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

658 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
where M(α) = Mα α (C), A(α) = Aαα (C) and ˆα ={1,...,m}\α.
More precisely, see [12, p. 573]; [13, Chapter 2.5, Problem 36]. See also [27, Corollary 3.2,
p. 34].
Let us set as before (see (25)) for λ = (λ1,...,λk) ∈ C k and C ∈ Mat(k, C)
Gk(λ) = det Ck(λ), where Ck(λ) = C +
In the following lemma we use the notation for λ = (λ1,...,λk) ∈ C k :
k
r=1
λrErr.
λ ]l[ = (λ1,...,λl−1, 0,λl+1,...,λk), 1 l k,
and Gl(λ) = M 12...l
12...l (Ck(λ)), 1 l k.Forα and β such that ∅⊆α ⊆{1, 2,...,l} and ∅⊆β ⊆
{l + 1,...,k}, with l