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Chapter 7. Growth relation cogenerator and inverse generator
Combining, we find
˜MA1
(n − 1)!
� ∞
e −t g(t)t n−1 dt
t1
� �n−1 �
˜MA1
∞
e
� e
2π(n − 1) n − 1 t1
−t g(t)t n−1 dt
� �n−1 �
˜MA1
∞
e
�
2π(n − 1) n − 1 t1
� �n−1 ˜MA1 e
� e
2π(n − 1) n − 1
−(1−ε)α(n−1) (α(n − 1)) n−1
≤
≤
≤
=
� ∞
t1
˜MA1
� 2π(n − 1)
e −εt g(t)dt
�
e (−(1−ε)α+1) α
e −(1−ε)t t (n−1) e −εt g(t)dt
� �
n−1 ∞
e −εt g(t)dt.
Using equation (7.6) we have that e (−(1−ε)α+1) α < 1. Thus we have that
˜MA1
(n − 1)!
� ∞
e −t g(t)t n−1 dt ≤
t1
˜MA1
� 2π(n − 1)
� ∞
t1
t1
e −εt g(t)dt = Mα (7.10)
with Mα independent of n. Combining the estimates (7.8), (7.9) and (7.10)
gives
�A n d � ≤ ˜ MA1 g(t1) + Mα.
Since t1 = α(n − 1) and since g is non-decreasing, we can find a constant
Mα,A such that �A n d � ≤ ˜ Mα,Ag(αn) which proves the result. �
7.2 Bounded semigroups on a Hilbert space
In the previous section, we have investigated the growth of (A n d ) n∈N and
�
eA−1 �
t
t≥0
under the condition that A generates an exponentially stable
C0-semigroup on Banach space X.
In this section we take X to be a Hilbert space. Furthermore, we assume
that A and A −1 generate a bounded semigroup.
In this section we use the notation R(A, s) = (sI − A) −1 .
For a bounded operator A that generates a bounded C0-semigroup on a Hilbert
space, we know that the power sequence of its cogenerator is bounded
as well, see Lemma 1.6. However, if A generates an bounded semigroup, but
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