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

For m = 3wehave
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 657
G2(λ) = det C2 + λ1A 1 1 (C2)
2
+ λ2 A2 (C2) + λ1A 12
12 (C2)
= det C2 + λ1A 1 {1}
1 C2 λ + λ2A 2 {2}
2 C2 λ .
G3(λ) = det C3 + λ1A 1 1 (C3) + λ2A 2 2 (C3) + λ3A 3 3 (C3) + λ1λ2A 12
12 (C3) + λ1λ3A 13
13 (C3)
+ λ2λ3A 23
23 (C3) + λ1λ2λ3A 123
123 (C3)
1
= det C2 + λ1 A1 (C3) + λ2A 12
12 (C3) + λ3A 13
13 (C3) + λ2λ3A 123
123 (C3)
2
+ λ2 A2 (C3) + λ3A 23
23 (C3) + λ3A 3 3 (C3)
= det C3 + λ1A 1 [1]
1 C3 λ + λ2A 2 [2]
2 C3 λ + λ3A 3 [3]
3 C3 λ ,
G3(λ) = det C3 + λ1A 1 1 (C3)
2
+ λ2 A2 (C3) + λ1A 12
12 (C3)
1
+ λ3 A1 (C3) + λ1A 12
13 (C3) + λ2A 23
23 (C3) + λ1λ2A 123
123 (C3)
= det C3 + λ1A 1 {1}
1 C3 λ + λ2A 2 {2}
2 C3 λ + λ3A 3 {3}
3 C3 λ .
For m>3 the proof of (28) and (30) is the same. The identity (29) follows from (28) and (31)
follows from (30). ✷
The proof of Lemma 16 is based on Lemmas A.4, A.6 and A.7 concerning the properties of a
positive matrices.
Lemma A.4. (Sylvester [10, Chapter II, Section 3]) Let C ∈ Mat(n, R) and 1 p