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Chapter 7. Growth relation cogenerator and inverse generator
Corollary 7.4 Let X be a Hilbert space and let A generate an exponentially
stable C0-semigroup on X, then the following growth estimate holds.
with M0 independent of t.
�e A−1 t � ≤ M0 ln(t + 2), t ≥ 0,
The following theorem can be seen as a partial reverse implication of Theorem
7.1.
Theorem 7.5 Assume that for every A that generates an exponentially
stable C0-semigroup the following estimate holds
�e A−1 t � ≤ ˜ MAg(t),
where g is a monotonically non-decreasing function, not depending on A,
i.e, 0 < g(α) ≤ g(β) for all 0 ≤ α ≤ β, and MA a constant not depending
on t. Then for every A which is the generator of an exponentially stable
semigroup satisfying �e At � ≤ Me −ωt for ω > 1, there exists for all α > 1
an Mα,A such that
�A n d � ≤ Mα,Ag(αn).
Proof: By Lemma 1.3 we see that without loss of generality we may assume
that g(t) ≤ 1 + M0t 1
4 .
Let α > 1 be given. Since for α > 1, we have that α − 1 − log(α) > 0, there
exists an ε ∈ (0, 1) such that
αε < α − 1 − log(α). (7.6)
Secondly, the function e −(1−ε)t t n−1 , t > 0 has a maximum at τ = n−1
1−ε and
is decreasing for t > τ. We choose now
t1 = α(1 − ε)τ = α(n − 1). (7.7)
Since α > 1, and since equation (7.6) holds, we have that t1 > τ.
We define A1 = 1
2 (A + I). By the assumption on A, we have that A1
generates an exponentially stable C0-semigroup. Furthermore, we have that
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Ad = (A + I)(A − I) −1 = 2A1(2A1 − 2I) −1 = (I − A −1
1 )−1 .