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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 641
and the series SL pq (μm ) and Σr pq (m) are defined in Lemmas 8 and 15 (see also (18)). This is done
in Appendices A–C.
In Appendix A we define the generalization of the characteristic polynomial for matrix C and
establish some its properties. These properties are used then in Appendices B and C. For a matrix
C ∈ Mat(k, C) we set
Gk(λ) = det Ck(λ), where Ck(λ) = C +
k
λrErr, λ= (λ1,...,λk) ∈ C k .
Lemma A. (See Appendix A, Lemma A.7) For a positive definite matrix C ∈ Mat(k, C), λ ∈ R k
with λr 0, r = 1,...,k, we have
r=1
∂ Gk(λ)
0,
∂λp Gl(λ)
where Gl(λ) = M 12...l
12...l (Ck(λ)) and 1 p l k.
The proof of Lemma A is based on the following inequality (see Lemma A.6).
Lemma B. (Hadamard–Ficher’s inequality [12,13], see also [27]) Let C ∈ Mat(m, R) be a positive
definite matrix and ∅⊆α, β ⊆{1,...,m}. Then
det Cα det
det Cα∪β
Cα∩β
det Cβ
=
M(α) M(α∩ β)
M(α∪ β) M(β) 0,
where Cα for α ={α1,...,αs} denotes the matrix which entries lie on the intersection of
α1,...,αs rows and α1,...,αs columns of the matrix C and M(α) = M α α (C) = det Cα are corresponding
minors of the matrix C.
The “best” approximation of xpq by the generators A R,m
kn is based on the exact computation
of the matrix elements
φp(t) = T R,μm B
t 1, 1 , t = I +
p
r=1
trErn, (tr) p
r=1 ∈ Rp ,
of the representation T R,μm B and their generalization (see Appendix B, Lemma B.1), and on
the finding the appropriate combinations of operator functions of the generators A R,m
kn (see Remark
13) to approximate the operators of multiplication by xpq.
Finally the proof of the inequality Σm >CSm, is based on Lemmas A, B and 16 dealing with
some inequalities involving the generalized characteristic polynomials. Lemma 16 is proved in
Appendix C.