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Chapter 4. Bergman distance
Furthermore, the following inequality holds
1
t
� t
0
�e As x0� 2 ds ≤ 1
t
� t
0
2�e As x0 − e Ãs x0� 2 ds + 1
t
Using this inequality and Lemma 2.10, we have that
1
lim
t→∞ t
� t
0
�e As x0� 2 1
ds ≤ lim
t→∞ t
+ lim
t→∞
where we have used inequality (4.11).
� t
2�e
0
Ãs x0� 2 ds.
� tε
2�e
0
As x0 − e Ãs x0� 2 ds
� t
1
2�e As x0 − e Ãs x0� 2 ds + 0
t
tε
� tε
tε 2
≤ lim
t→∞ t 0 s �eAsx0 − e Ãs x0� 2 ds
� t
2
+ lim
t→∞ s �eAsx0 − e Ãs x0� 2 ds ≤ 0 + 2ε,
Since this holds for all ε > 0, we have shown that
1
lim
t→∞ t
tε
� t
�e
0
As x0� 2 ds = 0,
and so by Lemma 2.10, (e At )t≥0 is strongly stable.
This third item concludes the proof. �
Note that the proof of Theorem 4.8 is done element-wise. So if the semigroup
(e At )t≥0 provides a stable solution for the initial condition x0, then
semigroup (e Ãt )t≥0 provides a stable solution for the same initial condition
as well. This is independent of their behaviour on other elements in X. We
examine this property of the Bergman distance in Lemma 5.7.
4.3 Properties of cogenerators with finite
Bergman distances
The discrete-time case is similar to the continuous-time case. The finite
Bergman distance also creates equivalence classes of sequences of bounded
operators, see Section 5.1. Elements within a class share the same stability
properties, as is shown next.
Theorem 4.9 Let (A n d )n≥0 and ( Ãn d )n≥0 be a power sequence of bounded
operators on the Hilbert space X. If they have a finite Bergman distance,
then the following assertions hold:
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