Théorie des Opérateurs : Cours M1 à l'Université de la Réunion

K R C
Hh·,·i
h∈H g 7→ hg, hi
g, h ∈Hhg, hi=hh, gi
g∈H\ {0} hg, gi>0
kgk=phg, giH
k·k
|hg, hi| 6kgk·khk.
kgk= sup
h∈H,khk61|hg, hi|.
g, h ∈H g h g ⊥hhg, hi= 0 M
H M
M⊥={h∈H∀g∈H, h ⊥g}.
M⊂H M⊥={0}
M H H =M⊕M⊥
f1, . . . , fn∈H
kf1+··· +fnk2=kf1k2+··· +kfnk2.
f, g ∈H
kf+gk2+kf−gk2= 2(kfk2+kgk2)
M⊂H M PM
h∈H PMh M h −PMh⊥M PM
P2
M=PMkPMhk6khkh∈Hker PM=M⊥Im PM=M
(ei)i∈IH
keik= 1
Vect{ei}H
ei⊥eji6=j

`
H y`∈H x ∈H
`(x) = hx, y`i.
Kn
H BHBH
H
Kn
hx, yi=
n
X
i=1
xiyi
Kn
`2(N) = {(un)n∈NPn∈N|un|2}
hu, vi=X
n∈N
unvn.
k∈Nekk1
(ek)k∈N`2(N)
`2(Z)Z
(X, Ω, µ)R
L2(X, Ω, µ) = f:X→KZX|f(x)|2dµ(x)<+∞/∼
∼
f∼g⇐⇒ f=g µ −
hf, gi=ZX f(x)g(x)dµ(x).
K=CL2([0,1], dx) (en)n∈Zen
en(x) = exp(2πinx).

H
H K
H A :H→H
A
A
x∈H A x
c > 0kAhk6ckhkh∈H
H H H
L(H)H A ∈L(H)
kAk= sup{kAhk:h∈H, khk61}.
A∈L(H)kAk= 0 A= 0
A, B ∈L(H)A+B∈L(H)kA+Bk6kAk+kBk
α∈KA∈L(H)αA ∈L(H)kαAk=|α|·kAk
A, B ∈L(H)AB ∈L(H)kABk6kAk·kBkAB A ◦B
k·k L(H)
H n H H
(e1, . . . , en)H A ∈L(H) (aij )
aij =hAei, eji
H=`2(e1, e2, . . . )A∈L(H)αij =hAei, eji
(αij )A
kAk
(αij )i,j∈N∗
A∈L(H)hAei, eji=αij
sup
n∈N
sup 
n
X
i,j=1
αij xiyj :|x1|2+··· +|xn|261,|y1|2+··· +|yn|261 <+∞.

L2[0,1]
V f(x) = Zx
0
f(y)dy.
S `2(N)
S(α1, α2, . . . ) = (0, α1, α2, . . . ).
kSk= 1
(X, Ω, µ)σ H =L2(µ)ϕ∈L∞(µ)
Mϕ∈L(H)Mϕf=ϕf kMϕk=kϕk∞
kϕk∞ϕ
kϕk∞= inf{c > 0µ(|ϕ|> c)=0}.
ϕ|ϕ(x)|6kϕk∞
x∈X f ∈L2(µ)
kMϕfk2=Z|ϕf|2dµ 6kϕk2
∞Z|f|2dµ 6kϕk2 ∞· kfk2,
Mϕ∈L(H)kMϕk6kϕk∞
ε > 0µ σ ∆∈Ω 0 < µ(∆) < µ(Ω)
|ϕ(x)|>kϕk∞−ε x ∈∆f= (µ(∆))−1/21∆f∈L2(µ)kfk2= 1
kMϕk2>kϕfk2
2= (µ(∆))−1Z∆|ϕ|2dµ >(kϕk∞−ε)2.
εkMϕk>kϕk∞
Mϕϕ
Mϕ
(X, Ω, µ)k:X×X7→ KΩ⊗Ω
c1c2
ZX|k(x, y)|dµ(y)6c1µ−p.p.
ZX|k(x, y)|dµ(x)6c2µ−p.p.
K:L2(µ)→L2(µ)
(Kf)(x) = Zk(x, y)f(y)dµ(y)
H=L2(µ)kKk6√c1c2
f∈L2(µ)
|Kf(x)|6Z|k(x, y)|·|f(y)|dµ(y)
=Z|k(x, y)|1/2· |k(x, y)|1/2· |f(y)|dµ(y)
6Z|k(x, y)|dµ(y)1/2Z|k(x, y)|·|f(y)|2dµ(y)1/2
6√c1Z|k(x, y)|·|f(y)|2dµ(y)1/2