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

A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 535
The function Wn,T thus has the power series expansion
Wn,T (z) =
−T ∗
n−1
(−1) k
k=0
n
T
k + 1
∗k T k
+
k1
1
μn;k
Dn,T T k−1
D
k
Qn,T z ,
∗
n,T
z ∈ D. (3.7)
We can now parametrize the wandering subspace En,T for In,T using the function Wn,T as
follows.
Theorem 3.3. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then a function f in
An(Dn,T ) belongs to the wandering subspace En,T for In,T if and only if it has the form
f(z)= Wn,T (z)x, z ∈ D, (3.8)
for some element x ∈ D∗ n,T . Furthermore, we have the norm equality
when f is of the form (3.8).
f 2 An =x2n =x2 n−1
+ Dk,T x 2 , x ∈ D ∗ n,T ,
Proof. Let f ∈ En,T be given by (3.6). By formulas (0.5) and (1.2) we have that
f(z)=−T ∗
n−1
(−1) k
k=0
k=1
n
T
k + 1
∗k T k
x +
k1
1
μn;k
By the power series expansion (3.7) of the function Wn,T we conclude that
f(z)= Wn,T (z)x, z ∈ D.
The conclusion of the theorem is now evident by Theorem 3.2. ✷
Dn,T T k−1 Qn,T x z k , z∈ D.
We remark that in the case n = 1 of a contraction operator T ∈ L(H) the L(DT ∗, DT )-valued
analytic function
WT (z) = W1,T (z) = −T ∗ + zDT (I − zT ) −1 DT ∗
DT , z∈D, ∗
is the characteristic operator function studied by Sz.-Nagy and Foias (see [26, Chapter VI]).
4. Multiplier properties of the function Wn,T
In this section we discuss some multiplier properties of the function Wn,T .Wefirstshow
that the function Wn,T acts as a contractive multiplier from the Hardy space A1(D∗ n,T ) into the
Bergman space An(Dn,T ).