110

K[X]
K K
K[X]
pgcd,ppcm r≥2,
P1,··· , Pr
U1,··· , Ur
r
k=1
UkPk= 1.
u E.
K[u]
u0=Id u
∀k∈N, uk+1 =uk
P=
p
k=0
akXk∈K[X],
P(u),
P(u) =
p
k=0
akuk
L(E)u v =P(u)
PK[X].K[u]
∀(P, Q)∈K[X]2,(P Q) (u) = P(u)◦Q(u) = Q(u)◦P(u) = (QP ) (u)
L(E)n2,dim (K[u]) ≤n2.
K[A]Mn(K)
A∈ Mn(K).

A∈ Mn(K)u∈ L(E)B,
P(A)P(u)B.
P∈K[X]. λ ∈Sp (u), P (λ)
P(u).
K
Sp (P(u)) = {P(λ)|λ∈Sp (u)}
λ∈Ku x ∈E\ {0}
P∈K[X], P (u)x=P(λ)x.
u(x) = λx, k ≥0uk(x) = λkx
k∈NP(u)x=P(λ)x P ∈K[X].
x P (u)P(λ).
P(X) = a0P(u) = a0Id a0,
E.
KPSp (u) Sp (P(u))
{P(λ)|λ∈Sp (u)} ⊂ Sp (P(u)) .
µ∈KP(u), Q (X) = P(X)−µ,
Q(u)Q(X) = α
p
i=1
(X−λi)miK
i u −λiId
λiu Q (λi)=0 µ=P(λi).
Sp (P(u)) = {P(λ)|λ∈Sp (u)}.
K{P(λ)|λ∈Sp (u)} ⊂ Sp (P(u))
P(X) = X2−1−I2=A2, A =0−1
1 0 ∈
M2(R)π
2,Sp (u) = ∅,
K=C,L(E)v7→ ∥v∥,
eu=
+∞
k=0
1
k!uk
eu∈C[u].
L(E)
L(E)E,
k≥0
uk
k!
≤∥u∥k
k!
+∞
k=0
∥u∥k
k!=e∥u∥<+∞uk k!
L(E),
C[u]L(E)
eu= lim
k→+∞
k
j=0
1
k!ukC[u].

L(E)n2,uk|0≤k≤n2
P∈K[X]\ {0}P(u) = 0.
Iu={P∈K[X]|P(u) = 0}
P7→
P(u),K[X].
u Iuu
πu
Iu={P∈K[X]|P(u) = 0}=K[X]πu
πuu.
πu
A∈ Mn(K).
u A BE, A u
P∈K[X], P (u)
BP(A)P(u) = 0 P(A) = 0, Iu=IAπu=πA
B=Q−1AQ P ∈K[X], P (B) = Q−1P(A)Q P (A)=0
P(B) = 0, IA=IBπA=πB
Iu
πuIu{0}.
D
f∈E=C∞(R,R)f′
Iu̸={0}, P =
p
k=0
akXk
D λ, fλ:t7→ eλt,
0 = P(D) (fλ) =
p
k=0
akDk(fλ) = p
k=0
akλkfλ=P(λ)fλ
P(λ) = 0. P
Iu={0}D
u= 0 πu(X) = X.
1,
u=λId πu(X) = X−λ.
u∈ L(E)
q≥1uq−1̸= 0 uq= 0 q u
u∈ L(E)q≥1πu(X) = Xq.

u E u ◦u=u.
X2−X, πu(X) = X u = 0,
πu(X) = X−1u=Id, πu(X) = X2−X
F E u,
u F u.
v u F. F F
u. πu(u) = 0 L(E), πu(v) = 0 L(F), πu
v v.
P∈Iu,
Sp (u)⊂P−1{0}
u
Sp (u) = π−1
u{0}
u
λ∈Ku x ∈E\ {0}
0 = πu(u) (x) = πu(λ)x
πu(λ) = 0, λ πuP∈Iu
πu.
λ∈Kπu, πu(X)=(X−λ)Q(X)
πu(u) = (u−λId)◦Q(u) = 0 πuu−λId
λ u.
0
u∈GL (E), u−1∈K[u].
FL(E)Id
G=F∩GL (E)
GL (E).
u∈GL (E), πu(0) ̸= 0. πu(X) = XQ (X)
πu(u) = u◦Q(u) = 0 u−1Q(u)=0,
πu. πu(u) =
p
k=0
akukp≥1a0̸= 0,
u◦
p
k=1
akuk−1=−a0Id
u−1=−1
a0
p
k=1
akuk−1∈K[u].

u, v G, u ◦v G F
u∈G, u−1F, u−1
u u ∈GL (E).
K[X],
K[u]puπu,
uk0≤k≤pu−1.
P∈K[X], P =πuQ+R R ∈
Kpu−1[X]πu(u) = 0, P (u) = R(u) =
pu−1
k=0
αkukR(X) =
pu−1
k=0
αkXk,
K[u] = Vect uk|0≤k≤pu−1.
RKpu−1[X]R(u) = 0, R ∈IuR πu,
R= 0 deg (R)<deg (πu).uk0≤k≤pu−1
K[u].
K[u]v=P(u)
P∈Kpu−1[X].
dim (K[u]) = pu.
φu:P7→ P(u)K[X]K[u]Iu=
K[X]πu= (πu)πuK[X]
(πu)
K[u],dim (K[u]) = dim K[X]
(πu)=pu
(K[u] ) ⇔(K[u] ) ⇔(πu)
K[u]
πuπu=P Q P, Q 0 = πu(u) =
P(u)◦Q(u)P(u)Q(u)πuK[u]
K[u]πu
πuP∈Kpu−1[X]πu
A, B Aπu+BP = 1
Id =B(u)◦P(u), P (u)K[u].
K[u]v=P(u)P∈Kpu−1[X]\ {0},
K[u]
E1,··· , ErE{0}, u
E=
r
k=1
Ek. k 1r, uk∈ L(Ek)u
Ekπkuk. πu=π1∨···∨πrppcm π1,··· , π2
P∈Iu, P (u) = 0, P (uk)=0 k
1r, P ∈
r
k=1
Iuk. Iu⊂
r
k=1
Iuk. P ∈
r
k=1
Iuk,