5.1 Multiple Systems
81
in the first n factors ϕ j . Such vectors k , with n ∈ , form a total set in H
. We
have
(( 1 , A 2 ) = lim
N →∞
1
N
⎛
⎝
n
j=1
(( 1 , π j (A)) 2 ) +
N
j=n+1
(( 1 , π j (A)) 2 )
⎞
⎠ =
= lim
N →∞
1
N
N
j=n+1
(ϕ j , u j Au
−1
j ϕ j )(( 1 , , 2 ) =
= lim
N →∞
1
N
N
j=1
((, π j (A)))(( 1 , , 2 ) = ((, A )(( 1 , , 2 ). (5.1.34)
By linearity, the obtained relation extends to all k ∈ P D (A). On that domain,
we obtain
A
= T r(P
◦
A
)P = T r(P
◦
A )P ,
(5.1.35)
where P
◦
is the projector onto the one-dimensional subspace of H
spanned by the
vector .
Note: Since A
is bounded on H (if ∈ D (A) is a product-vector), we shall
extend this operator to the whole H by continuity and we shall denote this extension
by the same symbol, hence: A
∈ L(H ).
5.1.9 Proposition. Let ∈ D (g) be an arbitrary vector from
D (g) :=
ξ∈g
D (X ξ ),
(5.1.36)
in the notation of 5.1.3 and 5.1.7. Then U (g)) ∈ D (g), for all g ∈ G. In particular, with g · := U (g)), we have for product-vectors ∈ D (g):
X
g·
ξ
= T r(P
◦
g· X ξ )P g· = T r(P
◦
X Ad(g −1 )ξ )P g· .
(5.1.37)
Proof. According to Lemma 3.1.4, U (g)X ξ U (g
−1
) = X Ad(g)ξ for any ξ ∈ g. Then,
according to 5.1.3, we have also
U (g
−1
)π j (X ξ )U (g) = π j (X Ad(g −1 )ξ ).
(5.1.38)
For ∈ D (g) there exist X
ξ for all ξ ∈ g. Because of continuity of unitary
operators U (g) for any fixed g ∈ G, there exist also the limits
lim
N →∞
U (g)X Ad(g −1 )ξ N = U (g)X
Ad(g −1 )ξ
(5.1.39)
for all ξ ∈ g. Rewriting the expression on the left hand side of (5.1.39) we get