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Chapter 5. Extension Bergman distance
Let yn ∈ Ran(e Ãt0 ) be a sequence converging to y. The sequence xn ∈ X
defined by e Ãt0 xm = ym is a Cauchy sequence as well, since by equation
(5.10),
�xn − xm� ≤ 1
δ �eÃt0 (xn − xm)� = 1
δ �yn − ym�. (5.12)
In the Hilbert space X, the Cauchy sequence xn converges. Let x be the
limit. Then
�y − e Ãt0 x� = �y − yn + e Ãt0 xn − e Ãt0 x�
≤ �y − yn� + �e Ãt0 ��xn − x� → 0, as n → ∞.
So y = e Ãt0 x and thus Ran(e Ãt0 ) is closed.
Summarizing we have that Ker(e Ãt0 ) = {0}, and Ran(e Ãt0 ) = Ran(e Ãt0) =
X. This means that e Ãt0 is invertible and thus à generates a C0-group on
X. �
Now we focus on the set of states for which the semigroup is stable.
Definition 5.6 Let A generate a semigroup on X, then we define the set
of stable states S(A) as follows,
S(A) := {x ∈ X | e At x → 0, if t → ∞}.
By Theorem 4.8 we know that stability properties which hold for all elements
of the space X, are shared by semigroups within the same Bergman class.
In the following lemma, we state that also stability properties on a subset
of X are shared within the Bergman classes.
Lemma 5.7 Let A B ∼ Ã, and let the semigroup (eÃt )t≥0 be bounded, then
S(A) = S( Ã)
Proof: Since A B ∼ Ã, and (eÃt )t≥0 is bounded, the semigroup (e At )t≥0
is bounded as well, see Theorem 4.8. Now, we show that S( Ã) ⊂ S(A).
Then by symmetry the equality holds. Let x0 ∈ S( Ã). Since � ∞ 1
0 s �eAsx0 −
eÃsx0�2ds < ∞, for every ε > 0, there exists a tε such that
� ∞
1
s �eAsx0 − e Ãs x0� 2 ds < ε. (5.13)
Furthermore, the following inequality holds
68
1
t
� t
0
tε
�e As x0� 2 ds ≤ 1
t
� t
2�e
0
As x0 − e Ãs x0� 2 ds
� t
2�e
0
Ãs x0� 2 ds.
+ 1
t