Introduction à la Théorie Spectrale

T X
X
sup||x||=1||T x|| <
+∞L(X)X
αx,l L(X)αx,l(T) = l(T(x)) x∈X, l ∈
X∗
T ρ(T) = {λ∈
C, λI −T∈GL(X)}C
σ(T) = C−ρ(T)
Rλ(T)=(λI −T)−1
T
X
λ µ ρ(T), Rλ(T)−Rµ(T) =
(λ−µ)Rλ(T)Rµ(T)
λ→Rλ(T)ρ(T)
λ∈ρ(T)R(z) = Rλ(T)(I+
∞
X
i=1
(λ−
z)nRn
λ(T)) |z−λ|<||Rλ(T)||−1R(z)(zI −
T)=(zI −T)R(z) = I R(z) = Rz(T)
X T ∈L(X)∃!V∈L(X)
(T(x), y) = (x, V (y)) V=T∗
T T ∗=T

T
T T T ∗=I
T
σ(T)R
T
K(X)
X Y
xnH
x T T (xn)T x
||xn||
TnT
T
T S T S
T
DC
f:→K(H)
∀z∈D, (I−f(z))
I−f(z)D
D
I−f(z)
TH
I−T I −T
T
H

H
D(T)
RH2
L2(X)V X C
V L2(X){f∈2(X),ZX V(x)f(x)dx <
∞} V
T
T∈L(H)T
∀x∈ H, Im(T x, x)≤0T
H
TH
∀ ∈ C, z /∈σ(T)
∃z∈C, z /∈σ(T)
T
CnT