SMA 5878 Functional Analysis II - icmc.usp.br · Spectral AnalysisofLinearOperators SMA 5878...
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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
April 01, 2019
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 each
x ∈ 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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Conversely, given x ∈ D(A) suppose that (1) holds for all λ > 0.
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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
‖λ(λ− A)−1‖L(X ) ≤ 1,∀λ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Proof: If λ > 0 and x ∈ D(A), from Lemma 1 we have that
‖(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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Corollary
Let A be a closed and densely defined linear operator. If both A
and 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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Proof: From Theorem (G. Lummer) it is enough to prove thatR(I − A) = X .
SinceA 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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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
‖(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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Example
Let H be a Hilbert space over K and A : D(A) ⊂ H → H be a
self-adjoint operator. It follows that A is closed and densely
defined. If A is bounded above; that is, 〈Au, u〉 ≤ a〈u, u〉 for some
a ∈ R, then C\(−∞, a] ⊂ ρ(A), and
‖(λ− A)−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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
ExerciseLet X be a Banach space such that X ∗ is uniformly convex and
A : D(A) ⊂ X → X be a closed, densely defined and dissipative
linear operator. If R(I − A) = X, show that
ρ(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 uniformly convex necessary?
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Proof: From the definition of M we have that〈Au, u〉 ≤ M‖u‖2, ∀u ∈ H. From this it follows that, if λ > M
then,〈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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 1
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
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 operator
with σ(A) = {0}, then A = 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Operational calculus in L(X )
Let X be a Banach space over C and A ∈ L(X ). We have alreadyseen that σ(A) is closed, non-empty and bounded.
In fact,
σ(A)⊂{λ ∈ C : |λ|≤ rσ(A)}
and
rσ= infn≥1
‖An‖1n
L(X )≤‖A‖L(X ).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Let γ : [0, 2π] → C be the curve given by γ(t) = re it , t ∈ [0, 2π],with r > rσ(A). We know that, for |λ| > rσ(A),
(λ− A)−1 =∞∑
n=0
λ−n−1An,
and, for j ∈ N,
Aj =1
2πi
∫
γλj (λ− A)−1dλ. (9)
Observe that, the curve γ can be chosen to be any closedrectifiable curve which is homotopic to the above curve in ρ(A).
Emphasise the case j = 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
So, if p : C → C is a polynomial,
p(A) =1
2πi
∫
γp(λ)(λ− A)−1dλ.
ExerciseLet X be a complex Banach space and A ∈ L(X ). Show that, if
r > ‖A‖L(X ) and γr (t) = re2πit , t ∈ [0, 1] then,
∞∑
n=0
An
n!=
1
2πi
∫
γr
eλ(λ− A)−1dλ.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
DefinitionIf X is a Banach space over C and A ∈ L(X ). The class of analytic
functions f : D(f ) ⊂ C → C such that D(f ) is a Cauchy domain
that contains σ(A) is denoted by U(A). For f ∈ U(A) we define
f (A) =1
2πi
∫
+∂Df (λ)(λ− A)−1dλ (10)
where D is a bounded Cauchy domain such that σ(A) ⊂ D and
D ⊂ D(f ).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
ExerciseLet X be a Banach space over C and A ∈ L(X ). Show that, if
f , g ∈ U(A) and f , g coincide in an open set that contains σ(A)then, f (A) = g(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
It is clear that, if f , g ∈ U(A) and α ∈ C, we have that f + g , fgand αf are in em U(A). Besides that, it is easy to see that
f (A) + g(A) = (f + g)(A) and αf (A) = (αf )(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Let us show that f (A) ◦ g(A)=(fg)(A). Let D1 and D2 be Cauchydomains such that σ(A) ⊂ D1 ⊂ D1 ⊂ D2 ⊂ D(f ) ∩ D(g). Withthis notation we have that
f (A) =1
2πi
∫
+∂D1
f (λ)(λ− A)−1dλ,
g(A) =1
2πi
∫
+∂D2
g(µ)(µ − A)−1dµ.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Hence
f (A) ◦ g(A) =1
(2πi)2
∫
+∂D1
∫
+∂D2
f (λ)g(µ) (λ−A)−1(µ−A)−1 dµ dλ
=1
(2πi)2
∫
+∂D1
∫
+∂D2
f (λ)g(µ)1
µ− λ[(λ− A)−1−(µ− A)−1] dµ dλ
=1
2πi
∫
+∂D1
f (λ)g(λ)(λ − A)−1 dλ = (fg)(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
ExerciseLet X be a Banach space over C, B ∈ L(X ) with ‖B‖L(X ) < 1and A = I + B. Show that, if 1 > r > ‖B‖L(X ), α > 0 and
γr (t) = 1 + re2πit , t ∈ [0, 1], then
A−α =∞∑
n=0
(α+ n − 1
n
)
(−1)nBn =1
2πi
∫
γr
λ−α(λ− A)−1dλ.
where
(α+ n− 1
n
)
:=Γ(α+ n)
n! Γ(α)=
α(α + 1) · · · (α+ n − 1)
n!.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Show that A−α−β = A−αA−β for all α, β ∈ (0,∞). In particular,
A−1 =∞∑
n=0
(−1)nBn =1
2πi
∫
γr
λ−1(λ− A)−1dλ and
A−2 =
∞∑
n=0
(n + 1)(−1)nBn =1
2πi
∫
γr
λ−2(λ− A)−1dλ.
Study the positive powers of A.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
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TheoremLet X be a Banach space over C and A ∈ L(X ). If f ∈ U(A) issuch that f (λ) 6= 0 for all λ ∈ σ(A), then f (A) one-to-one and
onto X with inverse g(A) where g is any function in U(A) thatcoincides with 1
fin an open set that contains σ(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
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Proof: If g = 1fin an open set that contains σ(A) then, g ∈ U(A)
and f (λ)g(λ) = 1 in an open set that contains σ(A). Hence
f (A)g(A) = g(A)f (A) = (fg)(A) = I .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
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Operational calculus for closed operators
Let X be a Banach space over C and A : D(A) ⊂ X → X a closedlinear operator with non-empty resolvent ρ(A).
Denote by U∞(A) the set of all analytic functions f with domainscontaining the union of σ(A) with the exterior of a compact setand that satisfying limλ→∞ f (λ) = f (∞).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
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ExerciseLet R > 0, A(0,R ,∞) = (B
C
R(0))c and f : A(0,R ,∞) → C a
bounded analytic function. Show that the limit below exists 1
limλ→∞
f (λ).
1Suggestion: Show that 0 is a removable singularity of the analytic functiong : B 1
R(0)\{0} → C defined by g(λ) = f ( 1
λ).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
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We define in U∞(A) the equivalence relation R by (f , g) ∈ R if fand g are equal in an open set that contains σ(A) and the exteriorof a ball. We write f ∼ g to denote that (f , g) ∈ R.
ExerciseShow that the relation R ⊂ U∞ × U∞ is an equivalence relation.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
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Observe that, if D is an unbounded Cauchy domain withD ⊃ A(0, r ,∞) and f : D(f ) ⊂ C → C is a function in U∞(A)with D(f ) ⊃ D, entao
f (ξ) =1
2πi
∫
γr
f (λ)
λ− ξdλ+
1
2πi
∫
∂D+
f (λ)
λ− ξdλ (11)
where r > 0 is such that Br (0) ⊃ Dc , ξ is a point of D with|ξ| < r and γr (t) = re2πit , t ∈ [0, 1].
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Hence, making r → ∞ in (11) and using that limλ→∞
f (λ) = f (∞),
we obtain
f (ξ) = f (∞) +1
2πi
∫
∂D+
f (λ)
λ− ξdλ (12)
for all ξ in D.
Using the same reasoning, if ξ is exterior to D, then
0 = f (∞) +1
2πi
∫
∂D+
f (λ)
λ− ξdλ (13)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
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When f ∈ U∞(A), we define
f (A) = f (∞)I +1
2πi
∫
+∂Df (λ)(λ− A)−1dλ, (14)
where D is an unbounded Cauchy domain such thatσ(A) ⊂ D ⊂ D ⊂ D(f ). Note that f (A) ∈ L(X ) even when A isnot a bounded operator.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
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ExerciseLet X be a Banach space over C and A : D(A) ⊂ X → X a closed
operator with non-empty resolvent.
a) Show that, if f , g ∈ U∞(A) and f ∼ g then, f (A) = g(A).
b) Show that, if f (λ) = 1 for all λ ∈ C then, f (A) = I .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Let X be a Banach space over C and A : D(A) ⊂ X → X a closedoperator with non-empty resolvent. Se f , g ∈ U∞(A), show thatf (A) ◦ g(A) = (fg)(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
As before, let D1 and D2 Cauchy domains such thatσ(A) ⊂ D1 ⊂ D1 ⊂ D2 ⊂ D(f ) ∩ D(g). With this notation wehave that
f (A) = f (∞)I +1
2πi
∫
+∂D1
f (λ)(λ − A)−1dλ
and
g(A) = g(∞)I +1
2πi
∫
+∂D2
g(µ)(µ − A)−1dµ.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Using (12) and (11), if λ ∈ ∂D1 and µ ∈ ∂D2, we have that
g(λ) = g(∞) +1
2πi
∫
+∂D2
g(µ)
µ− λdµ
and
0 = f (∞) +1
2πi
∫
+∂D1
f (λ)
λ− µdλ.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Consequently,
f (A) ◦ g(A) = f (∞)g(∞)I
+1
(2πi)2
∫
+∂D1
∫
+∂D2
f (λ)g(µ) (λ − A)−1(µ− A)−1 dµ dλ
+g(∞)
2πi
∫
+∂D1
f (λ) (λ − A)−1dλ+f (∞)
2πi
∫
+∂D1
g(µ) (µ − A)−1dµ
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
= f (∞)g(∞)I
+1
(2πi)2
∫
+∂D1
∫
+∂D2
f (λ)g(µ)(λ−A)−1−(µ−A)−1
µ− λdµ dλ
+g(∞)
2πi
∫
+∂D1
f (λ) (λ − A)−1dλ+f (∞)
2πi
∫
+∂D1
g(µ) (µ − A)−1dµ
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
= f (∞)g(∞)I +1
2πi
∫
+∂D1
f (λ)(λ − A)−1
(1
2πi
∫
+∂D2
g(µ)
µ− λdµ
)
dλ
+1
2πi
∫
+∂D2
g(µ)(µ − A)−1
(1
2πi
∫
+∂D1
f (λ)
λ− µdλ
)
dµ
+g(∞)
2πi
∫
+∂D1
f (λ) (λ − A)−1dλ+f (∞)
2πi
∫
+∂D1
g(µ) (µ − A)−1dµ
= f (∞)g(∞)I +1
2πi
∫
+∂D1
f (λ)g(λ)(λ − A)−1 dλ = (fg)(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
Proceeding exactly as in Theorem 3 we have that the followingresult holds.
TheoremLet X be a Banach space over C and A : D(A) ⊂ X → X be a
closed operator with non-empty resolvent. If f ∈ U∞(A) is suchthat f (λ) 6= 0 for all λ ∈ σ(A) ∪ {∞} then, f (A) is one-to-one and
onto X with inverse g(A) where g is any function of U∞(A) withg ∼ 1
f.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear Operators
Dissipative operators and numerical range
Operational calculus
Operational calculus for closed operators
ExerciseLet X be a Banach space over C and A ∈ L(X ). Show that, if
f ∈ U(A) ∩ U∞(A) then, (10) and (14) give rise to the same
operator f (A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II