λ∈ρ(A) and ρ(A)∩Σ is closed in Σ.
It follows that ρ(A) ∩Σ = Σ that is ρ(A) ⊃ Σ, as desired.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Example
Let H be a Hilbert space over K and A : D(A) ⊂ H → H be aself-adjoint operator. It follows that A is closed and denselydefined. If A is bounded above; that is, 〈Au, u〉 ≤ a〈u, u〉 for somea ∈ R, then C\(−∞, a] ⊂ ρ(A), and
‖(A − λ)−1‖L(X ) ≤M
|λ− a|,
for some constant M ≥ 1, depending only on ϕ, for allλ ∈ Σa,ϕ = {λ ∈ C : |arg(λ− a)| < ϕ}, ϕ < π.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: We start localising the numerical range of A. First notethat
W (A) = {〈Ax , x〉 : x ∈ D(A), ‖x‖ = 1} ⊂ (−∞, a].
Also, A− a = A∗ − a is dissipative and therefore, from a previousresult, ρ(A− a) ⊃ (0,∞).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
From Theorem (Numerical Range) we have thatC\(−∞, a] ⊂ ρ(A) and that
‖(λ− A)−1‖ ≤1
d(λ,W (A))≤
1
d(λ, (−∞, a]).
Besides that, if λ ∈ Σa,ϕ, we have that
1
d(λ, (−∞, a])≤
1
sinϕ
1
|λ− a|
and the result follows.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
ExerciseLet X be a Banach space such that X ∗ is strictly convex andA : D(A) ⊂ X → X be a closed, densely defined and dissipativelinear operator. Se R(I − A) = X, show thatρ(A) ⊃ {λ ∈ C : Reλ > 0} and that
‖(λ− A)−1‖L(X ) ≤1
Reλ, for all λ ∈ Σ0,π
2.
Is the hypothesis that X ∗ be strictly convex necessary?
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proposition
Let H be a Hilbert space over K with inner product 〈·, ·〉 andA ∈ L(H) be a self-adjoint operator. If
m = infu∈H‖u‖=1
〈Au, u〉, M = supu∈H‖u‖=1
〈Au, u〉,
then, {m,M} ⊂ σ(A) ⊂ [m,M].
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Proof: From the definition of M we have that〈Au, u〉 ≤ M‖u‖2, ∀u ∈ H. From this it follows that, if λ > Mthen,
〈λu − Au, u〉 ≥ (λ−M)︸ ︷︷ ︸
>0
‖u‖2. (8)
With that, it is easy to see that a(v , u) = 〈v , λu − Au〉 is asymmetric (a(u, v) = a(v , u) for all u, v ∈ H), continous andcoercive sesquilinear form.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It follows from Lax-Milgram theorem that
〈v , λu − Au〉 = 〈v , f 〉, ∀v ∈ H,
has a unique solution uf for each f ∈ H. It is easy to see that thissolution satisfies
(λ− A)uf = f .
From this it follows that (λ− A) is bijective and (M,∞) ⊂ ρ(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Let us show that M∈σ(A). Note that a(u, v)=(Mu−Au, v) is acontinuous, symmetric sesquilinear form and a(u, u) ≥ 0, ∀u ∈ H.Hence
|a(u, v)| ≤ a(u, u)1/2a(v , v)1/2, for all u, v ∈ H,
that is, the Cauchy-Schwarz inequality holds.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It follows that
|(Mu − Au, v)| ≤ (Mu − Au, u)1/2(Mv − Av , v)1/2, ∀u, v ∈ H
≤ C (Mu − Au, u)1/2 ‖v‖
and that
‖Mu − Au‖ ≤ C (Mu − Au, u)1/2, ∀u ∈ H.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
Let {un} be a sequence of vectors such that ‖un‖ = 1,〈Aun, un〉 → M. It follows that ‖Mun − Aun‖ → 0. If M ∈ ρ(A)
un = (MI − A)−1(Mun − Aun) → 0
which is in contradiction with ‖un‖ = 1, ∀n ∈ N. It follows thatM ∈ σ(A).
From the above result applied to −A we obtain that(−∞,m) ⊂ ρ(A) and m ∈ σ(A). The proof that σ(A) ⊂ R hasbeen given in Example 2
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsMin-Max Characterisation of Eigenvalues
Dissipative operators and numerical range
It follows directly from Proposition 1 (if A ∈ L(H) is self-adjoint,‖A‖ = sup{〈Au, u〉 : u ∈ H, ‖u‖H = 1}) that
Corollary
Let H be a Hilbert space and A ∈ L(H) be a self-adjoint operatorwith σ(A) = {0}, then A = 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
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