ISUP 2004

Rn
ERL(E)E
E E
u∈ L(E)
EE u0=Eu1=u u2=u◦u
u k uk= 0
α(u) = min k∈N|uk= 0;
α(u)>1
F E F u u(F)⊂F
pMp(R)p
Ip
R[X]
R[X]
Rn[X]n
Rn[X] = {P∈R[X]|deg P6n}.
Rn[X]R[X]
C= (Xk)k∈NR[X]Cn=Xk06k6nRn[X]
∆
R[X]−→ R[X]
P7−→ P(X+ 1) −P(X)D
R[X]−→ R[X]
P7−→ P0
∆R[X]
Rn[X] ∆
∆nRn[X] ∆ ∆n
Cn
∆n
∆n
∆
D
A B D2∆2C2=1, X, X2
R2[X]
k∈NAkBk

J1=
001
000
000
(E1)M2=J1M∈ M3(R)
(E1)
M(E1)fR2[X]M
g=f◦f
Ker gIm g f
f
(E1)
J2=
010
001
000
(E2)M2=J2M∈ M3(R)
(E2)
ERn>3
f∈ L(E)
Com (f) = {g∈ L(E)|f◦g=g◦f}Pol(f) = {P(f)|P∈R[X]}.
u∈ L(E)n
u0∈ L(E)n−1
Pol(f) Com (f)L(E)f∈ L(E)
L(E)
B= (e1, . . ., en)E u BN=
(ni,j )16i,j6nni,j = 1 j=i+ 1,
ni,j = 0
k∈ {1, . . ., n}Ker uk
(uk)06k6n−1Pol(u)
w∈Com (u)P∈R[X] Ker P(u)w
wB
Com (u) = Pol(u) Com (u)
B0= (e0
1, . . ., e0
n)E u0B0
N0=n0
i,j 16i,j6nn0
i,j = 1 j=i+ 1 j6n−1
n0
i,j = 0
Com (u0)
Mn(R)n>2
U U −In
A
exp (A) =
∞
X
k=0
1
k!Ak,(1)
ln (In+A) =
∞
X
k=1
(−1)k+1
kAk.(2)

A B AB =BA
AB α(AB)6min {α(A), α(B)}
A+B α(A+B)6α(A) + α(B)−1
A r>2
P Q d
rexp (A) = P(A) ln (In+A) = Q(A)
x0
Q(P(x)−1) = x+oxd, P (Q(x)) = 1 + x+oxd.
(1) (2)
A B exp (A+B)
exp (A) exp (B)
U V ln (UV ) ln (U)
ln (V)
Mn(R)
(1) exp (M1+M2) exp (M1) exp (M2)M1
M2