SMA 5878 Functional Analysis II

Spectral Analysis of Linear Operators

Alexandre Nolasco de Carvalho

Departamento de Matemática Instituto de Ciências Matemáticas and de Computa¸cão Universidade de São Paulo

March 27, 2019

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Example 2 2 Let X = L (0,π) and D(A0)= C0 (0,π) the set of functions which are twice continuously diﬀerentiable functions and have compact support 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

2 2 It is easy to see that A0 is symmetric and that hA0φ, φi≥ π2 kφkX for all φ ∈ D(A0).

From Friedrichs Theorem, A0 has a self-adjoint extension A that 2 2 satisﬁes hAφ, φi≥ π2 kφkX for all φ ∈ D(A).

1 Note that, the space X 2 from Friedrichs theorem is, in this example the closure of D(A) in the norm H1(0,π) and therefore 1 1 X 2 = H0 (0,π).

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

On the other hand D(A∗) is characterised by

∗ ∗ ′′ ∗ D(A0)= {φ ∈ X : ∃ φ ∈ X such that h−u , φi = hu, φ i, ∀u ∈ D(A0)}

∗ ′′ ∗ and A0φ=−φ for all φ∈D(A0).

2 1 ′′ Hence, D(A)=H (0,π)∩H0 (0,π) and Au =−u for all u ∈D(A).

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

1 Also from Friedrichs Theorem we know that (−∞, π ) ⊂ ρ(A). In particular 0 ∈ ρ(A) and if φ ∈ D(A), we have that

1 ′ 1 1 |φ(x) − φ(y)| ≤ |x − y| 2 kφ kL2(0,π) = |x − y| 2 hAφ, φi 2 .

Hence, if B is a bounded subset of D(A) with the graph norm, ′ then supφ∈B kφ kL2(0,π) < ∞ and the family B of functions is equicontinuous and bounded in C([0,π], R) with the uniform convergence topology.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

It follows from the Arzel´a-Ascoli Theorem that B is relatively compact in C([0,π], R) and consequently B is relatively compact in L2(0,π).

From a previous exercise it follows that A−1 is a compact operator.

2 It follows that σ(A)= {λ1, λ2, λ3, ···} where λn = n ∈ σp(A) 1 2 2 N with eigenfunctions φn(x)= π sen(nx), n ∈ .

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range Min-Max Characterisation of Eigenvalues

In this section we introduce min-max characterisations of eigenvalues of compact and self-adjoint operators. To present these characterisations we will employ the following result

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Lemma Let H be a Hilbert space over K and A ∈L(H) be a self-adjoint operator, then

kAkL(H) = sup |hAu, vi| = sup |hAu, ui|. kuk=1 kuk=1 kvk=1

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: It is enough to prove that

kAkL(H) = sup |hAu, vi| ≤ sup |hAu, ui| := a. kuk=1 kuk=1 kvk=1

If u, v ′ ∈ H, kuk = kv ′k = 1, |hAu, v ′i| eiα = hAu, v ′i and v = e−iαv ′, we have that 1 |hAu, v ′i| = hAu, vi = [hA(u + v), u + vi − hA(u − v), u − vi] 4 a ≤ [ku + vk2 + ku − vk2] ≤ a. 4 This completes the proof.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Exercise Show that, if 0 6= A ∈L(H) is self-adjoint, then A is not quasinilpotent.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Theorem Let H be a Banach space over K and A ∈ K(H) be a self-adjoint operator such that hAu, ui≥ 0 for all u ∈ H. Then, 1. λ1 :=sup{hAu, ui :kuk=1} is an eigenvalue and exists v1 ∈H, kv1k=1 such that λ1 =hAv1, v1i. Besides that Av1 =λ1v1. 2. Inductively,

λn :=sup{hAu, ui:kuk=1 and u ⊥vj , 1≤j ≤n−1}∈σp(A) (1) and exists vn ∈ H, kvnk = 1, vn ⊥ vj , 1 ≤ j ≤ n − 1, such that λn = hAvn, vni. Besides that Avn = λnvn.

3. If Vn = {F ⊂ H : F is a vec. subspace of dimension n of H}, λn = inf sup{hAu, ui : kuk = 1, u ⊥ F }, n ≥ 1 and (2) F ∈Vn−1

λn = sup inf{hAu, ui : kuk = 1, u ∈ F }, n ≥ 1. (3) F ∈Vn

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: We consider only the case K = C and λ1 > 0 leaving the remaining cases as exercises for the reader.

n→∞ 1.Let {un} be a sequence in H with kunk=1 and hAun, uni −→ λ1.

Taking subsequences if necessary, {un} converges weakly to v1 ∈ H and {Aun} converges strongly to Av1.

Hence hAv1, v1i = λ1.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Let us show that the sequence {un} converges strongly.

From the previous lemma we know that 0 < λ1 = kAkL(H) and from the fact that {un} converges weakly to v1 we have that 0 < kv1k≤ 1. Hence,

2 2 2 lim kAun − λ1unk = lim kAunk − 2λ1 lim hAun, uni + λ n→∞ n→∞ n→∞ 1 2 2 = kAv1k − λ1 ≤ 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Since {Aun} converges strongly to Av1, {Aun − λ1un} converges strongly to zero and λ1 > 0, it follows that {un} converges strongly to v1, kv1k = 1 and Av1 = λ1v1. This concludes the proof of 1.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

2. The proof of this item follows from 1. simply noting that the orthogonal of Hn−1 = span{v1, · · · , vn−1} is invariant by A and ⊥ repeating the procedure for the restriction of A to Hn−1, n ≥ 2. This concludes the proof of 2.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

3. We ﬁrst prove expression (2). If G = span{v1, · · · , vn−1} we have, from (1), that

λn = sup{hAu, ui : kuk = 1, u ⊥ G} ≥ inf sup{hAu, ui : kuk = 1, u ⊥ F }. F ∈Vn−1

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

On the other hand, let F ∈Vn−1 and w1, · · · , wn−1 an n orthornormal set of F . Choose u = Pi=1 αi vi such that kuk = 1 n 2 and u ⊥ wj , 1 ≤ j ≤ n − 1. Hence Pi=1 |αi | = 1 and n 2 hAu, ui = X |αi | λi ≥ λn. i=1

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

This implies

sup{hAu, ui : kuk = 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

We now prove (3). If G = span{v1, · · · , vn} and u ∈ G, kuk = 1, n n 2 we have that u = Pi=1 αi vi with Pi=1 |αi | = 1 e n 2 hAu, ui = X |αi | λi ≥ λn. i=1 This implies that

sup inf{hAu, ui : kuk = 1, u ∈ F }≥ λn. F ∈Vn

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Conversely, given F ∈Vn choose u ∈ F , kuk = 1, such that u ⊥ vj , 1 ≤ j ≤ n − 1. It follows, from 2., that hAu, ui≤ λn and consequently

inf{hAu, ui : kuk = 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Exercise If A : D(A) ⊂ H → H is positive, self-adjoint and (hAu, ui > 0 for all u ∈ D(A)) and has compact resolvent, ﬁnd the min-max characterisation for the eigenvalues of A.

Deﬁnition ∗ Let X be a Banach space over K. The duality map J : X → 2X is a multivalued function deﬁned by

J(x)= {x∗ ∈ X ∗ : Rehx, x∗i = kxk2, kx∗k = kxk}.

From the Hanh-Banach Theorem we have that J(x) 6= ∅.

A linear operator A : D(A) ⊂ X → X is dissipative if for each x ∈ D(A) there exists x∗ ∈ J(x) such that Re hAx, x∗i≤ 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Exercise Show that, if X ∗ is uniformly convex and x ∈ X, then J(x) is a unitary subset of X ∗.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Lemma The linear operator A is dissipative if and only if

k(λ − A)xk≥ λkxk (1)

for all x ∈ D(A) and λ> 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: If A is dissipative, λ> 0, x ∈ D(A), x∗ ∈ J(x) and RehAx, x∗i≤ 0,

kλx − Axkkxk ≥ |hλx − Ax, x∗i| ≥ Rehλx − Ax, x∗i≥ λkxk2

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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

∗ ∗ ∗ ∗ If yλ ∈ J((λ − A)x) and gλ = yλ/kyλ k we have that

∗ ∗ ∗ λkxk≤kλx − Axk=hλx −Ax, gλ i=λRehx, gλ i−RehAx, gλ i ∗ (2) ≤ λkxk− RehAx, gλ i

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Since the unit ball of X ∗ is compact in the weak∗-topology we have that there exists g ∗ ∈ X ∗ with kg ∗k≤ 1 such that g ∗ is a ∗ limit point of the sequence {gn } [there is a sub-net (see Apˆendix) ∗ ∗ of {gn } that converges to g ].

From (2) it follows that RehAx, g ∗i≤ 0 and Rehx, g ∗i ≥ kxk. But Rehx, g ∗i ≤ |hx, g ∗i| ≤ kxk and therefore Rehx, g ∗i = kxk.

Taking x∗ = kxkg ∗ we have that x∗ ∈ J(x) and RehAx, x∗i≤ 0. Thus, for all x ∈ D(A) there exists x∗ ∈ J(x) such that RehAx, x∗i≤ 0 and A ´edissipative.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Theorem (G. Lumer) Suppose that A is a linear operator in a Banach space X. If A is dissipative and R(λ0 − A)= X for some λ0 > 0, then A is closed, ρ(A) ⊃ (0, ∞) and

−1 kλ(λ − A) kL(X ) ≤ 1, ∀ λ> 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: If λ> 0 and x ∈ D(A), do Lemma 2 temos que

k(λ − A)xk≥ λkxk.

Now R(λ0 − A)= X , k(λ0 − A)xk≥ λ0kxk for x ∈ D(A), so λ0 is in ρ(A) and A is closed. Let Λ = ρ(A) ∩ (0, ∞).

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Λ is an open subset of (0, ∞) for ρ(A) is open, let us prove that Λ is 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, −1 −1 −1 k(λ−λn)(λn −A) k≤|λn −λ|λn ≤1/2 and I +(λ−λn)(λn −A) is in isomorphism of X .

Then −1 λ − A = I + (λ − λn)(λn − A) (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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Corollary Let A be a closed and densely deﬁned linear operator. If both A and A∗ are dissipative, then ρ(A) ⊃ (0, ∞) and

kλ(λ − A)−1k≤ 1, ∀ λ> 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: From Theorem (G. Lummer) it is enough to prove that R(I − A)= X .

Since A is dissipative and closed, R(I − A) is a closed subspace of X .

Let x∗ ∈ X ∗ be such that h(I − A)x, x∗i = 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 that x∗ = 0. Consequently R(I − A) is dense in X and since R(I − A) is closed, R(I − A)= X .

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

In several examples, the technique used to obtain estimates for the resolvent of a given operator and the localisation of its spectrum is the localisation of the numerical range (deﬁned next).

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

If A is a linear operator in a complex Banach space X its numerical range W (A) is the set

W (A):={hAx, x∗i:x ∈D(A), x∗ ∈X ∗, kxk=kx∗k= hx, x∗i=1}. (4)

When X is a Hilbert space

W (A)= {hAx, xi : x ∈ D(A), kxk = 1}.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Theorem (Numerical Range) Let A : D(A) ⊂ X → X be a closed densely deﬁned operator and W (A) be the numerical range of A. 1. If λ∈ / W (A) then λ − A is injective, has closed image and satisﬁes k(λ − A)xk≥ d(λ, W (A))kxk. (5) where d(λ, W (A)) is the distance of λ to W (A). Besides that, if λ ∈ ρ(A),

1 k(λ − A)−1k ≤ . (6) L(X ) d(λ, W (A))

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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: Let λ∈ / W (A). If x ∈ D(A), kxk = 1, x∗ ∈ X ∗, kx∗k = 1 and hx, x∗i = 1 then,

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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

It remains to show that, if Σ intersects ρ(A) then, ρ(A) ⊃ Σ. To that 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)).

−1 From this and (6) it follows that |λ − λn| k(λn − A) k < 1, for n suﬃciently 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Example Let H be a Hilbert space over K and A : D(A) ⊂ H → H bea self-adjoint operator. It follows that A is closed and densely deﬁned. If A is bounded above; that is, hAu, ui≤ ahu, ui for some a ∈ R, then C\(−∞, a] ⊂ ρ(A), and

M k(A − λ)−1k ≤ , L(X ) |λ − a|

for some constant M ≥ 1, depending only on ϕ, for all λ ∈ Σa,ϕ = {λ ∈ C : |arg(λ − a)| < ϕ}, ϕ<π.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: We start localising the numerical range of A. First note that

W (A)= {hAx, xi : x ∈ D(A), kxk = 1}⊂ (−∞, a].

Also, A − a = A∗ − a is dissipative and therefore, from a previous result, ρ(A − a) ⊃ (0, ∞).

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

From Theorem (Numerical Range) we have that C\(−∞, a] ⊂ ρ(A) and that

1 1 k(λ − A)−1k≤ ≤ . d(λ, W (A)) d(λ, (−∞, a])

Besides that, if λ ∈ Σa,ϕ, we have that 1 1 1 ≤ d(λ, (−∞, a]) sin ϕ |λ − a|

and the result follows.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Exercise Let X be a Banach space such that X ∗ is strictly convex and A : D(A) ⊂ X → X be a closed, densely deﬁned and dissipative linear operator. Se R(I − A)= X, show that ρ(A) ⊃ {λ ∈ C : Reλ> 0} and that

1 −1 π k(λ − A) kL(X ) ≤ , for all λ ∈ Σ0, . Reλ 2 Is the hypothesis that X ∗ be strictly convex necessary?

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proposition Let H be a Hilbert space over K with inner product h·, ·i and A ∈L(H) be a self-adjoint operator. If

m = inf hAu, ui, M = sup hAu, ui, u∈H u∈H kuk=1 kuk=1

then, {m, M}⊂ σ(A) ⊂ [m, M].

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Proof: From the deﬁnition of M we have that hAu, ui≤ Mkuk2, ∀u ∈ H. From this it follows that, if λ> M then, hλu − Au, ui≥ (λ − M) kuk2. (8) | >{z0 } With that, it is easy to see that a(v, u)= hv, λu − Aui is a symmetric (a(u, v)= a(v, u) for all u, v ∈ H), continous and coercive sesquilinear form.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

It follows from Lax-Milgram theorem that

hv, λu − Aui = hv, f i, ∀v ∈ H,

has a unique solution uf for each f ∈ H. It is easy to see that this solution 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Let us show that M ∈σ(A). Note that a(u, v)=(Mu−Au, v) is a continuous, 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 Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators 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 kvk

and that

kMu − Auk≤ C(Mu − Au, u)1/2, ∀u ∈ H.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

Let {un} be a sequence of vectors such that kunk = 1, hAun, uni→ M. It follows that kMun − Aunk→ 0. If M ∈ ρ(A)

−1 un = (MI − A) (Mun − Aun) → 0

which is in contradiction with kunk = 1, ∀n ∈ N. It follows that M ∈ 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 has been given in Example 2

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II Min-Max Characterisation of Eigenvalues Spectral Analysis of Linear Operators Dissipative operators and numerical range

It follows directly from Proposition 1 (if A ∈L(H) is self-adjoint, kAk = sup{hAu, ui : u ∈ H, kukH = 1}) that

Corollary Let H be a Hilbert space and A ∈L(H) be a self-adjoint operator with σ(A)= {0}, then A = 0.

Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II