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

H. Schultz / Journal of Functional Analysis 236 (2006) 457–489 479
Recall from [7] that the modified spectral radius of T ∈ M, r ′ (T ), is defined by
r ′ (T ) := max |z| z ∈ supp(μT ) . (6.4)
Also recall from [7, Corollary 2.6] that in fact
r ′
(T ) = lim lim T
p→∞ n→∞
n 1/n
. (6.5)
p/n
Lemma 6.3. Let S,T ∈ M be commuting operators. Then the modified spectral radii, r ′ (S),
r ′ (T ), r ′ (ST ) and r ′ (S + T), satisfy the inequalities
and
r ′ (ST ) r ′ (S) · r ′ (T ), (6.6)
r ′ (S + T) r ′ (S) + r ′ (T ). (6.7)
Proof. (6.6) follows from (6.5) and the generalized Hölder inequality (cf. [5]): for A,B ∈ M
and for 0