Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

H. Schultz / Journal of Functional Analysis 236 (2006) 457–489 475
μS|K,T |K
(B) =
=
∞
k=1
∞
k=1
τP MP PS
τP MP PS
(k)
B 1 ∧ PT
(k)
B 1 ∧ PT
(k)
B 2 ∧ P
(k)
B 2
= 1
∞
tr
τ(P)
k=1
(k) (k)
eS B 1
eT B 2
= 1
τ(P) tr
∞
(k) (k)
eS B 1
eT B 2
k=1
= 1
τ(P) τ
∞
(k) (k)
PS B 1 ∧ PT B 2
= 1.
k=1
Thus, (b) holds.
Now, suppose that Q ∈ M is an S- and T -invariant projection, and that μS|L,T |L is concentrated
on B, where L = Q(H). Then by Lemma 3.3 and Proposition 2.8,
P ∧ Q =
=
∞
∞
k=1
k=1
PS
PS|L
Hence, Proposition 2.8 and (5.6) imply that
τQMQ(P ∧ Q) = trQMQ
(k) (k)
B 1 ∧ PT B 2
∧ Q
(k) (k)
B 1 ∧ PT |L B 2
= P range( ∞k=1 eS| L (B (k)
1 )eT | L (B (k)
2 )).
=
=
∞
k=1
∞
k=1
∞
k=1
eS|L
trQMQ eS|L
τQMQ PS|L
= μS|L,T |L (B)
= 1.
Thus, P ∧ Q = Q, and this shows that (c) holds.
(k) (k)
B 1
eT |L B 2
(k)
B 1
eT |L
(k)
B 1 ∧ PT |L
(k)
B 2
(k)
B 2